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

Optimal. Leaf size=268 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {x \left (2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

[Out]

1/8*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)^2+1/32*(4*a*f+x*(7*b*c-a*g+2*(-a*h+3*b*d)*x+(
-3*a*i+5*b*e)*x^2))/a^2/b/(-b*x^4+a)+1/16*(-a*h+3*b*d)*arctanh(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(3/2)-1/64*arcta
n(b^(1/4)*x/a^(1/4))*(5*b*e-3*a*i-3*(-a*g+7*b*c)*b^(1/2)/a^(1/2))/a^(9/4)/b^(7/4)+1/64*arctanh(b^(1/4)*x/a^(1/
4))*(5*b*e-3*a*i+3*(-a*g+7*b*c)*b^(1/2)/a^(1/2))/a^(9/4)/b^(7/4)

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Rubi [A]  time = 0.43, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1858, 1854, 1876, 275, 208, 1167, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {x \left (2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

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)^3,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + b*f*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a*f + x*(7*b*c - a*g +
2*(3*b*d - a*h)*x + (5*b*e - 3*a*i)*x^2))/(32*a^2*b*(a - b*x^4)) - ((5*b*e - (3*Sqrt[b]*(7*b*c - a*g))/Sqrt[a]
- 3*a*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((5*b*e + (3*Sqrt[b]*(7*b*c - a*g))/Sqrt[a] - 3*
a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(9/4)*b^(7/4)) + ((3*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a
^(5/2)*b^(3/2))

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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 1167

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(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 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+199 x^6}{\left (a-b x^4\right )^3} \, dx &=\frac {x \left (b c+a g+(b d+a h) x+(199 a+b e) x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}-\frac {\int \frac {-b (7 b c-a g)-2 b (3 b d-a h) x+b (597 a-5 b e) x^2-4 b^2 f x^3}{\left (a-b x^4\right )^2} \, dx}{8 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(199 a+b e) x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)+4 b (3 b d-a h) x-b (597 a-5 b e) x^2}{a-b x^4} \, dx}{32 a^2 b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(199 a+b e) x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\int \left (\frac {4 b (3 b d-a h) x}{a-b x^4}+\frac {3 b (7 b c-a g)-b (597 a-5 b e) x^2}{a-b x^4}\right ) \, dx}{32 a^2 b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(199 a+b e) x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)-b (597 a-5 b e) x^2}{a-b x^4} \, dx}{32 a^2 b^2}+\frac {(3 b d-a h) \int \frac {x}{a-b x^4} \, dx}{8 a^2 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(199 a+b e) x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\left (597 a-5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2 b}-\frac {\left (597 a-5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2 b}+\frac {(3 b d-a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(199 a+b e) x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\left (597 a-5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}-\frac {\left (597 a-5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 359, normalized size = 1.34 \[ \frac {\frac {16 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^2}-\frac {4 a^{3/4} b^{3/4} x (a (g+x (2 h+3 i x))-b (7 c+x (6 d+5 e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt {a} b e+3 a \sqrt {b} g-21 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt {a} b e-3 a \sqrt {b} g+21 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-5 \sqrt {a} b e-3 a \sqrt {b} g+21 b^{3/2} c\right )-4 \sqrt [4]{a} \sqrt [4]{b} (a h-3 b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^{7/4}} \]

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)^3,x]

[Out]

((-4*a^(3/4)*b^(3/4)*x*(-(b*(7*c + x*(6*d + 5*e*x))) + a*(g + x*(2*h + 3*i*x))))/(a - b*x^4) + (16*a^(7/4)*b^(
3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + x*(h + i*x)))))/(a - b*x^4)^2 + 2*(21*b^(3/2)*c - 5*Sqrt[a]*b*e -
3*a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-21*b^(3/2)*c - 12*a^(1/4)*b^(5/4)*d - 5*Sqrt[a]*b
*e + 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + (21*b^(3/2)*c - 12*a^(1/4)*
b^(5/4)*d + 5*Sqrt[a]*b*e - 3*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 4*a^
(1/4)*b^(1/4)*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*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)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.28, size = 652, normalized size = 2.43 \[ -\frac {3}{256} \, 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^{2} 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^{2} b^{4}}\right )} - \frac {3}{256} \, 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^{2} 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^{2} b^{4}}\right )} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + 5 \, \sqrt {-a b} b 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 )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - 5 \, \sqrt {-a b} b 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 )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {3 \, a b i x^{7} - 5 \, b^{2} x^{7} e - 6 \, b^{2} d x^{6} + 2 \, a b h x^{6} - 7 \, b^{2} c x^{5} + a b g x^{5} + a^{2} i x^{3} + 9 \, a b x^{3} e + 10 \, a b d x^{2} + 2 \, a^{2} h x^{2} + 11 \, a b c x + 3 \, a^{2} g x + 4 \, a^{2} f}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} 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)^3,x, algorithm="giac")

