3.1593 \(\int \frac{1}{(a+b x)^{11/3} \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{27 d^2 (c+d x)^{2/3}}{40 (a+b x)^{2/3} (b c-a d)^3}+\frac{9 d (c+d x)^{2/3}}{20 (a+b x)^{5/3} (b c-a d)^2}-\frac{3 (c+d x)^{2/3}}{8 (a+b x)^{8/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(2/3))/(8*(b*c - a*d)*(a + b*x)^(8/3)) + (9*d*(c + d*x)^(2/3))/(20
*(b*c - a*d)^2*(a + b*x)^(5/3)) - (27*d^2*(c + d*x)^(2/3))/(40*(b*c - a*d)^3*(a
+ b*x)^(2/3))

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Rubi [A]  time = 0.0872245, 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 \[ -\frac{27 d^2 (c+d x)^{2/3}}{40 (a+b x)^{2/3} (b c-a d)^3}+\frac{9 d (c+d x)^{2/3}}{20 (a+b x)^{5/3} (b c-a d)^2}-\frac{3 (c+d x)^{2/3}}{8 (a+b x)^{8/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(11/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*(c + d*x)^(2/3))/(8*(b*c - a*d)*(a + b*x)^(8/3)) + (9*d*(c + d*x)^(2/3))/(20
*(b*c - a*d)^2*(a + b*x)^(5/3)) - (27*d^2*(c + d*x)^(2/3))/(40*(b*c - a*d)^3*(a
+ b*x)^(2/3))

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Rubi in Sympy [A]  time = 13.5638, size = 88, normalized size = 0.87 \[ \frac{27 d^{2} \left (c + d x\right )^{\frac{2}{3}}}{40 \left (a + b x\right )^{\frac{2}{3}} \left (a d - b c\right )^{3}} + \frac{9 d \left (c + d x\right )^{\frac{2}{3}}}{20 \left (a + b x\right )^{\frac{5}{3}} \left (a d - b c\right )^{2}} + \frac{3 \left (c + d x\right )^{\frac{2}{3}}}{8 \left (a + b x\right )^{\frac{8}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(11/3)/(d*x+c)**(1/3),x)

[Out]

27*d**2*(c + d*x)**(2/3)/(40*(a + b*x)**(2/3)*(a*d - b*c)**3) + 9*d*(c + d*x)**(
2/3)/(20*(a + b*x)**(5/3)*(a*d - b*c)**2) + 3*(c + d*x)**(2/3)/(8*(a + b*x)**(8/
3)*(a*d - b*c))

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Mathematica [A]  time = 0.0969337, size = 77, normalized size = 0.76 \[ -\frac{3 (c+d x)^{2/3} \left (20 a^2 d^2+8 a b d (3 d x-2 c)+b^2 \left (5 c^2-6 c d x+9 d^2 x^2\right )\right )}{40 (a+b x)^{8/3} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(11/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*(c + d*x)^(2/3)*(20*a^2*d^2 + 8*a*b*d*(-2*c + 3*d*x) + b^2*(5*c^2 - 6*c*d*x
+ 9*d^2*x^2)))/(40*(b*c - a*d)^3*(a + b*x)^(8/3))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[{\frac{27\,{b}^{2}{d}^{2}{x}^{2}+72\,ab{d}^{2}x-18\,{b}^{2}cdx+60\,{a}^{2}{d}^{2}-48\,abcd+15\,{b}^{2}{c}^{2}}{40\,{a}^{3}{d}^{3}-120\,{a}^{2}cb{d}^{2}+120\,a{b}^{2}{c}^{2}d-40\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( bx+a \right ) ^{-{\frac{8}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(11/3)/(d*x+c)^(1/3),x)

[Out]

3/40*(d*x+c)^(2/3)*(9*b^2*d^2*x^2+24*a*b*d^2*x-6*b^2*c*d*x+20*a^2*d^2-16*a*b*c*d
+5*b^2*c^2)/(b*x+a)^(8/3)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{11}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/3)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(11/3)*(d*x + c)^(1/3)), x)

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Fricas [A]  time = 0.212706, size = 317, normalized size = 3.14 \[ -\frac{3 \,{\left (9 \, b^{2} d^{3} x^{3} + 5 \, b^{2} c^{3} - 16 \, a b c^{2} d + 20 \, a^{2} c d^{2} + 3 \,{\left (b^{2} c d^{2} + 8 \, a b d^{3}\right )} x^{2} -{\left (b^{2} c^{2} d - 8 \, a b c d^{2} - 20 \, a^{2} d^{3}\right )} x\right )}}{40 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} +{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/3)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

-3/40*(9*b^2*d^3*x^3 + 5*b^2*c^3 - 16*a*b*c^2*d + 20*a^2*c*d^2 + 3*(b^2*c*d^2 +
8*a*b*d^3)*x^2 - (b^2*c^2*d - 8*a*b*c*d^2 - 20*a^2*d^3)*x)/((a^2*b^3*c^3 - 3*a^3
*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3 + (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^
2 - a^3*b^2*d^3)*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*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)**(11/3)/(d*x+c)**(1/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{11}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/3)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(11/3)*(d*x + c)^(1/3)), x)