3.208 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{(a+b x^4)^4} \, dx\)

Optimal. Leaf size=516 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {(a h+5 b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (2 x (a h+5 b d)+3 x^2 (a i+3 b e)+a g+11 b c\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]

[Out]

1/12*x*(b*c-a*g+(-a*h+b*d)*x+(-a*i+b*e)*x^2+b*f*x^3)/a/b/(b*x^4+a)^3+1/384*x*(7*a*g+77*b*c+12*(a*h+5*b*d)*x+15
*(a*i+3*b*e)*x^2)/a^3/b/(b*x^4+a)+1/96*(-8*a*f+x*(11*b*c+a*g+2*(a*h+5*b*d)*x+3*(a*i+3*b*e)*x^2))/a^2/b/(b*x^4+
a)^2+1/32*(a*h+5*b*d)*arctan(x^2*b^(1/2)/a^(1/2))/a^(7/2)/b^(3/2)-1/1024*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)
+x^2*b^(1/2))*(-5*(a*i+3*b*e)*a^(1/2)+7*(a*g+11*b*c)*b^(1/2))/a^(15/4)/b^(7/4)*2^(1/2)+1/1024*ln(a^(1/4)*b^(1/
4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-5*(a*i+3*b*e)*a^(1/2)+7*(a*g+11*b*c)*b^(1/2))/a^(15/4)/b^(7/4)*2^(1/2)+1/5
12*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*(5*(a*i+3*b*e)*a^(1/2)+7*(a*g+11*b*c)*b^(1/2))/a^(15/4)/b^(7/4)*2^(1/2
)+1/512*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(5*(a*i+3*b*e)*a^(1/2)+7*(a*g+11*b*c)*b^(1/2))/a^(15/4)/b^(7/4)*2^
(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.85, antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1858, 1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {(a h+5 b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}-\frac {8 a f-x \left (2 x (a h+5 b d)+3 x^2 (a i+3 b e)+a g+11 b c\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^4,x]

[Out]

(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(12*a*b*(a + b*x^4)^3) + (x*(7*(11*b*c + a*g) + 12
*(5*b*d + a*h)*x + 15*(3*b*e + a*i)*x^2))/(384*a^3*b*(a + b*x^4)) - (8*a*f - x*(11*b*c + a*g + 2*(5*b*d + a*h)
*x + 3*(3*b*e + a*i)*x^2))/(96*a^2*b*(a + b*x^4)^2) + ((5*b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2
)*b^(3/2)) - ((7*Sqrt[b]*(11*b*c + a*g) + 5*Sqrt[a]*(3*b*e + a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2
56*Sqrt[2]*a^(15/4)*b^(7/4)) + ((7*Sqrt[b]*(11*b*c + a*g) + 5*Sqrt[a]*(3*b*e + a*i))*ArcTan[1 + (Sqrt[2]*b^(1/
4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(7/4)) - ((7*Sqrt[b]*(11*b*c + a*g) - 5*Sqrt[a]*(3*b*e + a*i))*Log[Sqr
t[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(7/4)) + ((7*Sqrt[b]*(11*b*c + a*g) -
5*Sqrt[a]*(3*b*e + a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(7/4
))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1854

