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

Optimal. Leaf size=296 \[ -\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{4/3} (b c-a d)}+\frac{x}{b d} \]

[Out]

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

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Rubi [A]  time = 0.643533, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{4/3} (b c-a d)}+\frac{x}{b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 96.8317, size = 265, normalized size = 0.9 \[ - \frac{a^{\frac{4}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{4}{3}} \left (a d - b c\right )} + \frac{a^{\frac{4}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{4}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} a^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{4}{3}} \left (a d - b c\right )} + \frac{c^{\frac{4}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 d^{\frac{4}{3}} \left (a d - b c\right )} - \frac{c^{\frac{4}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{4}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} c^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 d^{\frac{4}{3}} \left (a d - b c\right )} + \frac{x}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(4/3)*log(a**(1/3) + b**(1/3)*x)/(3*b**(4/3)*(a*d - b*c)) + a**(4/3)*log(a**
(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(4/3)*(a*d - b*c)) + sqrt(3)*
a**(4/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*b**(4/3)*(a*d -
b*c)) + c**(4/3)*log(c**(1/3) + d**(1/3)*x)/(3*d**(4/3)*(a*d - b*c)) - c**(4/3)
*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*d**(4/3)*(a*d - b*c)) -
sqrt(3)*c**(4/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*d**(4/3
)*(a*d - b*c)) + x/(b*d)

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Mathematica [A]  time = 0.253131, size = 238, normalized size = 0.8 \[ \frac{-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{4/3}}+\frac{2 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}-\frac{2 \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{4/3}}-\frac{6 a x}{b}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{4/3}}-\frac{2 c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{4/3}}+\frac{2 \sqrt{3} c^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{4/3}}+\frac{6 c x}{d}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 266, normalized size = 0.9 \[{\frac{x}{bd}}-{\frac{{a}^{2}}{3\,{b}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}}{6\,{b}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}\sqrt{3}}{3\,{b}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{c}^{2}}{3\,{d}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{c}^{2}}{6\,{d}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{c}^{2}\sqrt{3}}{3\,{d}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263067, size = 327, normalized size = 1.1 \[ \frac{\sqrt{3}{\left (\sqrt{3} a d \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) + \sqrt{3} b c \left (\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{c}{d}\right )^{\frac{1}{3}} + \left (\frac{c}{d}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} a d \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} b c \left (\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{c}{d}\right )^{\frac{1}{3}}\right ) + 6 \, a d \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 6 \, b c \left (\frac{c}{d}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{c}{d}\right )^{\frac{1}{3}}}{3 \, \left (\frac{c}{d}\right )^{\frac{1}{3}}}\right ) + 6 \, \sqrt{3}{\left (b c - a d\right )} x\right )}}{18 \,{\left (b^{2} c d - a b d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(sqrt(3)*a*d*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))
+ sqrt(3)*b*c*(c/d)^(1/3)*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3)) - 2*sqrt(3)*a*d
*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) - 2*sqrt(3)*b*c*(c/d)^(1/3)*log(x + (c/d)^(1
/3)) + 6*a*d*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*x + sqrt(3)*(-a/b)^(1/3))/(-a/b)
^(1/3)) + 6*b*c*(c/d)^(1/3)*arctan(-1/3*(2*sqrt(3)*x - sqrt(3)*(c/d)^(1/3))/(c/d
)^(1/3)) + 6*sqrt(3)*(b*c - a*d)*x)/(b^2*c*d - a*b*d^2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228377, size = 416, normalized size = 1.41 \[ -\frac{a^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{2} c - a^{2} b d\right )}} + \frac{c^{2} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} d - a c d^{2}\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} a \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{3} c - \sqrt{3} a b^{2} d} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} c \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{2} - \sqrt{3} a d^{3}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} a{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{3} c - a b^{2} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} c{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c d^{2} - a d^{3}\right )}} + \frac{x}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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