[Out]

-3/256*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^2*b^4) - s
qrt(2)*(-a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^4)) - 3/256*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^2*b^4) + sqrt(2)*(-a*b^3)^(3/4)*log(x^2
- sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^4)) - 1/128*sqrt(2)*(21*b^2*c - 3*a*b*g - 12*sqrt(2)*(-a*b^3)^(
1/4)*b*d + 4*sqrt(2)*(-a*b^3)^(1/4)*a*h + 5*sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-
a/b)^(1/4))/((-a*b^3)^(3/4)*a^2) - 1/128*sqrt(2)*(21*b^2*c - 3*a*b*g + 12*sqrt(2)*(-a*b^3)^(1/4)*b*d - 4*sqrt(
2)*(-a*b^3)^(1/4)*a*h - 5*sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b
^3)^(3/4)*a^2) - 1/256*sqrt(2)*(21*b^2*c - 3*a*b*g - 5*sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt
(-a/b))/((-a*b^3)^(3/4)*a^2) + 1/256*sqrt(2)*(21*b^2*c - 3*a*b*g - 5*sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b
)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) + 1/32*(3*a*b*i*x^7 - 5*b^2*x^7*e - 6*b^2*d*x^6 + 2*a*b*h*x^6 - 7*b
^2*c*x^5 + a*b*g*x^5 + a^2*i*x^3 + 9*a*b*x^3*e + 10*a*b*d*x^2 + 2*a^2*h*x^2 + 11*a*b*c*x + 3*a^2*g*x + 4*a^2*f
)/((b*x^4 - a)^2*a^2*b)

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maple [B]  time = 0.06, size = 472, normalized size = 1.76 \[ \frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a b}-\frac {3 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a^{2}}+\frac {3 i \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}-\frac {3 i \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}-\frac {5 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{2} b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{2} b}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{3}}-\frac {-\frac {\left (3 a i -5 b e \right ) x^{7}}{32 a^{2}}-\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}-\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}-\frac {\left (a i +9 b e \right ) x^{3}}{32 a b}-\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}-\frac {f}{8 b}-\frac {\left (3 a g +11 b c \right ) x}{32 a b}}{\left (b \,x^{4}-a \right )^{2}} \]

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)^3,x)

[Out]

-(-1/32*(3*a*i-5*b*e)/a^2*x^7-1/16*(a*h-3*b*d)/a^2*x^6-1/32*(a*g-7*b*c)/a^2*x^5-1/32*(a*i+9*b*e)/a/b*x^3-1/16*
(a*h+5*b*d)/a/b*x^2-1/32*(3*a*g+11*b*c)/a/b*x-1/8/b*f)/(b*x^4-a)^2-3/64*(a/b)^(1/4)/a^2/b*g*arctan(1/(a/b)^(1/
4)*x)+21/64*(a/b)^(1/4)/a^3*c*arctan(1/(a/b)^(1/4)*x)-3/128*(a/b)^(1/4)/a^2/b*g*ln((x+(a/b)^(1/4))/(x-(a/b)^(1
/4)))+21/128*(a/b)^(1/4)/a^3*c*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/32/(a*b)^(1/2)/a/b*h*ln(((a*b)^(1/2)*x^2-
a)/(-(a*b)^(1/2)*x^2-a))-3/32/(a*b)^(1/2)/a^2*d*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))+3/64/a/b^2/(a/b)^
(1/4)*arctan(1/(a/b)^(1/4)*x)*i-5/64/(a/b)^(1/4)/a^2/b*e*arctan(1/(a/b)^(1/4)*x)-3/128/a/b^2/(a/b)^(1/4)*ln((x
+(a/b)^(1/4))/(x-(a/b)^(1/4)))*i+5/128/(a/b)^(1/4)/a^2/b*e*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))

