3.689 \(\int \frac{1}{x^{5/3} (a+b x)^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.115408, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/3)*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.4528, size = 126, normalized size = 0.98 \[ \frac{1}{a x^{\frac{2}{3}} \left (a + b x\right )} - \frac{5}{2 a^{2} x^{\frac{2}{3}}} - \frac{5 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{8}{3}}} + \frac{5 b^{\frac{2}{3}} \log{\left (a + b x \right )}}{6 a^{\frac{8}{3}}} + \frac{5 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/3)/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.174068, size = 147, normalized size = 1.15 \[ \frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-\frac{6 a^{2/3} b \sqrt [3]{x}}{a+b x}-\frac{9 a^{2/3}}{x^{2/3}}-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+10 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/3)*(a + b*x)^2),x]

[Out]

((-9*a^(2/3))/x^(2/3) - (6*a^(2/3)*b*x^(1/3))/(a + b*x) + 10*Sqrt[3]*b^(2/3)*Arc
Tan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] - 10*b^(2/3)*Log[a^(1/3) + b^(1/3
)*x^(1/3)] + 5*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])
/(6*a^(8/3))

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Maple [A]  time = 0.02, size = 121, normalized size = 1. \[ -{\frac{3}{2\,{a}^{2}}{x}^{-{\frac{2}{3}}}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }\sqrt [3]{x}}-{\frac{5}{3\,{a}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5}{6\,{a}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,\sqrt{3}}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/3)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232116, size = 270, normalized size = 2.11 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b x + a\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3}{\left (b x + a\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{\frac{1}{3}} - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left (b x + a\right )} x^{\frac{2}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x^{\frac{1}{3}} + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (5 \, b x + 3 \, a\right )}\right )}}{18 \,{\left (a^{2} b x + a^{3}\right )} x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^(5/3)),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(5*sqrt(3)*(b*x + a)*x^(2/3)*(-b^2/a^2)^(1/3)*log(b^2*x^(2/3) + a*
b*x^(1/3)*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 10*sqrt(3)*(b*x + a)*x^(2/3
)*(-b^2/a^2)^(1/3)*log(b*x^(1/3) - a*(-b^2/a^2)^(1/3)) + 30*(b*x + a)*x^(2/3)*(-
b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x^(1/3) + sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*
(-b^2/a^2)^(1/3))) + 3*sqrt(3)*(5*b*x + 3*a))/((a^2*b*x + a^3)*x^(2/3))

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Sympy [A]  time = 6.08757, size = 615, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/3)/(b*x+a)**2,x)

[Out]

9*a**(5/3)*gamma(-2/3)/(9*a**(11/3)*x**(2/3)*gamma(1/3) + 9*a**(8/3)*b*x**(5/3)*
gamma(1/3)) + 15*a**(2/3)*b*x*gamma(-2/3)/(9*a**(11/3)*x**(2/3)*gamma(1/3) + 9*a
**(8/3)*b*x**(5/3)*gamma(1/3)) - 10*a*b**(2/3)*x**(2/3)*exp(5*I*pi/3)*log(1 - b*
*(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-2/3)/(9*a**(11/3)*x**(2/3)*ga
mma(1/3) + 9*a**(8/3)*b*x**(5/3)*gamma(1/3)) + 10*a*b**(2/3)*x**(2/3)*log(1 - b*
*(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-2/3)/(9*a**(11/3)*x**(2/3)*gamm
a(1/3) + 9*a**(8/3)*b*x**(5/3)*gamma(1/3)) - 10*a*b**(2/3)*x**(2/3)*exp(I*pi/3)*
log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-2/3)/(9*a**(11/3)
*x**(2/3)*gamma(1/3) + 9*a**(8/3)*b*x**(5/3)*gamma(1/3)) - 10*b**(5/3)*x**(5/3)*
exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-2/3)/
(9*a**(11/3)*x**(2/3)*gamma(1/3) + 9*a**(8/3)*b*x**(5/3)*gamma(1/3)) + 10*b**(5/
3)*x**(5/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-2/3)/(9*a
**(11/3)*x**(2/3)*gamma(1/3) + 9*a**(8/3)*b*x**(5/3)*gamma(1/3)) - 10*b**(5/3)*x
**(5/3)*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamm
a(-2/3)/(9*a**(11/3)*x**(2/3)*gamma(1/3) + 9*a**(8/3)*b*x**(5/3)*gamma(1/3))

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GIAC/XCAS [A]  time = 0.218071, size = 185, normalized size = 1.45 \[ \frac{5 \, b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} - \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} - \frac{b x^{\frac{1}{3}}}{{\left (b x + a\right )} a^{2}} - \frac{5 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} - \frac{3}{2 \, a^{2} x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^(5/3)),x, algorithm="giac")

[Out]

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