3.1317 \(\int \frac{x^7}{a+b x^6} \, dx\)

Optimal. Leaf size=133 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} b^{4/3}}+\frac{x^2}{2 b} \]

[Out]

x^2/(2*b) + (a^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(2*Sqr
t[3]*b^(4/3)) - (a^(1/3)*Log[a^(1/3) + b^(1/3)*x^2])/(6*b^(4/3)) + (a^(1/3)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])/(12*b^(4/3))

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Rubi [A]  time = 0.228228, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} b^{4/3}}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^6),x]

[Out]

x^2/(2*b) + (a^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(2*Sqr
t[3]*b^(4/3)) - (a^(1/3)*Log[a^(1/3) + b^(1/3)*x^2])/(6*b^(4/3)) + (a^(1/3)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])/(12*b^(4/3))

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Rubi in Sympy [A]  time = 34.4201, size = 121, normalized size = 0.91 \[ - \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{2} \right )}}{6 b^{\frac{4}{3}}} + \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{2} + b^{\frac{2}{3}} x^{4} \right )}}{12 b^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{2}}{3}\right )}{\sqrt [3]{a}} \right )}}{6 b^{\frac{4}{3}}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**6+a),x)

[Out]

-a**(1/3)*log(a**(1/3) + b**(1/3)*x**2)/(6*b**(4/3)) + a**(1/3)*log(a**(2/3) - a
**(1/3)*b**(1/3)*x**2 + b**(2/3)*x**4)/(12*b**(4/3)) + sqrt(3)*a**(1/3)*atan(sqr
t(3)*(a**(1/3)/3 - 2*b**(1/3)*x**2/3)/a**(1/3))/(6*b**(4/3)) + x**2/(2*b)

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Mathematica [A]  time = 0.0920674, size = 186, normalized size = 1.4 \[ \frac{-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt [3]{a} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )+6 \sqrt [3]{b} x^2}{12 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^6),x]

[Out]

(6*b^(1/3)*x^2 + 2*Sqrt[3]*a^(1/3)*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*S
qrt[3]*a^(1/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] - 2*a^(1/3)*Log[a^(1/3) +
b^(1/3)*x^2] + a^(1/3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] +
a^(1/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*b^(4/3))

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Maple [A]  time = 0.004, size = 108, normalized size = 0.8 \[{\frac{{x}^{2}}{2\,b}}-{\frac{a}{6\,{b}^{2}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{12\,{b}^{2}}\ln \left ({x}^{4}-{x}^{2}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}}{6\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^6+a),x)

[Out]

1/2*x^2/b-1/6*a/b^2/(a/b)^(2/3)*ln(x^2+(a/b)^(1/3))+1/12*a/b^2/(a/b)^(2/3)*ln(x^
4-x^2*(a/b)^(1/3)+(a/b)^(2/3))-1/6*a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*
(2/(a/b)^(1/3)*x^2-1))

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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^6 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223302, size = 169, normalized size = 1.27 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} x^{2} - \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x^{2} + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )\right )}}{36 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^6 + a),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(6*sqrt(3)*x^2 - sqrt(3)*(-a/b)^(1/3)*log(x^4 + x^2*(-a/b)^(1/3) +
(-a/b)^(2/3)) + 2*sqrt(3)*(-a/b)^(1/3)*log(x^2 - (-a/b)^(1/3)) - 6*(-a/b)^(1/3)*
arctan(1/3*(2*sqrt(3)*x^2 + sqrt(3)*(-a/b)^(1/3))/(-a/b)^(1/3)))/b

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Sympy [A]  time = 1.49776, size = 27, normalized size = 0.2 \[ \operatorname{RootSum}{\left (216 t^{3} b^{4} + a, \left ( t \mapsto t \log{\left (- 6 t b + x^{2} \right )} \right )\right )} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**6+a),x)

[Out]

RootSum(216*_t**3*b**4 + a, Lambda(_t, _t*log(-6*_t*b + x**2))) + x**2/(2*b)

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GIAC/XCAS [A]  time = 0.227445, size = 162, normalized size = 1.22 \[ \frac{x^{2}}{2 \, b} + \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{6 \, b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{6 \, b^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{12 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^6 + a),x, algorithm="giac")

[Out]

1/2*x^2/b + 1/6*(-a/b)^(1/3)*ln(abs(x^2 - (-a/b)^(1/3)))/b - 1/6*sqrt(3)*(-a*b^2
)^(1/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/b^2 - 1/12*(-a*b
^2)^(1/3)*ln(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/b^2