3.1621 \(\int \frac{1}{(a+b x)^{8/3} (c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=101 \[ \frac{27 d^2 \sqrt [3]{a+b x}}{5 \sqrt [3]{c+d x} (b c-a d)^3}+\frac{9 d}{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{5 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)} \]

[Out]

-3/(5*(b*c - a*d)*(a + b*x)^(5/3)*(c + d*x)^(1/3)) + (9*d)/(5*(b*c - a*d)^2*(a +
b*x)^(2/3)*(c + d*x)^(1/3)) + (27*d^2*(a + b*x)^(1/3))/(5*(b*c - a*d)^3*(c + d*
x)^(1/3))

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Rubi [A]  time = 0.0752693, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{27 d^2 \sqrt [3]{a+b x}}{5 \sqrt [3]{c+d x} (b c-a d)^3}+\frac{9 d}{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{5 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(8/3)*(c + d*x)^(4/3)),x]

[Out]

-3/(5*(b*c - a*d)*(a + b*x)^(5/3)*(c + d*x)^(1/3)) + (9*d)/(5*(b*c - a*d)^2*(a +
b*x)^(2/3)*(c + d*x)^(1/3)) + (27*d^2*(a + b*x)^(1/3))/(5*(b*c - a*d)^3*(c + d*
x)^(1/3))

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Rubi in Sympy [A]  time = 13.4129, size = 88, normalized size = 0.87 \[ - \frac{27 d^{2} \sqrt [3]{a + b x}}{5 \sqrt [3]{c + d x} \left (a d - b c\right )^{3}} + \frac{9 d}{5 \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )^{2}} + \frac{3}{5 \left (a + b x\right )^{\frac{5}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(8/3)/(d*x+c)**(4/3),x)

[Out]

-27*d**2*(a + b*x)**(1/3)/(5*(c + d*x)**(1/3)*(a*d - b*c)**3) + 9*d/(5*(a + b*x)
**(2/3)*(c + d*x)**(1/3)*(a*d - b*c)**2) + 3/(5*(a + b*x)**(5/3)*(c + d*x)**(1/3
)*(a*d - b*c))

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Mathematica [A]  time = 0.116215, size = 75, normalized size = 0.74 \[ \frac{3 \left (5 a^2 d^2+5 a b d (c+3 d x)+b^2 \left (-c^2+3 c d x+9 d^2 x^2\right )\right )}{5 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(8/3)*(c + d*x)^(4/3)),x]

[Out]

(3*(5*a^2*d^2 + 5*a*b*d*(c + 3*d*x) + b^2*(-c^2 + 3*c*d*x + 9*d^2*x^2)))/(5*(b*c
- a*d)^3*(a + b*x)^(5/3)*(c + d*x)^(1/3))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[ -{\frac{27\,{b}^{2}{d}^{2}{x}^{2}+45\,ab{d}^{2}x+9\,{b}^{2}cdx+15\,{a}^{2}{d}^{2}+15\,abcd-3\,{b}^{2}{c}^{2}}{5\,{a}^{3}{d}^{3}-15\,{a}^{2}cb{d}^{2}+15\,a{b}^{2}{c}^{2}d-5\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{-{\frac{5}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(8/3)/(d*x+c)^(4/3),x)

[Out]

-3/5*(9*b^2*d^2*x^2+15*a*b*d^2*x+3*b^2*c*d*x+5*a^2*d^2+5*a*b*c*d-b^2*c^2)/(b*x+a
)^(5/3)/(d*x+c)^(1/3)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(4/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(8/3)*(d*x + c)^(4/3)), x)

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Fricas [A]  time = 0.211508, size = 201, normalized size = 1.99 \[ \frac{3 \,{\left (9 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 5 \, a^{2} d^{2} + 3 \,{\left (b^{2} c d + 5 \, a b d^{2}\right )} x\right )}}{5 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(4/3)),x, algorithm="fricas")

[Out]

3/5*(9*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 5*a^2*d^2 + 3*(b^2*c*d + 5*a*b*d^2)*x
)/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3 + (b^4*c^3 - 3*a*b^3*c
^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*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)**(4/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{8}{3}}{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(8/3)*(d*x + c)^(4/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(8/3)*(d*x + c)^(4/3)), x)