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rubi steps

\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+208 x^6}{\left (a+b x^4\right )^4} \, dx &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-b (11 b c+a g)-2 b (5 b d+a h) x-3 b (208 a+3 b e) x^2-8 b^2 f x^3}{\left (a+b x^4\right )^3} \, dx}{12 a b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {\int \frac {7 b (11 b c+a g)+12 b (5 b d+a h) x+15 b (208 a+3 b e) x^2}{\left (a+b x^4\right )^2} \, dx}{96 a^2 b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c+a g)-24 b (5 b d+a h) x-15 b (208 a+3 b e) x^2}{a+b x^4} \, dx}{384 a^3 b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\int \left (-\frac {24 b (5 b d+a h) x}{a+b x^4}+\frac {-21 b (11 b c+a g)-15 b (208 a+3 b e) x^2}{a+b x^4}\right ) \, dx}{384 a^3 b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c+a g)-15 b (208 a+3 b e) x^2}{a+b x^4} \, dx}{384 a^3 b^2}+\frac {(5 b d+a h) \int \frac {x}{a+b x^4} \, dx}{16 a^3 b}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{256 a^3 b^2}+\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{256 a^3 b^2}+\frac {(5 b d+a h) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3 b}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt {2} a^{13/4} b^{7/4}}+\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt {2} a^{13/4} b^{7/4}}+\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^3 b^2}+\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^3 b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{13/4} b^{7/4}}-\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{13/4} b^{7/4}}+\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{13/4} b^{7/4}}-\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{13/4} b^{7/4}}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(208 a-b e) x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac {x \left (7 (11 b c+a g)+12 (5 b d+a h) x+15 (208 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {8 a f-x \left (11 b c+a g+2 (5 b d+a h) x+3 (208 a+3 b e) x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac {(5 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}-\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{13/4} b^{7/4}}+\frac {\left (5 (208 a+3 b e)+\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{13/4} b^{7/4}}+\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{13/4} b^{7/4}}-\frac {\left (5 (208 a+3 b e)-\frac {7 \sqrt {b} (11 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{13/4} b^{7/4}}\\ \end {align*}

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Mathematica [A]  time = 1.01, size = 530, normalized size = 1.03 \[ \frac {-\frac {256 a^{11/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^3}+\frac {32 a^{7/4} b^{3/4} x (a g+a x (2 h+3 i x)+11 b c+b x (10 d+9 e x))}{\left (a+b x^4\right )^2}+\frac {8 a^{3/4} b^{3/4} x (7 a g+3 a x (4 h+5 i x)+77 b c+15 b x (4 d+3 e x))}{a+b x^4}-6 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 a^{5/4} \sqrt [4]{b} h+5 \sqrt {2} a^{3/2} i+80 \sqrt [4]{a} b^{5/4} d+15 \sqrt {2} \sqrt {a} b e+7 \sqrt {2} a \sqrt {b} g+77 \sqrt {2} b^{3/2} c\right )+6 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 a^{5/4} \sqrt [4]{b} h+5 \sqrt {2} a^{3/2} i-80 \sqrt [4]{a} b^{5/4} d+15 \sqrt {2} \sqrt {a} b e+7 \sqrt {2} a \sqrt {b} g+77 \sqrt {2} b^{3/2} c\right )+3 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (5 a^{3/2} i+15 \sqrt {a} b e-7 a \sqrt {b} g-77 b^{3/2} c\right )+3 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-5 a^{3/2} i-15 \sqrt {a} b e+7 a \sqrt {b} g+77 b^{3/2} c\right )}{3072 a^{15/4} b^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^4,x]

[Out]

((32*a^(7/4)*b^(3/4)*x*(11*b*c + a*g + b*x*(10*d + 9*e*x) + a*x*(2*h + 3*i*x)))/(a + b*x^4)^2 + (8*a^(3/4)*b^(
3/4)*x*(77*b*c + 7*a*g + 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))/(a + b*x^4) - (256*a^(11/4)*b^(3/4)*(-(b
*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^3 - 6*(77*Sqrt[2]*b^(3/2)*c + 80*a^(1/4)*b^(
5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*Sqrt[2]*a*Sqrt[b]*g + 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1
- (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*(77*Sqrt[2]*b^(3/2)*c - 80*a^(1/4)*b^(5/4)*d + 15*Sqrt[2]*Sqrt[a]*b*e + 7*
Sqrt[2]*a*Sqrt[b]*g - 16*a^(5/4)*b^(1/4)*h + 5*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 3*
Sqrt[2]*(-77*b^(3/2)*c + 15*Sqrt[a]*b*e - 7*a*Sqrt[b]*g + 5*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x
+ Sqrt[b]*x^2] + 3*Sqrt[2]*(77*b^(3/2)*c - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g - 5*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b^(7/4))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^4,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.21, size = 735, normalized size = 1.42 \[ \frac {5}{1024} \, i {\left (\frac {2 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4}} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{a^{3} b^{4}}\right )} + \frac {5}{1024} \, i {\left (\frac {2 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4}} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{a^{3} b^{4}}\right )} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 8 \, \sqrt {2} \sqrt {a b} a b h + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 8 \, \sqrt {2} \sqrt {a b} a b h + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac {15 \, a b^{2} i x^{11} + 45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 12 \, a b^{2} h x^{10} + 77 \, b^{3} c x^{9} + 7 \, a b^{2} g x^{9} + 42 \, a^{2} b i x^{7} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 32 \, a^{2} b h x^{6} + 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} - 5 \, a^{3} i x^{3} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} - 12 \, a^{3} h x^{2} + 153 \, a^{2} b c x - 21 \, a^{3} g x - 32 \, a^{3} f}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^4,x, algorithm="giac")

