3.123 \(\int \frac{1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=352 \[ -\frac{b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac{b^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3} (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac{d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{d^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{10/3} (b c-a d)}-\frac{1}{7 a c x^7} \]

[Out]

-1/(7*a*c*x^7) + (b*c + a*d)/(4*a^2*c^2*x^4) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^
3*c^3*x) + (b^(10/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]
*a^(10/3)*(b*c - a*d)) - (d^(10/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/
3))])/(Sqrt[3]*c^(10/3)*(b*c - a*d)) + (b^(10/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^
(10/3)*(b*c - a*d)) - (d^(10/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(10/3)*(b*c - a*d
)) - (b^(10/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(10/3)*(b*c
- a*d)) + (d^(10/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(10/3)*
(b*c - a*d))

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Rubi [A]  time = 1.3383, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac{b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac{b^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3} (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac{d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac{d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac{d^{10/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{10/3} (b c-a d)}-\frac{1}{7 a c x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(7*a*c*x^7) + (b*c + a*d)/(4*a^2*c^2*x^4) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^
3*c^3*x) + (b^(10/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]
*a^(10/3)*(b*c - a*d)) - (d^(10/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/
3))])/(Sqrt[3]*c^(10/3)*(b*c - a*d)) + (b^(10/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^
(10/3)*(b*c - a*d)) - (d^(10/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(10/3)*(b*c - a*d
)) - (b^(10/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(10/3)*(b*c
- a*d)) + (d^(10/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(10/3)*
(b*c - a*d))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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Mathematica [A]  time = 0.494183, size = 304, normalized size = 0.86 \[ \frac{-\frac{28 b^{10/3} x^7 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{10/3}}-\frac{28 \sqrt{3} b^{10/3} x^7 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{10/3}}+\frac{14 b^{10/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{10/3}}+\frac{84 b^3 x^6}{a^3}-\frac{21 b^2 x^3}{a^2}+\frac{12 b}{a}+\frac{28 d^{10/3} x^7 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{10/3}}+\frac{28 \sqrt{3} d^{10/3} x^7 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{10/3}}-\frac{14 d^{10/3} x^7 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{10/3}}-\frac{84 d^3 x^6}{c^3}+\frac{21 d^2 x^3}{c^2}-\frac{12 d}{c}}{84 x^7 (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((12*b)/a - (12*d)/c - (21*b^2*x^3)/a^2 + (21*d^2*x^3)/c^2 + (84*b^3*x^6)/a^3 -
(84*d^3*x^6)/c^3 - (28*Sqrt[3]*b^(10/3)*x^7*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]])/a^(10/3) + (28*Sqrt[3]*d^(10/3)*x^7*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/
Sqrt[3]])/c^(10/3) - (28*b^(10/3)*x^7*Log[a^(1/3) + b^(1/3)*x])/a^(10/3) + (28*d
^(10/3)*x^7*Log[c^(1/3) + d^(1/3)*x])/c^(10/3) + (14*b^(10/3)*x^7*Log[a^(2/3) -
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(10/3) - (14*d^(10/3)*x^7*Log[c^(2/3) - c^(1
/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(10/3))/(84*(-(b*c) + a*d)*x^7)

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Maple [A]  time = 0.018, size = 334, normalized size = 1. \[ -{\frac{1}{7\,ac{x}^{7}}}+{\frac{d}{4\,a{x}^{4}{c}^{2}}}+{\frac{b}{4\,{x}^{4}{a}^{2}c}}-{\frac{{d}^{2}}{a{c}^{3}x}}-{\frac{bd}{{a}^{2}{c}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}cx}}-{\frac{{b}^{3}}{3\,{a}^{3} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{b}^{3}}{6\,{a}^{3} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{b}^{3}\sqrt{3}}{3\,{a}^{3} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{d}^{3}}{3\,{c}^{3} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{{d}^{3}}{6\,{c}^{3} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{{d}^{3}\sqrt{3}}{3\,{c}^{3} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(b*x^3+a)/(d*x^3+c),x)

[Out]

