3.1592 \(\int \frac{1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=66 \[ \frac{9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(2/3))/(5*(b*c - a*d)*(a + b*x)^(5/3)) + (9*d*(c + d*x)^(2/3))/(10
*(b*c - a*d)^2*(a + b*x)^(2/3))

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Rubi [A]  time = 0.0531332, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(8/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*(c + d*x)^(2/3))/(5*(b*c - a*d)*(a + b*x)^(5/3)) + (9*d*(c + d*x)^(2/3))/(10
*(b*c - a*d)^2*(a + b*x)^(2/3))

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Rubi in Sympy [A]  time = 7.389, size = 56, normalized size = 0.85 \[ \frac{9 d \left (c + d x\right )^{\frac{2}{3}}}{10 \left (a + b x\right )^{\frac{2}{3}} \left (a d - b c\right )^{2}} + \frac{3 \left (c + d x\right )^{\frac{2}{3}}}{5 \left (a + b x\right )^{\frac{5}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(8/3)/(d*x+c)**(1/3),x)

[Out]

9*d*(c + d*x)**(2/3)/(10*(a + b*x)**(2/3)*(a*d - b*c)**2) + 3*(c + d*x)**(2/3)/(
5*(a + b*x)**(5/3)*(a*d - b*c))

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Mathematica [A]  time = 0.0619596, size = 46, normalized size = 0.7 \[ \frac{3 (c+d x)^{2/3} (5 a d-2 b c+3 b d x)}{10 (a+b x)^{5/3} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(8/3)*(c + d*x)^(1/3)),x]

[Out]

(3*(c + d*x)^(2/3)*(-2*b*c + 5*a*d + 3*b*d*x))/(10*(b*c - a*d)^2*(a + b*x)^(5/3)
)

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Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \[{\frac{9\,bdx+15\,ad-6\,bc}{10\,{a}^{2}{d}^{2}-20\,abcd+10\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( bx+a \right ) ^{-{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(8/3)/(d*x+c)^(1/3),x)

[Out]

3/10*(d*x+c)^(2/3)*(3*b*d*x+5*a*d-2*b*c)/(b*x+a)^(5/3)/(a^2*d^2-2*a*b*c*d+b^2*c^
2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(8/3)*(d*x + c)^(1/3)), x)

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Fricas [A]  time = 0.211737, size = 138, normalized size = 2.09 \[ \frac{3 \,{\left (3 \, b d^{2} x^{2} - 2 \, b c^{2} + 5 \, a c d +{\left (b c d + 5 \, a d^{2}\right )} x\right )}}{10 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

3/10*(3*b*d^2*x^2 - 2*b*c^2 + 5*a*c*d + (b*c*d + 5*a*d^2)*x)/((a*b^2*c^2 - 2*a^2
*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x)*(b*x + a)^(2/3)*(d*x +
c)^(1/3))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(8/3)/(d*x+c)**(1/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(8/3)*(d*x + c)^(1/3)), x)