[Out]

5/1024*i*(2*sqrt(2)*(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) - sqrt
(2)*(a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4)) + 5/1024*i*(2*sqrt(2)*(a*b^3)^(3/4)*
arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^4) + sqrt(2)*(a*b^3)^(3/4)*log(x^2 - sqrt(2
)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4)) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 8*sqrt(2)*sqrt(a*b)*a*b*
h + 77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/
b)^(1/4))/(a/b)^(1/4))/(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 8*sqrt(2)*sqrt(a*b)*a*b*h + 77*
(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4
))/(a/b)^(1/4))/(a^4*b^3) + 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g - 15*(a*b^3)^(3/4)*
e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) - 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)
^(1/4)*a*b*g - 15*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) + 1/384*(15*a*b^2*i*
x^11 + 45*b^3*x^11*e + 60*b^3*d*x^10 + 12*a*b^2*h*x^10 + 77*b^3*c*x^9 + 7*a*b^2*g*x^9 + 42*a^2*b*i*x^7 + 126*a
*b^2*x^7*e + 160*a*b^2*d*x^6 + 32*a^2*b*h*x^6 + 198*a*b^2*c*x^5 + 18*a^2*b*g*x^5 - 5*a^3*i*x^3 + 113*a^2*b*x^3
*e + 132*a^2*b*d*x^2 - 12*a^3*h*x^2 + 153*a^2*b*c*x - 21*a^3*g*x - 32*a^3*f)/((b*x^4 + a)^3*a^3*b)

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maple [A]  time = 0.07, size = 767, normalized size = 1.49 \[ \frac {h \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{32 \sqrt {a b}\, a^{2} b}+\frac {5 d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{32 \sqrt {a b}\, a^{3}}+\frac {5 \sqrt {2}\, i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}+\frac {5 \sqrt {2}\, i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}+\frac {5 \sqrt {2}\, i \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}+\frac {15 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 \sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 a^{3} b}+\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 a^{3} b}+\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 a^{3} b}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 a^{4}}+\frac {\frac {5 \left (a i +3 b e \right ) b \,x^{11}}{128 a^{3}}+\frac {\left (a h +5 b d \right ) b \,x^{10}}{32 a^{3}}+\frac {7 \left (a g +11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {7 \left (a i +3 b e \right ) x^{7}}{64 a^{2}}+\frac {\left (a h +5 b d \right ) x^{6}}{12 a^{2}}+\frac {3 \left (a g +11 b c \right ) x^{5}}{64 a^{2}}-\frac {\left (5 a i -113 b e \right ) x^{3}}{384 a b}-\frac {\left (a h -11 b d \right ) x^{2}}{32 a b}-\frac {f}{12 b}-\frac {\left (7 a g -51 b c \right ) x}{128 a b}}{\left (b \,x^{4}+a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^4,x)

[Out]

