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

Optimal. Leaf size=124 \[ -\frac{b^2 \log (x)}{9 a^{2/3}}+\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\frac{2 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{(a+b x)^{4/3}}{2 x^2}-\frac{2 b \sqrt [3]{a+b x}}{3 x} \]

[Out]

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

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Rubi [A]  time = 0.113236, antiderivative size = 124, 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 \[ -\frac{b^2 \log (x)}{9 a^{2/3}}+\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\frac{2 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{(a+b x)^{4/3}}{2 x^2}-\frac{2 b \sqrt [3]{a+b x}}{3 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(4/3)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7195, size = 114, normalized size = 0.92 \[ - \frac{2 b \sqrt [3]{a + b x}}{3 x} - \frac{\left (a + b x\right )^{\frac{4}{3}}}{2 x^{2}} - \frac{b^{2} \log{\left (x \right )}}{9 a^{\frac{2}{3}}} + \frac{b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{3 a^{\frac{2}{3}}} - \frac{2 \sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0355968, size = 76, normalized size = 0.61 \[ \frac{-3 a^2-2 b^2 x^2 \left (\frac{a}{b x}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )-10 a b x-7 b^2 x^2}{6 x^2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^2 - 10*a*b*x - 7*b^2*x^2 - 2*b^2*(1 + a/(b*x))^(2/3)*x^2*Hypergeometric2F1
[2/3, 2/3, 5/3, -(a/(b*x))])/(6*x^2*(a + b*x)^(2/3))

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Maple [A]  time = 0.017, size = 111, normalized size = 0.9 \[ -{\frac{7}{6\,{x}^{2}} \left ( bx+a \right ) ^{{\frac{4}{3}}}}+{\frac{2\,a}{3\,{x}^{2}}\sqrt [3]{bx+a}}+{\frac{2\,{b}^{2}}{9}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{{b}^{2}}{9}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\frac{2\,{b}^{2}\sqrt{3}}{9}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(4/3)/x^3,x)

[Out]

-7/6*(b*x+a)^(4/3)/x^2+2/3/x^2*(b*x+a)^(1/3)*a+2/9*b^2/a^(2/3)*ln((b*x+a)^(1/3)-
a^(1/3))-1/9*b^2/a^(2/3)*ln((b*x+a)^(2/3)+(b*x+a)^(1/3)*a^(1/3)+a^(2/3))-2/9*b^2
/a^(2/3)*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((b*x + a)^(4/3)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238549, size = 200, normalized size = 1.61 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} b^{2} x^{2} \log \left (a^{2} +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 4 \, \sqrt{3} b^{2} x^{2} \log \left (-a +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) + 12 \, b^{2} x^{2} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (7 \, b x + 3 \, a\right )}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2}\right )}^{\frac{1}{3}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.56438, size = 1731, normalized size = 13.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.588976, size = 171, normalized size = 1.38 \[ -\frac{\frac{4 \, \sqrt{3} b^{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}}} + \frac{2 \, b^{3}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{2}{3}}} - \frac{4 \, b^{3}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} + \frac{3 \,{\left (7 \,{\left (b x + a\right )}^{\frac{4}{3}} b^{3} - 4 \,{\left (b x + a\right )}^{\frac{1}{3}} a b^{3}\right )}}{b^{2} x^{2}}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(4*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a
^(2/3) + 2*b^3*ln((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(2/3) -
4*b^3*ln(abs((b*x + a)^(1/3) - a^(1/3)))/a^(2/3) + 3*(7*(b*x + a)^(4/3)*b^3 - 4
*(b*x + a)^(1/3)*a*b^3)/(b^2*x^2))/b