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

Optimal. Leaf size=155 \[ \frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}}-\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{8/3}}-\frac{5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac{x^5}{6 b \left (a+b x^3\right )^2} \]

[Out]

-x^5/(6*b*(a + b*x^3)^2) - (5*x^2)/(18*b^2*(a + b*x^3)) - (5*ArcTan[(a^(1/3) - 2
*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(8/3)) - (5*Log[a^(1/3) + b
^(1/3)*x])/(27*a^(1/3)*b^(8/3)) + (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x
^2])/(54*a^(1/3)*b^(8/3))

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Rubi [A]  time = 0.191731, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}}-\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{8/3}}-\frac{5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac{x^5}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^3)^3,x]

[Out]

-x^5/(6*b*(a + b*x^3)^2) - (5*x^2)/(18*b^2*(a + b*x^3)) - (5*ArcTan[(a^(1/3) - 2
*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(8/3)) - (5*Log[a^(1/3) + b
^(1/3)*x])/(27*a^(1/3)*b^(8/3)) + (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x
^2])/(54*a^(1/3)*b^(8/3))

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Rubi in Sympy [A]  time = 35.8815, size = 146, normalized size = 0.94 \[ - \frac{x^{5}}{6 b \left (a + b x^{3}\right )^{2}} - \frac{5 x^{2}}{18 b^{2} \left (a + b x^{3}\right )} - \frac{5 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 \sqrt [3]{a} b^{\frac{8}{3}}} + \frac{5 \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 \sqrt [3]{a} b^{\frac{8}{3}}} - \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 \sqrt [3]{a} b^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**5/(6*b*(a + b*x**3)**2) - 5*x**2/(18*b**2*(a + b*x**3)) - 5*log(a**(1/3) + b
**(1/3)*x)/(27*a**(1/3)*b**(8/3)) + 5*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2
/3)*x**2)/(54*a**(1/3)*b**(8/3)) - 5*sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/
3)*x/3)/a**(1/3))/(27*a**(1/3)*b**(8/3))

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Mathematica [A]  time = 0.160078, size = 140, normalized size = 0.9 \[ \frac{\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{24 b^{2/3} x^2}{a+b x^3}+\frac{9 a b^{2/3} x^2}{\left (a+b x^3\right )^2}-\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}}{54 b^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^3)^3,x]

[Out]

((9*a*b^(2/3)*x^2)/(a + b*x^3)^2 - (24*b^(2/3)*x^2)/(a + b*x^3) - (10*Sqrt[3]*Ar
cTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) - (10*Log[a^(1/3) + b^(1/3)*x
])/a^(1/3) + (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(54*b^(
8/3))

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Maple [A]  time = 0.013, size = 119, normalized size = 0.8 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ( -{\frac{4\,{x}^{5}}{9\,b}}-{\frac{5\,a{x}^{2}}{18\,{b}^{2}}} \right ) }-{\frac{5}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5}{54\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{3}}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^3+a)^3,x)

[Out]

(-4/9*x^5/b-5/18*a*x^2/b^2)/(b*x^3+a)^2-5/27/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+5
/54/b^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/27/b^3*3^(1/2)/(a/b)^(1/
3)*arctan(1/3*3^(1/2)*(2/(a/b)^(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22308, size = 278, normalized size = 1.79 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 10 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 30 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (8 \, b x^{5} + 5 \, a x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(5*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((-a*b^2)^(1/3)*b*x^2 -
a*b + (-a*b^2)^(2/3)*x) - 10*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log(a*b + (-a*
b^2)^(2/3)*x) + 30*(b^2*x^6 + 2*a*b*x^3 + a^2)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt
(3)*(-a*b^2)^(2/3)*x)/(a*b)) + 3*sqrt(3)*(8*b*x^5 + 5*a*x^2)*(-a*b^2)^(1/3))/((b
^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)*(-a*b^2)^(1/3))

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Sympy [A]  time = 2.5454, size = 68, normalized size = 0.44 \[ - \frac{5 a x^{2} + 8 b x^{5}}{18 a^{2} b^{2} + 36 a b^{3} x^{3} + 18 b^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a b^{8} + 125, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a b^{5}}{25} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**3+a)**3,x)

[Out]

-(5*a*x**2 + 8*b*x**5)/(18*a**2*b**2 + 36*a*b**3*x**3 + 18*b**4*x**6) + RootSum(
19683*_t**3*a*b**8 + 125, Lambda(_t, _t*log(729*_t**2*a*b**5/25 + x)))

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GIAC/XCAS [A]  time = 0.237299, size = 192, normalized size = 1.24 \[ -\frac{5 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{2}} - \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{4}} - \frac{8 \, b x^{5} + 5 \, a x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} b^{2}} + \frac{5 \, \left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/27*(-a/b)^(2/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^2) - 5/27*sqrt(3)*(-a*b^2)^(2/
3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) - 1/18*(8*b*x^5
+ 5*a*x^2)/((b*x^3 + a)^2*b^2) + 5/54*(-a*b^2)^(2/3)*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a*b^4)