(5/128*(a*i+3*b*e)/a^3*b*x^11+1/32*(a*h+5*b*d)/a^3*b*x^10+7/384*(a*g+11*b*c)/a^3*b*x^9+7/64*(a*i+3*b*e)/a^2*x^
7+1/12*(a*h+5*b*d)/a^2*x^6+3/64*(a*g+11*b*c)/a^2*x^5-1/384*(5*a*i-113*b*e)/a/b*x^3-1/32*(a*h-11*b*d)/a/b*x^2-1
/128*(7*a*g-51*b*c)/a/b*x-1/12/b*f)/(b*x^4+a)^3+7/1024*(a/b)^(1/4)*2^(1/2)/a^3/b*g*ln((x^2+(a/b)^(1/4)*2^(1/2)
*x+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2)))+77/1024*(a/b)^(1/4)*2^(1/2)/a^4*c*ln((x^2+(a/b)^(1/4)
*2^(1/2)*x+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2)))+7/512*(a/b)^(1/4)*2^(1/2)/a^3/b*g*arctan(2^(1
/2)/(a/b)^(1/4)*x-1)+77/512*(a/b)^(1/4)*2^(1/2)/a^4*c*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+7/512*(a/b)^(1/4)*2^(1/2
)/a^3/b*g*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+77/512*(a/b)^(1/4)*2^(1/2)/a^4*c*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/3
2/(a*b)^(1/2)/a^2/b*h*arctan((1/a*b)^(1/2)*x^2)+5/32/(a*b)^(1/2)/a^3*d*arctan((1/a*b)^(1/2)*x^2)+5/1024/a^2/b^
2/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2)))*i+15
/1024/(a/b)^(1/4)*2^(1/2)/a^3/b*e*ln((x^2-(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*2^(1/2)*x+(a/b)^
(1/2)))+5/512/a^2/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*i+15/512/(a/b)^(1/4)*2^(1/2)/a^3/b*e
*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+5/512/a^2/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*i+15/512/(a
/b)^(1/4)*2^(1/2)/a^3/b*e*arctan(2^(1/2)/(a/b)^(1/4)*x+1)

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maxima [A]  time = 3.16, size = 579, normalized size = 1.12 \[ \frac {15 \, {\left (3 \, b^{3} e + a b^{2} i\right )} x^{11} + 12 \, {\left (5 \, b^{3} d + a b^{2} h\right )} x^{10} + 7 \, {\left (11 \, b^{3} c + a b^{2} g\right )} x^{9} + 42 \, {\left (3 \, a b^{2} e + a^{2} b i\right )} x^{7} + 32 \, {\left (5 \, a b^{2} d + a^{2} b h\right )} x^{6} + 18 \, {\left (11 \, a b^{2} c + a^{2} b g\right )} x^{5} - 32 \, a^{3} f + {\left (113 \, a^{2} b e - 5 \, a^{3} i\right )} x^{3} + 12 \, {\left (11 \, a^{2} b d - a^{3} h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b c - 7 \, a^{3} g\right )} x}{384 \, {\left (a^{3} b^{4} x^{12} + 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} + a^{6} b\right )}} + \frac {\frac {\sqrt {2} {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e + 7 \, a \sqrt {b} g - 5 \, a^{\frac {3}{2}} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e + 7 \, a \sqrt {b} g - 5 \, a^{\frac {3}{2}} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 7 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 5 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i - 80 \, \sqrt {a} b^{\frac {3}{2}} d - 16 \, a^{\frac {3}{2}} \sqrt {b} h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 7 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 5 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i + 80 \, \sqrt {a} b^{\frac {3}{2}} d + 16 \, a^{\frac {3}{2}} \sqrt {b} h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^4,x, algorithm="maxima")

[Out]

1/384*(15*(3*b^3*e + a*b^2*i)*x^11 + 12*(5*b^3*d + a*b^2*h)*x^10 + 7*(11*b^3*c + a*b^2*g)*x^9 + 42*(3*a*b^2*e
+ a^2*b*i)*x^7 + 32*(5*a*b^2*d + a^2*b*h)*x^6 + 18*(11*a*b^2*c + a^2*b*g)*x^5 - 32*a^3*f + (113*a^2*b*e - 5*a^
3*i)*x^3 + 12*(11*a^2*b*d - a^3*h)*x^2 + 3*(51*a^2*b*c - 7*a^3*g)*x)/(a^3*b^4*x^12 + 3*a^4*b^3*x^8 + 3*a^5*b^2
*x^4 + a^6*b) + 1/1024*(sqrt(2)*(77*b^(3/2)*c - 15*sqrt(a)*b*e + 7*a*sqrt(b)*g - 5*a^(3/2)*i)*log(sqrt(b)*x^2
+ sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) - sqrt(2)*(77*b^(3/2)*c - 15*sqrt(a)*b*e + 7*a*sqrt(b
)*g - 5*a^(3/2)*i)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(77*sqrt(2)*a^
(1/4)*b^(7/4)*c + 15*sqrt(2)*a^(3/4)*b^(5/4)*e + 7*sqrt(2)*a^(5/4)*b^(3/4)*g + 5*sqrt(2)*a^(7/4)*b^(1/4)*i - 8
0*sqrt(a)*b^(3/2)*d - 16*a^(3/2)*sqrt(b)*h)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sq
rt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(3/4)) + 2*(77*sqrt(2)*a^(1/4)*b^(7/4)*c + 15*sqrt(2)*a^(3/4)
*b^(5/4)*e + 7*sqrt(2)*a^(5/4)*b^(3/4)*g + 5*sqrt(2)*a^(7/4)*b^(1/4)*i + 80*sqrt(a)*b^(3/2)*d + 16*a^(3/2)*sqr
t(b)*h)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a
)*sqrt(b))*b^(3/4)))/(a^3*b)

