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

Optimal. Leaf size=125 \[ -\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac{2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}+\frac{4 \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}-\frac{x^{4/3}}{b (a+b x)}+\frac{4 \sqrt [3]{x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.112812, antiderivative size = 125, 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 \[ -\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac{2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}+\frac{4 \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}-\frac{x^{4/3}}{b (a+b x)}+\frac{4 \sqrt [3]{x}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(4/3)/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.1072, size = 121, normalized size = 0.97 \[ - \frac{2 \sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{b^{\frac{7}{3}}} + \frac{2 \sqrt [3]{a} \log{\left (a + b x \right )}}{3 b^{\frac{7}{3}}} + \frac{4 \sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{7}{3}}} - \frac{x^{\frac{4}{3}}}{b \left (a + b x\right )} + \frac{4 \sqrt [3]{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.141827, size = 147, normalized size = 1.18 \[ \frac{2 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )+\frac{3 a \sqrt [3]{b} \sqrt [3]{x}}{a+b x}-4 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+4 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )+9 \sqrt [3]{b} \sqrt [3]{x}}{3 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 123, normalized size = 1. \[ 3\,{\frac{\sqrt [3]{x}}{{b}^{2}}}+{\frac{a}{{b}^{2} \left ( bx+a \right ) }\sqrt [3]{x}}-{\frac{4\,a}{3\,{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,a}{3\,{b}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,a\sqrt{3}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(4/3)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224728, size = 213, normalized size = 1.7 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b x + a\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3}{\left (b x + a\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 12 \,{\left (b x + a\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x^{\frac{1}{3}} + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (3 \, b x + 4 \, a\right )} x^{\frac{1}{3}}\right )}}{9 \,{\left (b^{3} x + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.17492, size = 578, normalized size = 4.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28*a**(10/3)*b**3*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(
1/3))*gamma(7/3)/(9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*gamma(10/3))
- 28*a**(10/3)*b**3*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(7
/3)/(9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*gamma(10/3)) + 28*a**(10/
3)*b**3*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamm
a(7/3)/(9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*gamma(10/3)) + 28*a**(
7/3)*b**4*x*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*
gamma(7/3)/(9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*gamma(10/3)) - 28*
a**(7/3)*b**4*x*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(7/3)/(
9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*gamma(10/3)) + 28*a**(7/3)*b**
4*x*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(7/
3)/(9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*gamma(10/3)) + 84*a**3*b**
(10/3)*x**(1/3)*gamma(7/3)/(9*a**3*b**(16/3)*gamma(10/3) + 9*a**2*b**(19/3)*x*ga
mma(10/3)) + 63*a**2*b**(13/3)*x**(4/3)*gamma(7/3)/(9*a**3*b**(16/3)*gamma(10/3)
+ 9*a**2*b**(19/3)*x*gamma(10/3))

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GIAC/XCAS [A]  time = 0.224804, size = 182, normalized size = 1.46 \[ \frac{4 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b^{2}} - \frac{4 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{3}} + \frac{a x^{\frac{1}{3}}}{{\left (b x + a\right )} b^{2}} + \frac{3 \, x^{\frac{1}{3}}}{b^{2}} - \frac{2 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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