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maxima [A]  time = 3.08, size = 343, normalized size = 1.28 \[ -\frac {{\left (5 \, b^{2} e - 3 \, a b i\right )} x^{7} + 2 \, {\left (3 \, b^{2} d - a b h\right )} x^{6} + {\left (7 \, b^{2} c - a b g\right )} x^{5} - {\left (9 \, a b e + a^{2} i\right )} x^{3} - 4 \, a^{2} f - 2 \, {\left (5 \, a b d + a^{2} h\right )} x^{2} - {\left (11 \, a b c + 3 \, a^{2} g\right )} x}{32 \, {\left (a^{2} b^{3} x^{8} - 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g + 3 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]

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)^3,x, algorithm="maxima")

[Out]

-1/32*((5*b^2*e - 3*a*b*i)*x^7 + 2*(3*b^2*d - a*b*h)*x^6 + (7*b^2*c - a*b*g)*x^5 - (9*a*b*e + a^2*i)*x^3 - 4*a
^2*f - 2*(5*a*b*d + a^2*h)*x^2 - (11*a*b*c + 3*a^2*g)*x)/(a^2*b^3*x^8 - 2*a^3*b^2*x^4 + a^4*b) + 1/128*(4*(3*b
*d - a*h)*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 4*(3*b*d - a*h)*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*s
qrt(b)) + 2*(21*b^(3/2)*c - 5*sqrt(a)*b*e - 3*a*sqrt(b)*g + 3*a^(3/2)*i)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)
))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (21*b^(3/2)*c + 5*sqrt(a)*b*e - 3*a*sqrt(b)*g - 3*a^(3/2)*i)*log(
(sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b
)))/(a^2*b)

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mupad [B]  time = 5.80, size = 2680, normalized size = 10.00 \[ \text {result too large to display} \]

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)^3,x)

[Out]