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mupad [B]  time = 6.08, size = 2741, normalized size = 5.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a + b*x^4)^4,x)

[Out]

((3*x^5*(11*b*c + a*g))/(64*a^2) - f/(12*b) + (x^6*(5*b*d + a*h))/(12*a^2) + (7*x^7*(3*b*e + a*i))/(64*a^2) +
(7*b*x^9*(11*b*c + a*g))/(384*a^3) + (x*(51*b*c - 7*a*g))/(128*a*b) + (b*x^10*(5*b*d + a*h))/(32*a^3) + (5*b*x
^11*(3*b*e + a*i))/(128*a^3) + (x^2*(11*b*d - a*h))/(32*a*b) + (x^3*(113*b*e - 5*a*i))/(384*a*b))/(a^3 + b^3*x
^12 + 3*a^2*b*x^4 + 3*a*b^2*x^8) + symsum(log(- root(68719476736*a^15*b^7*z^4 + 1211105280*a^8*b^6*c*e*z^2 + 4
03701760*a^9*b^5*c*i*z^2 + 335544320*a^9*b^5*d*h*z^2 + 110100480*a^9*b^5*e*g*z^2 + 36700160*a^10*b^4*g*i*z^2 +
838860800*a^8*b^6*d^2*z^2 + 33554432*a^10*b^4*h^2*z^2 + 2457600*a^7*b^3*e*h*i*z - 88309760*a^5*b^5*c*d*g*z -
17661952*a^6*b^4*c*g*h*z + 12288000*a^6*b^4*d*e*i*z - 485703680*a^4*b^6*c^2*d*z + 409600*a^8*b^2*h*i^2*z - 971
40736*a^5*b^5*c^2*h*z - 802816*a^7*b^3*g^2*h*z + 3686400*a^6*b^4*e^2*h*z + 2048000*a^7*b^3*d*i^2*z - 4014080*a
^6*b^4*d*g^2*z + 18432000*a^5*b^5*d*e^2*z - 89600*a^4*b^2*d*g*h*i - 985600*a^3*b^3*c*d*h*i + 323400*a^3*b^3*c*
e*g*i - 268800*a^3*b^3*d*e*g*h - 2956800*a^2*b^4*c*d*e*h + 14700*a^4*b^2*e*g^2*i - 224000*a^3*b^3*d^2*g*i - 98
560*a^4*b^2*c*h^2*i - 26880*a^4*b^2*e*g*h^2 + 53900*a^4*b^2*c*g*i^2 + 1778700*a^2*b^4*c^2*e*i - 2464000*a^2*b^
4*c*d^2*i - 672000*a^2*b^4*d^2*e*g - 295680*a^3*b^3*c*e*h^2 + 485100*a^2*b^4*c*e^2*g - 8960*a^5*b*g*h^2*i - 73
92000*a*b^5*c*d^2*e + 7500*a^5*b*e*i^3 + 12782924*a*b^5*c^3*g + 33750*a^4*b^2*e^2*i^2 + 614400*a^3*b^3*d^2*h^2
+ 296450*a^3*b^3*c^2*i^2 + 22050*a^3*b^3*e^2*g^2 + 1743126*a^2*b^4*c^2*g^2 + 2450*a^5*b*g^2*i^2 + 67500*a^3*b
^3*e^3*i + 2048000*a^2*b^4*d^3*h + 81920*a^4*b^2*d*h^3 + 105644*a^3*b^3*c*g^3 + 2668050*a*b^5*c^2*e^2 + 2401*a
^4*b^2*g^4 + 50625*a^2*b^4*e^4 + 4096*a^5*b*h^4 + 2560000*a*b^5*d^4 + 625*a^6*i^4 + 35153041*b^6*c^4, z, l)*(r
