3.1591 \(\int \frac{1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(2/3))/(2*(b*c - a*d)*(a + b*x)^(2/3))

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Rubi [A]  time = 0.0236, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*(c + d*x)^(2/3))/(2*(b*c - a*d)*(a + b*x)^(2/3))

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Rubi in Sympy [A]  time = 3.53857, size = 26, normalized size = 0.81 \[ \frac{3 \left (c + d x\right )^{\frac{2}{3}}}{2 \left (a + b x\right )^{\frac{2}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/3)/(d*x+c)**(1/3),x)

[Out]

3*(c + d*x)**(2/3)/(2*(a + b*x)**(2/3)*(a*d - b*c))

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Mathematica [A]  time = 0.0375807, size = 32, normalized size = 1. \[ \frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/3)*(c + d*x)^(1/3)),x]

[Out]

(3*(c + d*x)^(2/3))/(2*(-(b*c) + a*d)*(a + b*x)^(2/3))

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Maple [A]  time = 0.008, size = 27, normalized size = 0.8 \[{\frac{3}{2\,ad-2\,bc} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( bx+a \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/3)/(d*x+c)^(1/3),x)

[Out]

3/2/(b*x+a)^(2/3)*(d*x+c)^(2/3)/(a*d-b*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)), x)

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Fricas [A]  time = 0.209897, size = 35, normalized size = 1.09 \[ -\frac{3 \,{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

-3/2*(d*x + c)^(2/3)/((b*c - a*d)*(b*x + a)^(2/3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{3}} \sqrt [3]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/3)/(d*x+c)**(1/3),x)

[Out]

Integral(1/((a + b*x)**(5/3)*(c + d*x)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(1/3)), x)