-1/7/a/c/x^7+1/4/x^4/a/c^2*d+1/4/x^4/a^2/c*b-1/a/c^3/x*d^2-1/a^2/c^2/x*b*d-1/a^3
/c/x*b^2-1/3*b^3/a^3/(a*d-b*c)/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6*b^3/a^3/(a*d-b*
c)/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*b^3/a^3/(a*d-b*c)*3^(1/2)/(
a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*d^3/c^3/(a*d-b*c)/(c/d)^(
1/3)*ln(x+(c/d)^(1/3))-1/6*d^3/c^3/(a*d-b*c)/(c/d)^(1/3)*ln(x^2-x*(c/d)^(1/3)+(c
/d)^(2/3))-1/3*d^3/c^3/(a*d-b*c)*3^(1/2)/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)
^(1/3)*x-1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.41818, size = 508, normalized size = 1.44 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3} b^{3} c^{3} x^{7} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 14 \, \sqrt{3} a^{3} d^{3} x^{7} \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (\frac{d}{c}\right )^{\frac{2}{3}} + c \left (\frac{d}{c}\right )^{\frac{1}{3}}\right ) - 28 \, \sqrt{3} b^{3} c^{3} x^{7} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 28 \, \sqrt{3} a^{3} d^{3} x^{7} \left (\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d}{c}\right )^{\frac{2}{3}}\right ) - 84 \, b^{3} c^{3} x^{7} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 84 \, a^{3} d^{3} x^{7} \left (\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d}{c}\right )^{\frac{2}{3}}}{3 \, c \left (\frac{d}{c}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (28 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{6} + 4 \, a^{2} b c^{3} - 4 \, a^{3} c^{2} d - 7 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{3}\right )}\right )}}{252 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^8),x, algorithm="fricas")

[Out]

1/252*sqrt(3)*(14*sqrt(3)*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3)
- a*(-b/a)^(1/3)) + 14*sqrt(3)*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x^2 - c*x*(d/c)^(2/
3) + c*(d/c)^(1/3)) - 28*sqrt(3)*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/
3)) - 28*sqrt(3)*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x + c*(d/c)^(2/3)) - 84*b^3*c^3*x
^7*(-b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(-b/a)^(2/3))/(a*(-b/a)^(
2/3))) - 84*a^3*d^3*x^7*(d/c)^(1/3)*arctan(-1/3*(2*sqrt(3)*d*x - sqrt(3)*c*(d/c)
^(2/3))/(c*(d/c)^(2/3))) - 3*sqrt(3)*(28*(b^3*c^3 - a^3*d^3)*x^6 + 4*a^2*b*c^3 -
4*a^3*c^2*d - 7*(a*b^2*c^3 - a^3*c*d^2)*x^3))/((a^3*b*c^4 - a^4*c^3*d)*x^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231721, size = 509, normalized size = 1.45 \[ \frac{b^{4} \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{4} b c - a^{5} d\right )}} - \frac{d^{4} \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{5} - a c^{4} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{4} b c - \sqrt{3} a^{5} d} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} d^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{5} - \sqrt{3} a c^{4} d} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} b^{2}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{4} b c - a^{5} d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} d^{2}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{5} - a c^{4} d\right )}} - \frac{28 \, b^{2} c^{2} x^{6} + 28 \, a b c d x^{6} + 28 \, a^{2} d^{2} x^{6} - 7 \, a b c^{2} x^{3} - 7 \, a^{2} c d x^{3} + 4 \, a^{2} c^{2}}{28 \, a^{3} c^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^8),x, algorithm="giac")

[Out]

1/3*b^4*(-a/b)^(2/3)*ln(abs(x - (-a/b)^(1/3)))/(a^4*b*c - a^5*d) - 1/3*d^4*(-c/d
)^(2/3)*ln(abs(x - (-c/d)^(1/3)))/(b*c^5 - a*c^4*d) + (-a*b^2)^(2/3)*b^2*arctan(
1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^4*b*c - sqrt(3)*a^5*d)
- (-c*d^2)^(2/3)*d^2*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqr
t(3)*b*c^5 - sqrt(3)*a*c^4*d) - 1/6*(-a*b^2)^(2/3)*b^2*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a^4*b*c - a^5*d) + 1/6*(-c*d^2)^(2/3)*d^2*ln(x^2 + x*(-c/d)^(1/3
) + (-c/d)^(2/3))/(b*c^5 - a*c^4*d) - 1/28*(28*b^2*c^2*x^6 + 28*a*b*c*d*x^6 + 28
*a^2*d^2*x^6 - 7*a*b*c^2*x^3 - 7*a^2*c*d*x^3 + 4*a^2*c^2)/(a^3*c^3*x^7)