3.121 \(\int \frac{1}{x^6 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=321 \[ -\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5} \]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(2*a^2*c^2*x^2) - (b^(8/3)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(8/3)*(b*c - a*d)) + (d^(8/3)*ArcTan[(c^(1
/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(8/3)*(b*c - a*d)) + (b^(8/3)*
Log[a^(1/3) + b^(1/3)*x])/(3*a^(8/3)*(b*c - a*d)) - (d^(8/3)*Log[c^(1/3) + d^(1/
3)*x])/(3*c^(8/3)*(b*c - a*d)) - (b^(8/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/(6*a^(8/3)*(b*c - a*d)) + (d^(8/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d
^(2/3)*x^2])/(6*c^(8/3)*(b*c - a*d))

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Rubi [A]  time = 1.10016, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac{b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac{b^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}+\frac{d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac{d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac{d^{8/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{8/3} (b c-a d)}-\frac{1}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(2*a^2*c^2*x^2) - (b^(8/3)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(8/3)*(b*c - a*d)) + (d^(8/3)*ArcTan[(c^(1
/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(8/3)*(b*c - a*d)) + (b^(8/3)*
Log[a^(1/3) + b^(1/3)*x])/(3*a^(8/3)*(b*c - a*d)) - (d^(8/3)*Log[c^(1/3) + d^(1/
3)*x])/(3*c^(8/3)*(b*c - a*d)) - (b^(8/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/(6*a^(8/3)*(b*c - a*d)) + (d^(8/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d
^(2/3)*x^2])/(6*c^(8/3)*(b*c - a*d))

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Rubi in Sympy [A]  time = 157.701, size = 289, normalized size = 0.9 \[ \frac{d^{\frac{8}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{8}{3}} \left (a d - b c\right )} - \frac{d^{\frac{8}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{8}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} d^{\frac{8}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{8}{3}} \left (a d - b c\right )} - \frac{1}{5 a c x^{5}} + \frac{a d + b c}{2 a^{2} c^{2} x^{2}} - \frac{b^{\frac{8}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{8}{3}} \left (a d - b c\right )} + \frac{b^{\frac{8}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{8}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} b^{\frac{8}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**3+a)/(d*x**3+c),x)

[Out]

d**(8/3)*log(c**(1/3) + d**(1/3)*x)/(3*c**(8/3)*(a*d - b*c)) - d**(8/3)*log(c**(
2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(8/3)*(a*d - b*c)) - sqrt(3)*d
**(8/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(8/3)*(a*d -
b*c)) - 1/(5*a*c*x**5) + (a*d + b*c)/(2*a**2*c**2*x**2) - b**(8/3)*log(a**(1/3)
+ b**(1/3)*x)/(3*a**(8/3)*(a*d - b*c)) + b**(8/3)*log(a**(2/3) - a**(1/3)*b**(1/
3)*x + b**(2/3)*x**2)/(6*a**(8/3)*(a*d - b*c)) + sqrt(3)*b**(8/3)*atan(sqrt(3)*(
a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(8/3)*(a*d - b*c))

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Mathematica [A]  time = 0.417279, size = 282, normalized size = 0.88 \[ \frac{-\frac{10 b^{8/3} x^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}+\frac{10 \sqrt{3} b^{8/3} x^5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{8/3}}+\frac{5 b^{8/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}-\frac{15 b^2 x^3}{a^2}+\frac{6 b}{a}+\frac{10 d^{8/3} x^5 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{8/3}}-\frac{10 \sqrt{3} d^{8/3} x^5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{8/3}}-\frac{5 d^{8/3} x^5 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{8/3}}+\frac{15 d^2 x^3}{c^2}-\frac{6 d}{c}}{30 x^5 (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((6*b)/a - (6*d)/c - (15*b^2*x^3)/a^2 + (15*d^2*x^3)/c^2 + (10*Sqrt[3]*b^(8/3)*x
^5*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) - (10*Sqrt[3]*d^(8/3)*x^
5*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(8/3) - (10*b^(8/3)*x^5*Log[a^(
1/3) + b^(1/3)*x])/a^(8/3) + (10*d^(8/3)*x^5*Log[c^(1/3) + d^(1/3)*x])/c^(8/3) +
(5*b^(8/3)*x^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3) - (5*d^(
8/3)*x^5*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(8/3))/(30*(-(b*c) +
a*d)*x^5)