symsum(log((27*a^4*i^3 - 125*a*b^3*e^3 - 3024*b^4*c*d^2 + 2205*b^4*c^2*e - 336*a^2*b^2*c*h^2 + 45*a^2*b^2*e*g^
2 + 225*a^2*b^2*e^2*i + 432*a*b^3*d^2*g - 1323*a*b^3*c^2*i - 135*a^3*b*e*i^2 + 48*a^3*b*g*h^2 - 27*a^3*b*g^2*i
+ 378*a^2*b^2*c*g*i - 288*a^2*b^2*d*g*h + 2016*a*b^3*c*d*h - 630*a*b^3*c*e*g)/(32768*a^6*b^2) - root(26843545
6*a^11*b^7*z^4 - 589824*a^8*b^4*g*i*z^2 + 4128768*a^7*b^5*c*i*z^2 + 3145728*a^7*b^5*d*h*z^2 + 983040*a^7*b^5*e
*g*z^2 - 6881280*a^6*b^6*c*e*z^2 - 524288*a^8*b^4*h^2*z^2 - 4718592*a^6*b^6*d^2*z^2 + 61440*a^6*b^3*e*h*i*z +
258048*a^5*b^4*c*g*h*z - 184320*a^5*b^4*d*e*i*z - 774144*a^4*b^5*c*d*g*z - 18432*a^7*b^2*h*i^2*z - 18432*a^6*b
^3*g^2*h*z + 55296*a^6*b^3*d*i^2*z - 51200*a^5*b^4*e^2*h*z - 903168*a^4*b^5*c^2*h*z + 55296*a^5*b^4*d*g^2*z +
153600*a^4*b^5*d*e^2*z + 2709504*a^3*b^6*c^2*d*z + 3456*a^4*b^2*d*g*h*i - 24192*a^3*b^3*c*d*h*i + 7560*a^3*b^3
*c*e*g*i - 5760*a^3*b^3*d*e*g*h + 40320*a^2*b^4*c*d*e*h - 540*a^4*b^2*e*g^2*i - 5184*a^3*b^3*d^2*g*i + 4032*a^
4*b^2*c*h^2*i + 960*a^4*b^2*e*g*h^2 - 2268*a^4*b^2*c*g*i^2 - 26460*a^2*b^4*c^2*e*i + 36288*a^2*b^4*c*d^2*i + 8
640*a^2*b^4*d^2*e*g - 6720*a^3*b^3*c*e*h^2 - 6300*a^2*b^4*c*e^2*g - 576*a^5*b*g*h^2*i - 60480*a*b^5*c*d^2*e +
540*a^5*b*e*i^3 + 111132*a*b^5*c^3*g - 1350*a^4*b^2*e^2*i^2 + 13824*a^3*b^3*d^2*h^2 + 7938*a^3*b^3*c^2*i^2 + 4
50*a^3*b^3*e^2*g^2 - 23814*a^2*b^4*c^2*g^2 + 162*a^5*b*g^2*i^2 + 1500*a^3*b^3*e^3*i - 27648*a^2*b^4*d^3*h - 30
72*a^4*b^2*d*h^3 + 2268*a^3*b^3*c*g^3 + 22050*a*b^5*c^2*e^2 - 81*a^4*b^2*g^4 - 625*a^2*b^4*e^4 + 256*a^5*b*h^4
+ 20736*a*b^5*d^4 - 81*a^6*i^4 - 194481*b^6*c^4, z, l)*(root(268435456*a^11*b^7*z^4 - 589824*a^8*b^4*g*i*z^2
+ 4128768*a^7*b^5*c*i*z^2 + 3145728*a^7*b^5*d*h*z^2 + 983040*a^7*b^5*e*g*z^2 - 6881280*a^6*b^6*c*e*z^2 - 52428
8*a^8*b^4*h^2*z^2 - 4718592*a^6*b^6*d^2*z^2 + 61440*a^6*b^3*e*h*i*z + 258048*a^5*b^4*c*g*h*z - 184320*a^5*b^4*
d*e*i*z - 774144*a^4*b^5*c*d*g*z - 18432*a^7*b^2*h*i^2*z - 18432*a^6*b^3*g^2*h*z + 55296*a^6*b^3*d*i^2*z - 512
00*a^5*b^4*e^2*h*z - 903168*a^4*b^5*c^2*h*z + 55296*a^5*b^4*d*g^2*z + 153600*a^4*b^5*d*e^2*z + 2709504*a^3*b^6
*c^2*d*z + 3456*a^4*b^2*d*g*h*i - 24192*a^3*b^3*c*d*h*i + 7560*a^3*b^3*c*e*g*i - 5760*a^3*b^3*d*e*g*h + 40320*
a^2*b^4*c*d*e*h - 540*a^4*b^2*e*g^2*i - 5184*a^3*b^3*d^2*g*i + 4032*a^4*b^2*c*h^2*i + 960*a^4*b^2*e*g*h^2 - 22
68*a^4*b^2*c*g*i^2 - 26460*a^2*b^4*c^2*e*i + 36288*a^2*b^4*c*d^2*i + 8640*a^2*b^4*d^2*e*g - 6720*a^3*b^3*c*e*h
^2 - 6300*a^2*b^4*c*e^2*g - 576*a^5*b*g*h^2*i - 60480*a*b^5*c*d^2*e + 540*a^5*b*e*i^3 + 111132*a*b^5*c^3*g - 1
