3.376 \(\int \frac{\sqrt [3]{a+b x}}{x^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{b \log (x)}{6 a^{2/3}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\sqrt [3]{a+b x}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0882571, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{b \log (x)}{6 a^{2/3}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\sqrt [3]{a+b x}}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/3)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.42704, size = 88, normalized size = 0.91 \[ - \frac{\sqrt [3]{a + b x}}{x} - \frac{b \log{\left (x \right )}}{6 a^{\frac{2}{3}}} + \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{2 a^{\frac{2}{3}}} - \frac{\sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0329435, size = 61, normalized size = 0.63 \[ \frac{-b x \left (\frac{a}{b x}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )-2 (a+b x)}{2 x (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 92, normalized size = 1. \[ -{\frac{1}{x}\sqrt [3]{bx+a}}+{\frac{b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{b}{6}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\frac{b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/3)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223554, size = 171, normalized size = 1.76 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x \log \left (a^{2} +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b x \log \left (-a +{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) + 6 \, b x \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 6 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{18 \,{\left (a^{2}\right )}^{\frac{1}{3}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*b*x*log(a^2 + (a^2)^(1/3)*(b*x + a)^(1/3)*a + (a^2)^(2/3)
*(b*x + a)^(2/3)) - 2*sqrt(3)*b*x*log(-a + (a^2)^(1/3)*(b*x + a)^(1/3)) + 6*b*x*
arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(a^2)^(1/3)*(b*x + a)^(1/3))/a) + 6*sqrt(3)*(a
^2)^(1/3)*(b*x + a)^(1/3))/((a^2)^(1/3)*x)

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Sympy [A]  time = 6.55475, size = 515, normalized size = 5.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(7/3)*b*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(4/3)/(9*a**3*gamm
a(7/3) - 9*a**2*b*(a/b + x)*gamma(7/3)) + 4*a**(7/3)*b*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(4/3)/(9*a**3*gamma(7/
3) - 9*a**2*b*(a/b + x)*gamma(7/3)) + 4*a**(7/3)*b*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(4/3)/(9*a**3*gamma(7/3) -
9*a**2*b*(a/b + x)*gamma(7/3)) - 4*a**(4/3)*b**2*(a/b + x)*log(1 - b**(1/3)*(a/
b + x)**(1/3)/a**(1/3))*gamma(4/3)/(9*a**3*gamma(7/3) - 9*a**2*b*(a/b + x)*gamma
(7/3)) - 4*a**(4/3)*b**2*(a/b + x)*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(4/3)/(9*a**3*gamma(7/3) - 9*a**2*b*(a/b +
x)*gamma(7/3)) - 4*a**(4/3)*b**2*(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(4/3)/(9*a**3*gamma(7/3) - 9*a**2
*b*(a/b + x)*gamma(7/3)) + 12*a**2*b**(4/3)*(a/b + x)**(1/3)*gamma(4/3)/(9*a**3*
gamma(7/3) - 9*a**2*b*(a/b + x)*gamma(7/3))

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GIAC/XCAS [A]  time = 0.527125, size = 142, normalized size = 1.46 \[ -\frac{\frac{2 \, \sqrt{3} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}}} + \frac{b^{2}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{2}{3}}} - \frac{2 \, b^{2}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} + \frac{6 \,{\left (b x + a\right )}^{\frac{1}{3}} b}{x}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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