oot(68719476736*a^15*b^7*z^4 + 1211105280*a^8*b^6*c*e*z^2 + 403701760*a^9*b^5*c*i*z^2 + 335544320*a^9*b^5*d*h*
z^2 + 110100480*a^9*b^5*e*g*z^2 + 36700160*a^10*b^4*g*i*z^2 + 838860800*a^8*b^6*d^2*z^2 + 33554432*a^10*b^4*h^
2*z^2 + 2457600*a^7*b^3*e*h*i*z - 88309760*a^5*b^5*c*d*g*z - 17661952*a^6*b^4*c*g*h*z + 12288000*a^6*b^4*d*e*i
*z - 485703680*a^4*b^6*c^2*d*z + 409600*a^8*b^2*h*i^2*z - 97140736*a^5*b^5*c^2*h*z - 802816*a^7*b^3*g^2*h*z +
3686400*a^6*b^4*e^2*h*z + 2048000*a^7*b^3*d*i^2*z - 4014080*a^6*b^4*d*g^2*z + 18432000*a^5*b^5*d*e^2*z - 89600
*a^4*b^2*d*g*h*i - 985600*a^3*b^3*c*d*h*i + 323400*a^3*b^3*c*e*g*i - 268800*a^3*b^3*d*e*g*h - 2956800*a^2*b^4*
c*d*e*h + 14700*a^4*b^2*e*g^2*i - 224000*a^3*b^3*d^2*g*i - 98560*a^4*b^2*c*h^2*i - 26880*a^4*b^2*e*g*h^2 + 539
00*a^4*b^2*c*g*i^2 + 1778700*a^2*b^4*c^2*e*i - 2464000*a^2*b^4*c*d^2*i - 672000*a^2*b^4*d^2*e*g - 295680*a^3*b
^3*c*e*h^2 + 485100*a^2*b^4*c*e^2*g - 8960*a^5*b*g*h^2*i - 7392000*a*b^5*c*d^2*e + 7500*a^5*b*e*i^3 + 12782924
*a*b^5*c^3*g + 33750*a^4*b^2*e^2*i^2 + 614400*a^3*b^3*d^2*h^2 + 296450*a^3*b^3*c^2*i^2 + 22050*a^3*b^3*e^2*g^2
+ 1743126*a^2*b^4*c^2*g^2 + 2450*a^5*b*g^2*i^2 + 67500*a^3*b^3*e^3*i + 2048000*a^2*b^4*d^3*h + 81920*a^4*b^2*
d*h^3 + 105644*a^3*b^3*c*g^3 + 2668050*a*b^5*c^2*e^2 + 2401*a^4*b^2*g^4 + 50625*a^2*b^4*e^4 + 4096*a^5*b*h^4 +
2560000*a*b^5*d^4 + 625*a^6*i^4 + 35153041*b^6*c^4, z, l)*((20185088*a^7*b^5*c + 1835008*a^8*b^4*g)/(2097152*
a^9*b^2) - (x*(655360*a^7*b^4*d + 131072*a^8*b^3*h))/(131072*a^9*b)) + (614400*a^4*b^4*d*e + 204800*a^5*b^3*d*
i + 122880*a^5*b^3*e*h + 40960*a^6*b^2*h*i)/(2097152*a^9*b^2) - (x*(800*a^6*b*i^2 - 189728*a^3*b^4*c^2 + 7200*
a^4*b^3*e^2 - 1568*a^5*b^2*g^2 - 34496*a^4*b^3*c*g + 4800*a^5*b^2*e*i))/(131072*a^9*b)) - (125*a^4*i^3 + 3375*
a*b^3*e^3 - 123200*b^4*c*d^2 + 88935*b^4*c^2*e - 4928*a^2*b^2*c*h^2 + 735*a^2*b^2*e*g^2 + 3375*a^2*b^2*e^2*i -