350*a^4*b^2*e^2*i^2 + 13824*a^3*b^3*d^2*h^2 + 7938*a^3*b^3*c^2*i^2 + 450*a^3*b^3*e^2*g^2 - 23814*a^2*b^4*c^2*g
^2 + 162*a^5*b*g^2*i^2 + 1500*a^3*b^3*e^3*i - 27648*a^2*b^4*d^3*h - 3072*a^4*b^2*d*h^3 + 2268*a^3*b^3*c*g^3 +
22050*a*b^5*c^2*e^2 - 81*a^4*b^2*g^4 - 625*a^2*b^4*e^4 + 256*a^5*b*h^4 + 20736*a*b^5*d^4 - 81*a^6*i^4 - 194481
*b^6*c^4, z, l)*((344064*a^5*b^5*c - 49152*a^6*b^4*g)/(32768*a^6*b^2) - (x*(24576*a^5*b^4*d - 8192*a^6*b^3*h))
/(4096*a^6*b)) - (15360*a^3*b^4*d*e - 9216*a^4*b^3*d*i - 5120*a^4*b^3*e*h + 3072*a^5*b^2*h*i)/(32768*a^6*b^2)
+ (x*(144*a^5*b*i^2 + 7056*a^2*b^4*c^2 + 400*a^3*b^3*e^2 + 144*a^4*b^2*g^2 - 2016*a^3*b^3*c*g - 480*a^4*b^2*e*
i))/(4096*a^6*b)) - (x*(216*b^3*d^3 - 8*a^3*h^3 - 315*b^3*c*d*e + 9*a^3*g*h*i - 216*a*b^2*d^2*h + 72*a^2*b*d*h
^2 + 189*a*b^2*c*d*i + 105*a*b^2*c*e*h + 45*a*b^2*d*e*g - 63*a^2*b*c*h*i - 27*a^2*b*d*g*i - 15*a^2*b*e*g*h))/(
4096*a^6*b))*root(268435456*a^11*b^7*z^4 - 589824*a^8*b^4*g*i*z^2 + 4128768*a^7*b^5*c*i*z^2 + 3145728*a^7*b^5*
d*h*z^2 + 983040*a^7*b^5*e*g*z^2 - 6881280*a^6*b^6*c*e*z^2 - 524288*a^8*b^4*h^2*z^2 - 4718592*a^6*b^6*d^2*z^2
+ 61440*a^6*b^3*e*h*i*z + 258048*a^5*b^4*c*g*h*z - 184320*a^5*b^4*d*e*i*z - 774144*a^4*b^5*c*d*g*z - 18432*a^7
*b^2*h*i^2*z - 18432*a^6*b^3*g^2*h*z + 55296*a^6*b^3*d*i^2*z - 51200*a^5*b^4*e^2*h*z - 903168*a^4*b^5*c^2*h*z
+ 55296*a^5*b^4*d*g^2*z + 153600*a^4*b^5*d*e^2*z + 2709504*a^3*b^6*c^2*d*z + 3456*a^4*b^2*d*g*h*i - 24192*a^3*
b^3*c*d*h*i + 7560*a^3*b^3*c*e*g*i - 5760*a^3*b^3*d*e*g*h + 40320*a^2*b^4*c*d*e*h - 540*a^4*b^2*e*g^2*i - 5184
*a^3*b^3*d^2*g*i + 4032*a^4*b^2*c*h^2*i + 960*a^4*b^2*e*g*h^2 - 2268*a^4*b^2*c*g*i^2 - 26460*a^2*b^4*c^2*e*i +
36288*a^2*b^4*c*d^2*i + 8640*a^2*b^4*d^2*e*g - 6720*a^3*b^3*c*e*h^2 - 6300*a^2*b^4*c*e^2*g - 576*a^5*b*g*h^2*
i - 60480*a*b^5*c*d^2*e + 540*a^5*b*e*i^3 + 111132*a*b^5*c^3*g - 1350*a^4*b^2*e^2*i^2 + 13824*a^3*b^3*d^2*h^2
+ 7938*a^3*b^3*c^2*i^2 + 450*a^3*b^3*e^2*g^2 - 23814*a^2*b^4*c^2*g^2 + 162*a^5*b*g^2*i^2 + 1500*a^3*b^3*e^3*i
- 27648*a^2*b^4*d^3*h - 3072*a^4*b^2*d*h^3 + 2268*a^3*b^3*c*g^3 + 22050*a*b^5*c^2*e^2 - 81*a^4*b^2*g^4 - 625*a
^2*b^4*e^4 + 256*a^5*b*h^4 + 20736*a*b^5*d^4 - 81*a^6*i^4 - 194481*b^6*c^4, z, l), l, 1, 4) + (f/(8*b) - (x^5*
(7*b*c - a*g))/(32*a^2) - (x^6*(3*b*d - a*h))/(16*a^2) - (x^7*(5*b*e - 3*a*i))/(32*a^2) + (x*(11*b*c + 3*a*g))
/(32*a*b) + (x^2*(5*b*d + a*h))/(16*a*b) + (x^3*(9*b*e + a*i))/(32*a*b))/(a^2 + b^2*x^8 - 2*a*b*x^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)**3,x)

[Out]

Timed out

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