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Maple [A]  time = 0.017, size = 293, normalized size = 0.9 \[ -{\frac{1}{5\,ac{x}^{5}}}+{\frac{d}{2\,a{c}^{2}{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}}{6\,{a}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{d}^{2}}{3\,{c}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{d}^{2}}{6\,{c}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{d}^{2}\sqrt{3}}{3\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^3+a)/(d*x^3+c),x)

[Out]

-1/5/a/c/x^5+1/2/a/c^2/x^2*d+1/2/a^2/c/x^2*b-1/3/a^2*b^2/(a*d-b*c)/(a/b)^(2/3)*l
n(x+(a/b)^(1/3))+1/6/a^2*b^2/(a*d-b*c)/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2
/3))-1/3/a^2*b^2/(a*d-b*c)/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)
*x-1))+1/3/c^2*d^2/(a*d-b*c)/(c/d)^(2/3)*ln(x+(c/d)^(1/3))-1/6/c^2*d^2/(a*d-b*c)
/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))+1/3/c^2*d^2/(a*d-b*c)/(c/d)^(2/3)
*3^(1/2)*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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.35578, size = 525, normalized size = 1.64 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3} b^{2} c^{2} x^{5} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + 5 \, \sqrt{3} a^{2} d^{2} x^{5} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3} b^{2} c^{2} x^{5} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3} a^{2} d^{2} x^{5} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 30 \, b^{2} c^{2} x^{5} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 30 \, a^{2} d^{2} x^{5} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (2 \, a b c^{2} - 2 \, a^{2} c d - 5 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}\right )}\right )}}{90 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^6),x, algorithm="fricas")

[Out]

1/90*sqrt(3)*(5*sqrt(3)*b^2*c^2*x^5*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a
^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) + 5*sqrt(3)*a^2*d^2*x^5*(d^2/c^2)^(1/3)*log(d^
2*x^2 - c*d*x*(d^2/c^2)^(1/3) + c^2*(d^2/c^2)^(2/3)) - 10*sqrt(3)*b^2*c^2*x^5*(-
b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)) - 10*sqrt(3)*a^2*d^2*x^5*(d^2/c^2)^
(1/3)*log(d*x + c*(d^2/c^2)^(1/3)) + 30*b^2*c^2*x^5*(-b^2/a^2)^(1/3)*arctan(1/3*
(2*sqrt(3)*b*x + sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 30*a^2*d^2*
x^5*(d^2/c^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*d*x - sqrt(3)*c*(d^2/c^2)^(1/3))/(c*(
d^2/c^2)^(1/3))) - 3*sqrt(3)*(2*a*b*c^2 - 2*a^2*c*d - 5*(b^2*c^2 - a^2*d^2)*x^3)
)/((a^2*b*c^3 - a^3*c^2*d)*x^5)

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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**6/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227652, size = 454, normalized size = 1.41 \[ -\frac{b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{3} b c - \sqrt{3} a^{4} d} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{4} - \sqrt{3} a c^{3} d} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b^{2}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{3} b c - a^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d^{2}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{5 \, b c x^{3} + 5 \, a d x^{3} - 2 \, a c}{10 \, a^{2} c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^6),x, algorithm="giac")

[Out]

-1/3*b^3*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^3*b*c - a^4*d) + 1/3*d^3*(-c/
d)^(1/3)*ln(abs(x - (-c/d)^(1/3)))/(b*c^4 - a*c^3*d) + (-a*b^2)^(1/3)*b^2*arctan
(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^3*b*c - sqrt(3)*a^4*d
) - (-c*d^2)^(1/3)*d^2*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sq
rt(3)*b*c^4 - sqrt(3)*a*c^3*d) + 1/6*(-a*b^2)^(1/3)*b^2*ln(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a^3*b*c - a^4*d) - 1/6*(-c*d^2)^(1/3)*d^2*ln(x^2 + x*(-c/d)^(1/
3) + (-c/d)^(2/3))/(b*c^4 - a*c^3*d) + 1/10*(5*b*c*x^3 + 5*a*d*x^3 - 2*a*c)/(a^2
*c^2*x^5)