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

Optimal. Leaf size=140 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac{x^{4/3}}{2 b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.116019, antiderivative size = 140, 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 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac{x^{4/3}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(4/3)/(a + b*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.1039, size = 129, normalized size = 0.92 \[ - \frac{x^{\frac{4}{3}}}{2 b \left (a + b x\right )^{2}} - \frac{2 \sqrt [3]{x}}{3 b^{2} \left (a + b x\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{3 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\log{\left (a + b x \right )}}{9 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(4/3)/(2*b*(a + b*x)**2) - 2*x**(1/3)/(3*b**2*(a + b*x)) + log(a**(1/3) + b*
*(1/3)*x**(1/3))/(3*a**(2/3)*b**(7/3)) - log(a + b*x)/(9*a**(2/3)*b**(7/3)) - 2*
sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(1/3)/3)/a**(1/3))/(9*a**(2/3)*
b**(7/3))

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Mathematica [A]  time = 0.111452, size = 154, normalized size = 1.1 \[ \frac{-\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{a^{2/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}+\frac{9 a \sqrt [3]{b} \sqrt [3]{x}}{(a+b x)^2}-\frac{21 \sqrt [3]{b} \sqrt [3]{x}}{a+b x}}{18 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

((9*a*b^(1/3)*x^(1/3))/(a + b*x)^2 - (21*b^(1/3)*x^(1/3))/(a + b*x) - (4*Sqrt[3]
*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3) + (4*Log[a^(1/3) + b
^(1/3)*x^(1/3)])/a^(2/3) - (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^
(2/3)])/a^(2/3))/(18*b^(7/3))

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Maple [A]  time = 0.018, size = 124, normalized size = 0.9 \[ 3\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( -{\frac{7\,{x}^{4/3}}{18\,b}}-2/9\,{\frac{a\sqrt [3]{x}}{{b}^{2}}} \right ) }+{\frac{2}{9\,{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{9\,{b}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(4/3)/(b*x+a)^3,x)

[Out]

3*(-7/18*x^(4/3)/b-2/9*a*x^(1/3)/b^2)/(b*x+a)^2+2/9/b^3/(a/b)^(2/3)*ln(x^(1/3)+(
a/b)^(1/3))-1/9/b^3/(a/b)^(2/3)*ln(x^(2/3)-x^(1/3)*(a/b)^(1/3)+(a/b)^(2/3))+2/9/
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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22676, size = 257, normalized size = 1.84 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (a^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right ) - 4 \, \sqrt{3}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (a + \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}\right ) + 3 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (7 \, b x + 4 \, a\right )} x^{\frac{1}{3}} - 12 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{\frac{1}{3}}}{3 \, a}\right )\right )}}{54 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(2*sqrt(3)*(b^2*x^2 + 2*a*b*x + a^2)*log(a^2 - (a^2*b)^(1/3)*a*x^(
1/3) + (a^2*b)^(2/3)*x^(2/3)) - 4*sqrt(3)*(b^2*x^2 + 2*a*b*x + a^2)*log(a + (a^2
*b)^(1/3)*x^(1/3)) + 3*sqrt(3)*(a^2*b)^(1/3)*(7*b*x + 4*a)*x^(1/3) - 12*(b^2*x^2
+ 2*a*b*x + a^2)*arctan(-1/3*(sqrt(3)*a - 2*sqrt(3)*(a^2*b)^(1/3)*x^(1/3))/a))/
((b^4*x^2 + 2*a*b^3*x + a^2*b^2)*(a^2*b)^(1/3))

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Sympy [A]  time = 7.78021, size = 1681, normalized size = 12.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-28*a**(19/3)*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)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10
/3) + 162*a**5*b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3))
+ 28*a**(19/3)*b**3*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(7/
3)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*
b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 28*a**(19/3)*
b**3*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(7
/3)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5
*b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 84*a**(16/3)
*b**4*x*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamm
a(7/3)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a
**5*b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) + 84*a**(16
/3)*b**4*x*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(7/3)/(54*a*
*7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(22/3)
*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 84*a**(16/3)*b**4*x*ex
p(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(7/3)/(54
*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(22
/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 84*a**(13/3)*b**5*x
**2*exp(5*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(7/
3)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*
b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) + 84*a**(13/3)*
b**5*x**2*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(7/3)/(54*a**
7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(22/3)*
x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 84*a**(13/3)*b**5*x**2*
exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(7/3)/(
54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(
22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 28*a**(10/3)*b**6
*x**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)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**
5*b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) + 28*a**(10/3
)*b**6*x**3*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(7/3)/(54*a
**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(22/3
)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 28*a**(10/3)*b**6*x**
3*exp(I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(7/3)
/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b*
*(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3)) - 84*a**6*b**(10/
3)*x**(1/3)*gamma(7/3)/(54*a**7*b**(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gam
ma(10/3) + 162*a**5*b**(22/3)*x**2*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10
/3)) - 231*a**5*b**(13/3)*x**(4/3)*gamma(7/3)/(54*a**7*b**(16/3)*gamma(10/3) + 1
62*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(22/3)*x**2*gamma(10/3) + 54*a**4*
b**(25/3)*x**3*gamma(10/3)) - 147*a**4*b**(16/3)*x**(7/3)*gamma(7/3)/(54*a**7*b*
*(16/3)*gamma(10/3) + 162*a**6*b**(19/3)*x*gamma(10/3) + 162*a**5*b**(22/3)*x**2
*gamma(10/3) + 54*a**4*b**(25/3)*x**3*gamma(10/3))

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GIAC/XCAS [A]  time = 0.225741, size = 197, normalized size = 1.41 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{2}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{3}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a b^{3}} - \frac{7 \, b x^{\frac{4}{3}} + 4 \, a x^{\frac{1}{3}}}{6 \,{\left (b x + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(-a/b)^(1/3)*ln(abs(x^(1/3) - (-a/b)^(1/3)))/(a*b^2) + 2/9*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*b^3) + 1/9
*(-a*b^2)^(1/3)*ln(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^3) - 1/6*
(7*b*x^(4/3) + 4*a*x^(1/3))/((b*x + a)^2*b^2)