3.418 \(\int \frac{1}{x^2 (a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=115 \[ \frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac{4 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{4 (a+b x)^{2/3}}{a^2 x}+\frac{3}{a x \sqrt [3]{a+b x}} \]

[Out]

3/(a*x*(a + b*x)^(1/3)) - (4*(a + b*x)^(2/3))/(a^2*x) - (4*b*ArcTan[(a^(1/3) + 2
*(a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) + (2*b*Log[x])/(3*a^(7/3
)) - (2*b*Log[a^(1/3) - (a + b*x)^(1/3)])/a^(7/3)

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Rubi [A]  time = 0.108373, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac{4 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{4 (a+b x)^{2/3}}{a^2 x}+\frac{3}{a x \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^(4/3)),x]

[Out]

3/(a*x*(a + b*x)^(1/3)) - (4*(a + b*x)^(2/3))/(a^2*x) - (4*b*ArcTan[(a^(1/3) + 2
*(a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) + (2*b*Log[x])/(3*a^(7/3
)) - (2*b*Log[a^(1/3) - (a + b*x)^(1/3)])/a^(7/3)

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Rubi in Sympy [A]  time = 11.0953, size = 110, normalized size = 0.96 \[ \frac{3}{a x \sqrt [3]{a + b x}} - \frac{4 \left (a + b x\right )^{\frac{2}{3}}}{a^{2} x} + \frac{2 b \log{\left (x \right )}}{3 a^{\frac{7}{3}}} - \frac{2 b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{a^{\frac{7}{3}}} - \frac{4 \sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)**(4/3),x)

[Out]

3/(a*x*(a + b*x)**(1/3)) - 4*(a + b*x)**(2/3)/(a**2*x) + 2*b*log(x)/(3*a**(7/3))
- 2*b*log(a**(1/3) - (a + b*x)**(1/3))/a**(7/3) - 4*sqrt(3)*b*atan(sqrt(3)*(a**
(1/3)/3 + 2*(a + b*x)**(1/3)/3)/a**(1/3))/(3*a**(7/3))

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Mathematica [C]  time = 0.0387122, size = 61, normalized size = 0.53 \[ \frac{4 b x \sqrt [3]{\frac{a}{b x}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x}\right )-a-4 b x}{a^2 x \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)^(4/3)),x]

[Out]

(-a - 4*b*x + 4*b*(1 + a/(b*x))^(1/3)*x*Hypergeometric2F1[1/3, 1/3, 4/3, -(a/(b*
x))])/(a^2*x*(a + b*x)^(1/3))

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Maple [A]  time = 0.019, size = 108, normalized size = 0.9 \[ -3\,{\frac{b}{{a}^{2}\sqrt [3]{bx+a}}}-{\frac{1}{{a}^{2}x} \left ( bx+a \right ) ^{{\frac{2}{3}}}}-{\frac{4\,b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{7}{3}}}}+{\frac{2\,b}{3}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{7}{3}}}}-{\frac{4\,b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(4/3),x)

[Out]

-3*b/a^2/(b*x+a)^(1/3)-(b*x+a)^(2/3)/a^2/x-4/3*b/a^(7/3)*ln((b*x+a)^(1/3)-a^(1/3
))+2/3*b/a^(7/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a^(1/3)+a^(2/3))-4/3*b/a^(7/3)*3
^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(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)^(4/3)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223195, size = 213, normalized size = 1.85 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 4 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 12 \,{\left (b x + a\right )}^{\frac{1}{3}} b x \arctan \left (\frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (4 \, b x + a\right )} \left (-a\right )^{\frac{1}{3}}\right )}}{9 \,{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*x^2),x, algorithm="fricas")

[Out]

-1/9*sqrt(3)*(2*sqrt(3)*(b*x + a)^(1/3)*b*x*log((b*x + a)^(2/3)*(-a)^(1/3) - (b*
x + a)^(1/3)*(-a)^(2/3) - a) - 4*sqrt(3)*(b*x + a)^(1/3)*b*x*log((b*x + a)^(1/3)
*(-a)^(2/3) - a) - 12*(b*x + a)^(1/3)*b*x*arctan(1/3*(2*sqrt(3)*(b*x + a)^(1/3)*
(-a)^(2/3) + sqrt(3)*a)/a) + 3*sqrt(3)*(4*b*x + a)*(-a)^(1/3))/((b*x + a)^(1/3)*
(-a)^(1/3)*a^2*x)

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Sympy [A]  time = 7.67848, size = 704, normalized size = 6.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a)**(4/3),x)

[Out]

-9*a**(4/3)*b**(2/3)*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1/3)*gamma(2/3) + 9*a
**(7/3)*b*(a/b + x)**(4/3)*gamma(2/3)) + 12*a**(1/3)*b**(5/3)*(a/b + x)*gamma(-1
/3)/(-9*a**(10/3)*(a/b + x)**(1/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*ga
mma(2/3)) - 4*a*b*(a/b + x)**(1/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*g
amma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(
4/3)*gamma(2/3)) - 4*a*b*(a/b + x)**(1/3)*exp(8*I*pi/3)*log(1 - b**(1/3)*(a/b +
x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1/
3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*gamma(2/3)) - 4*a*b*(a/b + x)**(1/
3)*exp(4*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))
*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)*
*(4/3)*gamma(2/3)) + 4*b**2*(a/b + x)**(4/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a
**(1/3))*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1/3)*gamma(2/3) + 9*a**(7/3)*b*(a
/b + x)**(4/3)*gamma(2/3)) + 4*b**2*(a/b + x)**(4/3)*exp(8*I*pi/3)*log(1 - b**(1
/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(-1/3)/(-9*a**(10/3)*(a/
b + x)**(1/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*gamma(2/3)) + 4*b**2*(a
/b + x)**(4/3)*exp(4*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/
3)/a**(1/3))*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1/3)*gamma(2/3) + 9*a**(7/3)*
b*(a/b + x)**(4/3)*gamma(2/3))

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GIAC/XCAS [A]  time = 0.524059, size = 162, normalized size = 1.41 \[ -\frac{4 \, \sqrt{3} b \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{3 \, a^{\frac{7}{3}}} + \frac{2 \, b{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{3 \, a^{\frac{7}{3}}} - \frac{4 \, b{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{3 \, a^{\frac{7}{3}}} - \frac{4 \,{\left (b x + a\right )} b - 3 \, a b}{{\left ({\left (b x + a\right )}^{\frac{4}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(4/3)*x^2),x, algorithm="giac")

[Out]

-4/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3)
+ 2/3*b*ln((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) - 4/3*b
*ln(abs((b*x + a)^(1/3) - a^(1/3)))/a^(7/3) - (4*(b*x + a)*b - 3*a*b)/(((b*x + a
)^(4/3) - (b*x + a)^(1/3)*a)*a^2)