3.108 \(\int \frac{x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=301 \[ \frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.728367, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 114.541, size = 269, normalized size = 0.89 \[ \frac{a^{\frac{5}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{5}{3}} \left (a d - b c\right )} - \frac{a^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{5}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} a^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}} \left (a d - b c\right )} - \frac{c^{\frac{5}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 d^{\frac{5}{3}} \left (a d - b c\right )} + \frac{c^{\frac{5}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{5}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} c^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 d^{\frac{5}{3}} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**3+a)/(d*x**3+c),x)

[Out]

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

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Mathematica [A]  time = 0.278244, size = 242, normalized size = 0.8 \[ \frac{\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-\frac{2 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}-\frac{2 \sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{5/3}}-\frac{3 a x^2}{b}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{5/3}}+\frac{2 c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{5/3}}+\frac{2 \sqrt{3} c^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{5/3}}+\frac{3 c x^2}{d}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Maple [A]  time = 0.011, size = 269, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,bd}}+{\frac{{a}^{2}}{ \left ( 3\,ad-3\,bc \right ){b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}}{ \left ( 6\,ad-6\,bc \right ){b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ){b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}}{ \left ( 3\,ad-3\,bc \right ){d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{c}^{2}}{ \left ( 6\,ad-6\,bc \right ){d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{c}^{2}\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ){d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^3+a)/(d*x^3+c),x)

[Out]

1/2*x^2/b/d+1/3*a^2/(a*d-b*c)/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/6*a^2/(a*d-b*c
)/b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3*a^2/(a*d-b*c)/b^2*3^(1/2
)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3*c^2/(a*d-b*c)/d^2/(c/d
)^(1/3)*ln(x+(c/d)^(1/3))+1/6*c^2/(a*d-b*c)/d^2/(c/d)^(1/3)*ln(x^2-x*(c/d)^(1/3)
+(c/d)^(2/3))+1/3*c^2/(a*d-b*c)/d^2*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(x^7/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.486651, size = 412, normalized size = 1.37 \[ \frac{\sqrt{3}{\left (\sqrt{3} a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + \sqrt{3} b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x^{2} - d x \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}} - c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x + d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}\right ) + 3 \, \sqrt{3}{\left (b c - a d\right )} x^{2} - 6 \, a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) - 6 \, b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} c x - \sqrt{3} d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}}{3 \, d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \,{\left (b^{2} c d - a b d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(sqrt(3)*a*d*(a^2/b^2)^(1/3)*log(a*x^2 - b*x*(a^2/b^2)^(2/3) + a*(a
^2/b^2)^(1/3)) + sqrt(3)*b*c*(-c^2/d^2)^(1/3)*log(c*x^2 - d*x*(-c^2/d^2)^(2/3) -
c*(-c^2/d^2)^(1/3)) - 2*sqrt(3)*a*d*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3)
) - 2*sqrt(3)*b*c*(-c^2/d^2)^(1/3)*log(c*x + d*(-c^2/d^2)^(2/3)) + 3*sqrt(3)*(b*
c - a*d)*x^2 - 6*a*d*(a^2/b^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*a*x - sqrt(3)*b*(a^2
/b^2)^(2/3))/(b*(a^2/b^2)^(2/3))) - 6*b*c*(-c^2/d^2)^(1/3)*arctan(-1/3*(2*sqrt(3
)*c*x - sqrt(3)*d*(-c^2/d^2)^(2/3))/(d*(-c^2/d^2)^(2/3))))/(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**7/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228266, size = 436, normalized size = 1.45 \[ \frac{\left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b^{5} c - a^{2} b^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} c^{2} d{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{2} d^{4} - a c d^{5}\right )}} - \frac{a^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{2} c - a^{2} b d\right )}} + \frac{c^{2} \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} d - a c d^{2}\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} a \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{4} c - \sqrt{3} a b^{3} d} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} c \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{3} - \sqrt{3} a d^{4}} + \frac{x^{2}}{2 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")

[Out]

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