3.209 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{(a+b x^4)^4} \, dx\)

Optimal. Leaf size=534 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {(a h+5 b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (a j+2 b f)-x \left (b (a g+11 b c)+2 b x (a h+5 b d)+3 b x^2 (a i+3 b e)\right )}{96 a^2 b^2 \left (a+b x^4\right )^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{12 a b \left (a+b x^4\right )^3} \]

[Out]

1/12*x*(b*c-a*g+(-a*h+b*d)*x+(-a*i+b*e)*x^2+(-a*j+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*(-4*a*(a*j+2*b*f)+x*(b*(a*g+11*b*c)+2*b*(a*h+5*b*d)*x+3*b*(a*i+3
*b*e)*x^2))/a^2/b^2/(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)+1/512*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)+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 = 0.82, antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1858, 1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {4 a (a j+2 b f)-x \left (b (a g+11 b c)+2 b x (a h+5 b d)+3 b x^2 (a i+3 b e)\right )}{96 a^2 b^2 \left (a+b x^4\right )^2}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (7 \sqrt {b} (a g+11 b c)-5 \sqrt {a} (a i+3 b e)\right )}{512 \sqrt {2} a^{15/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt {b} (a g+11 b c)+5 \sqrt {a} (a i+3 b e)\right )}{256 \sqrt {2} a^{15/4} b^{7/4}}+\frac {(a h+5 b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{12 a b \left (a+b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(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 - a*j)*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)) - (4*a*(2*b*f + a*j) - x*(b*(11*b*c
+ a*g) + 2*b*(5*b*d + a*h)*x + 3*b*(3*b*e + a*i)*x^2))/(96*a^2*b^2*(a + b*x^4)^2) + ((5*b*d + a*h)*ArcTan[(Sqr
t[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 - (S
qrt[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))*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[Sqrt[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+209 x^6+j x^7}{\left (a+b x^4\right )^4} \, dx &=\frac {x \left (b c-a g+(b d-a h) x-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2-4 b (2 b f+a j) 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-(209 a-b e) x^2+(b f-a j) x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \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 (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \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 (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \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 (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \left (a+b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c+a g)-15 b (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \left (a+b x^4\right )^2}-\frac {\left (5 (209 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 (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \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 (209 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 (209 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 (209 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 (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \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 (209 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 (209 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 (209 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 (209 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-(209 a-b e) x^2+(b f-a j) 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 (209 a+3 b e) x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac {4 a (2 b f+a j)-x \left (b (11 b c+a g)+2 b (5 b d+a h) x+3 b (209 a+3 b e) x^2\right )}{96 a^2 b^2 \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 (209 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 (209 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 (209 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 (209 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 = 0.71, size = 555, normalized size = 1.04 \[ \frac {-6 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 a^{5/4} \sqrt [4]{b} h+5 \sqrt {2} a^{3/2} i+80 \sqrt [4]{a} b^{5/4} d+15 \sqrt {2} \sqrt {a} b e+7 \sqrt {2} a \sqrt {b} g+77 \sqrt {2} b^{3/2} c\right )+6 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 a^{5/4} \sqrt [4]{b} h+5 \sqrt {2} a^{3/2} i-80 \sqrt [4]{a} b^{5/4} d+15 \sqrt {2} \sqrt {a} b e+7 \sqrt {2} a \sqrt {b} g+77 \sqrt {2} b^{3/2} c\right )+3 \sqrt {2} \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (5 a^{3/2} i+15 \sqrt {a} b e-7 a \sqrt {b} g-77 b^{3/2} c\right )+3 \sqrt {2} \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-5 a^{3/2} i-15 \sqrt {a} b e+7 a \sqrt {b} g+77 b^{3/2} c\right )+\frac {8 a^{3/4} b x (7 a g+3 a x (4 h+5 i x)+77 b c+15 b x (4 d+3 e x))}{a+b x^4}+\frac {256 a^{11/4} \left (a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a+b x^4\right )^3}-\frac {32 a^{7/4} \left (12 a^2 j-a b x (g+x (2 h+3 i x))-b^2 x (11 c+x (10 d+9 e x))\right )}{\left (a+b x^4\right )^2}}{3072 a^{15/4} b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a + b*x^4)^4,x]

[Out]

((8*a^(3/4)*b*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) - (32*a^(7/4)*(12*a
^2*j - b^2*x*(11*c + x*(10*d + 9*e*x)) - a*b*x*(g + x*(2*h + 3*i*x))))/(a + b*x^4)^2 + (256*a^(11/4)*(a^2*j +
b^2*x*(c + x*(d + e*x)) - a*b*(f + x*(g + x*(h + i*x)))))/(a + b*x^4)^3 - 6*b^(1/4)*(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*b^(1/4)*(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]*b^(1/4)*(-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]*b^(1/4)*(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^2)

