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

Optimal. Leaf size=109 \[ \frac{3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac{\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}+\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{3}{a \sqrt [3]{x}} \]

[Out]

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

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Rubi [A]  time = 0.0879252, antiderivative size = 109, 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 \[ \frac{3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac{\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}}+\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{3}{a \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(4/3)*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.1336, size = 104, normalized size = 0.95 \[ - \frac{3}{a \sqrt [3]{x}} + \frac{3 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{2 a^{\frac{4}{3}}} - \frac{\sqrt [3]{b} \log{\left (a + b x \right )}}{2 a^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0671427, size = 127, normalized size = 1.17 \[ \frac{-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{6 \sqrt [3]{a}}{\sqrt [3]{x}}}{2 a^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-6*a^(1/3))/x^(1/3) + 2*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3
))/Sqrt[3]] + 2*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] - b^(1/3)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(2*a^(4/3))

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Maple [A]  time = 0.012, size = 104, normalized size = 1. \[ -3\,{\frac{1}{a\sqrt [3]{x}}}+{\frac{1}{a}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{2\,a}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(4/3)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232659, size = 169, normalized size = 1.55 \[ -\frac{2 \, \sqrt{3} x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (a \left (\frac{b}{a}\right )^{\frac{2}{3}} - 2 \, b x^{\frac{1}{3}}\right )}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (-a x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{2}{3}} + b x^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (a \left (\frac{b}{a}\right )^{\frac{2}{3}} + b x^{\frac{1}{3}}\right ) + 6}{2 \, a x^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.91452, size = 182, normalized size = 1.67 \[ \frac{\Gamma \left (- \frac{1}{3}\right )}{a \sqrt [3]{x} \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} e^{\frac{10 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{\sqrt [3]{b} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{x} e^{\frac{5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(4/3)/(b*x+a),x)

[Out]

gamma(-1/3)/(a*x**(1/3)*gamma(2/3)) - b**(1/3)*exp(10*I*pi/3)*log(1 - b**(1/3)*x
**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*gamma(2/3)) - b**(1/
3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*g
amma(2/3)) - b**(1/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3
)/a**(1/3))*gamma(-1/3)/(3*a**(4/3)*gamma(2/3))

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GIAC/XCAS [A]  time = 0.221063, size = 169, normalized size = 1.55 \[ \frac{b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a^{2} b} - \frac{3}{a x^{\frac{1}{3}}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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