11200*a*b^3*d^2*g + 29645*a*b^3*c^2*i + 1125*a^3*b*e*i^2 - 448*a^3*b*g*h^2 + 245*a^3*b*g^2*i + 5390*a^2*b^2*c
*g*i - 4480*a^2*b^2*d*g*h - 49280*a*b^3*c*d*h + 16170*a*b^3*c*e*g)/(2097152*a^9*b^2) - (x*(5775*b^3*c*d*e - 32
*a^3*h^3 - 4000*b^3*d^3 + 35*a^3*g*h*i - 2400*a*b^2*d^2*h - 480*a^2*b*d*h^2 + 1925*a*b^2*c*d*i + 1155*a*b^2*c*
e*h + 525*a*b^2*d*e*g + 385*a^2*b*c*h*i + 175*a^2*b*d*g*i + 105*a^2*b*e*g*h))/(131072*a^9*b))*root(68719476736
*a^15*b^7*z^4 + 1211105280*a^8*b^6*c*e*z^2 + 403701760*a^9*b^5*c*i*z^2 + 335544320*a^9*b^5*d*h*z^2 + 110100480
*a^9*b^5*e*g*z^2 + 36700160*a^10*b^4*g*i*z^2 + 838860800*a^8*b^6*d^2*z^2 + 33554432*a^10*b^4*h^2*z^2 + 2457600
*a^7*b^3*e*h*i*z - 88309760*a^5*b^5*c*d*g*z - 17661952*a^6*b^4*c*g*h*z + 12288000*a^6*b^4*d*e*i*z - 485703680*
a^4*b^6*c^2*d*z + 409600*a^8*b^2*h*i^2*z - 97140736*a^5*b^5*c^2*h*z - 802816*a^7*b^3*g^2*h*z + 3686400*a^6*b^4
*e^2*h*z + 2048000*a^7*b^3*d*i^2*z - 4014080*a^6*b^4*d*g^2*z + 18432000*a^5*b^5*d*e^2*z - 89600*a^4*b^2*d*g*h*
i - 985600*a^3*b^3*c*d*h*i + 323400*a^3*b^3*c*e*g*i - 268800*a^3*b^3*d*e*g*h - 2956800*a^2*b^4*c*d*e*h + 14700
*a^4*b^2*e*g^2*i - 224000*a^3*b^3*d^2*g*i - 98560*a^4*b^2*c*h^2*i - 26880*a^4*b^2*e*g*h^2 + 53900*a^4*b^2*c*g*
i^2 + 1778700*a^2*b^4*c^2*e*i - 2464000*a^2*b^4*c*d^2*i - 672000*a^2*b^4*d^2*e*g - 295680*a^3*b^3*c*e*h^2 + 48
5100*a^2*b^4*c*e^2*g - 8960*a^5*b*g*h^2*i - 7392000*a*b^5*c*d^2*e + 7500*a^5*b*e*i^3 + 12782924*a*b^5*c^3*g +
33750*a^4*b^2*e^2*i^2 + 614400*a^3*b^3*d^2*h^2 + 296450*a^3*b^3*c^2*i^2 + 22050*a^3*b^3*e^2*g^2 + 1743126*a^2*
b^4*c^2*g^2 + 2450*a^5*b*g^2*i^2 + 67500*a^3*b^3*e^3*i + 2048000*a^2*b^4*d^3*h + 81920*a^4*b^2*d*h^3 + 105644*
a^3*b^3*c*g^3 + 2668050*a*b^5*c^2*e^2 + 2401*a^4*b^2*g^4 + 50625*a^2*b^4*e^4 + 4096*a^5*b*h^4 + 2560000*a*b^5*
d^4 + 625*a^6*i^4 + 35153041*b^6*c^4, z, l), l, 1, 4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**4,x)

[Out]

Timed out

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