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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((j*x^7+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 = 767, normalized size = 1.44 \[ \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^{3} i x^{11} + 45 \, b^{4} x^{11} e + 60 \, b^{4} d x^{10} + 12 \, a b^{3} h x^{10} + 77 \, b^{4} c x^{9} + 7 \, a b^{3} g x^{9} + 42 \, a^{2} b^{2} i x^{7} + 126 \, a b^{3} x^{7} e + 160 \, a b^{3} d x^{6} + 32 \, a^{2} b^{2} h x^{6} + 198 \, a b^{3} c x^{5} + 18 \, a^{2} b^{2} g x^{5} - 48 \, a^{3} b j x^{4} - 5 \, a^{3} b i x^{3} + 113 \, a^{2} b^{2} x^{3} e + 132 \, a^{2} b^{2} d x^{2} - 12 \, a^{3} b h x^{2} + 153 \, a^{2} b^{2} c x - 21 \, a^{3} b g x - 32 \, a^{3} b f - 16 \, a^{4} j}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x^7+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^3*i*
x^11 + 45*b^4*x^11*e + 60*b^4*d*x^10 + 12*a*b^3*h*x^10 + 77*b^4*c*x^9 + 7*a*b^3*g*x^9 + 42*a^2*b^2*i*x^7 + 126
*a*b^3*x^7*e + 160*a*b^3*d*x^6 + 32*a^2*b^2*h*x^6 + 198*a*b^3*c*x^5 + 18*a^2*b^2*g*x^5 - 48*a^3*b*j*x^4 - 5*a^
3*b*i*x^3 + 113*a^2*b^2*x^3*e + 132*a^2*b^2*d*x^2 - 12*a^3*b*h*x^2 + 153*a^2*b^2*c*x - 21*a^3*b*g*x - 32*a^3*b
*f - 16*a^4*j)/((b*x^4 + a)^3*a^3*b^2)

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maple [A]  time = 0.07, size = 783, normalized size = 1.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((j*x^7+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^5-1/8/b*j*x^4-1/384*(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-1/24*(a*j+2*b*f)/b^2)/(b*x^4+a)^3+7/512*(a/b)^(1/4)*2^(1/2)/a^3/b*g*arct
an(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)+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)))+1/32/(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/b)^(1/4)*2^(1/2)/a^2/b^2*i*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)))+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/b)^(1/4)*2^(1/2)/a^2/b^2*i*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+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/b)^(1/4)*2^(1/2)/a^2/b^2*i*arctan(2^(1/2)/(a
/b)^(1/4)*x-1)+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 = 3.23, size = 613, normalized size = 1.15 \[ \frac {15 \, {\left (3 \, b^{4} e + a b^{3} i\right )} x^{11} + 12 \, {\left (5 \, b^{4} d + a b^{3} h\right )} x^{10} + 7 \, {\left (11 \, b^{4} c + a b^{3} g\right )} x^{9} - 48 \, a^{3} b j x^{4} + 42 \, {\left (3 \, a b^{3} e + a^{2} b^{2} i\right )} x^{7} + 32 \, {\left (5 \, a b^{3} d + a^{2} b^{2} h\right )} x^{6} + 18 \, {\left (11 \, a b^{3} c + a^{2} b^{2} g\right )} x^{5} - 32 \, a^{3} b f - 16 \, a^{4} j + {\left (113 \, a^{2} b^{2} e - 5 \, a^{3} b i\right )} x^{3} + 12 \, {\left (11 \, a^{2} b^{2} d - a^{3} b h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b^{2} c - 7 \, a^{3} b g\right )} x}{384 \, {\left (a^{3} b^{5} x^{12} + 3 \, a^{4} b^{4} x^{8} + 3 \, a^{5} b^{3} x^{4} + a^{6} b^{2}\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((j*x^7+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^4*e + a*b^3*i)*x^11 + 12*(5*b^4*d + a*b^3*h)*x^10 + 7*(11*b^4*c + a*b^3*g)*x^9 - 48*a^3*b*j*x^4
+ 42*(3*a*b^3*e + a^2*b^2*i)*x^7 + 32*(5*a*b^3*d + a^2*b^2*h)*x^6 + 18*(11*a*b^3*c + a^2*b^2*g)*x^5 - 32*a^3*
b*f - 16*a^4*j + (113*a^2*b^2*e - 5*a^3*b*i)*x^3 + 12*(11*a^2*b^2*d - a^3*b*h)*x^2 + 3*(51*a^2*b^2*c - 7*a^3*b
*g)*x)/(a^3*b^5*x^12 + 3*a^4*b^4*x^8 + 3*a^5*b^3*x^4 + a^6*b^2) + 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)) - s
qrt(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 - 80*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(sqrt(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)*sqrt(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 = 6.48, size = 2757, normalized size = 5.16 \[ \text {result too large to display} \]

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 + j*x^7)/(a + b*x^4)^4,x)

[Out]

symsum(log(- 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 + 33554432
0*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 + 335544
32*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 - 29
56800*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 + 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 + 40
96*a^5*b*h^4 + 2560000*a*b^5*d^4 + 625*a^6*i^4 + 35153041*b^6*c^4, z, m)*(root(68719476736*a^15*b^7*z^4 + 1211
105280*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 + 3
6700160*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 - 8
8309760*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 + 40
9600*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 - 2
24000*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 + 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 + 2668
050*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, m)*((20185088*a^7*b^5*c + 1835008*a^8*b^4*g)/(2097152*a^9*b^2) - (x*(655360*a^7*b^4*d + 1
31072*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 - 34
496*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 + 8893
5*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 - 246
4000*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, m), m, 1, 4) + ((3*x^5*(11*b*c + a*g))/(64*a^2) - (j*x^4)/(8*b) - (2*b*f + a*j)/(24*b^2) + (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)

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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((j*x**7+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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