3.1622 \(\int \frac{1}{(a+b x)^{11/3} (c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{243 d^3 \sqrt [3]{a+b x}}{40 \sqrt [3]{c+d x} (b c-a d)^4}-\frac{81 d^2}{40 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^3}+\frac{27 d}{40 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{8 (a+b x)^{8/3} \sqrt [3]{c+d x} (b c-a d)} \]

[Out]

-3/(8*(b*c - a*d)*(a + b*x)^(8/3)*(c + d*x)^(1/3)) + (27*d)/(40*(b*c - a*d)^2*(a
+ b*x)^(5/3)*(c + d*x)^(1/3)) - (81*d^2)/(40*(b*c - a*d)^3*(a + b*x)^(2/3)*(c +
d*x)^(1/3)) - (243*d^3*(a + b*x)^(1/3))/(40*(b*c - a*d)^4*(c + d*x)^(1/3))

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Rubi [A]  time = 0.110903, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{243 d^3 \sqrt [3]{a+b x}}{40 \sqrt [3]{c+d x} (b c-a d)^4}-\frac{81 d^2}{40 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^3}+\frac{27 d}{40 (a+b x)^{5/3} \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{8 (a+b x)^{8/3} \sqrt [3]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(11/3)*(c + d*x)^(4/3)),x]

[Out]

-3/(8*(b*c - a*d)*(a + b*x)^(8/3)*(c + d*x)^(1/3)) + (27*d)/(40*(b*c - a*d)^2*(a
+ b*x)^(5/3)*(c + d*x)^(1/3)) - (81*d^2)/(40*(b*c - a*d)^3*(a + b*x)^(2/3)*(c +
d*x)^(1/3)) - (243*d^3*(a + b*x)^(1/3))/(40*(b*c - a*d)^4*(c + d*x)^(1/3))

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Rubi in Sympy [A]  time = 21.0168, size = 121, normalized size = 0.89 \[ - \frac{243 d^{3} \sqrt [3]{a + b x}}{40 \sqrt [3]{c + d x} \left (a d - b c\right )^{4}} + \frac{81 d^{2}}{40 \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )^{3}} + \frac{27 d}{40 \left (a + b x\right )^{\frac{5}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )^{2}} + \frac{3}{8 \left (a + b x\right )^{\frac{8}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(11/3)/(d*x+c)**(4/3),x)

[Out]

-243*d**3*(a + b*x)**(1/3)/(40*(c + d*x)**(1/3)*(a*d - b*c)**4) + 81*d**2/(40*(a
+ b*x)**(2/3)*(c + d*x)**(1/3)*(a*d - b*c)**3) + 27*d/(40*(a + b*x)**(5/3)*(c +
d*x)**(1/3)*(a*d - b*c)**2) + 3/(8*(a + b*x)**(8/3)*(c + d*x)**(1/3)*(a*d - b*c
))

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Mathematica [A]  time = 0.180667, size = 112, normalized size = 0.82 \[ \sqrt [3]{a+b x} (c+d x)^{2/3} \left (-\frac{3 d^3}{(c+d x) (b c-a d)^4}-\frac{123 b d^2}{40 (a+b x) (b c-a d)^4}+\frac{21 b d}{20 (a+b x)^2 (b c-a d)^3}-\frac{3 b}{8 (a+b x)^3 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(11/3)*(c + d*x)^(4/3)),x]

[Out]

(a + b*x)^(1/3)*(c + d*x)^(2/3)*((-3*b)/(8*(b*c - a*d)^2*(a + b*x)^3) + (21*b*d)
/(20*(b*c - a*d)^3*(a + b*x)^2) - (123*b*d^2)/(40*(b*c - a*d)^4*(a + b*x)) - (3*
d^3)/((b*c - a*d)^4*(c + d*x)))

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Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[ -{\frac{243\,{b}^{3}{d}^{3}{x}^{3}+648\,a{b}^{2}{d}^{3}{x}^{2}+81\,{b}^{3}c{d}^{2}{x}^{2}+540\,{a}^{2}b{d}^{3}x+216\,a{b}^{2}c{d}^{2}x-27\,{b}^{3}{c}^{2}dx+120\,{a}^{3}{d}^{3}+180\,{a}^{2}cb{d}^{2}-72\,a{b}^{2}{c}^{2}d+15\,{b}^{3}{c}^{3}}{40\,{d}^{4}{a}^{4}-160\,b{d}^{3}c{a}^{3}+240\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-160\,{b}^{3}d{c}^{3}a+40\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{8}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(11/3)/(d*x+c)^(4/3),x)

[Out]

-3/40*(81*b^3*d^3*x^3+216*a*b^2*d^3*x^2+27*b^3*c*d^2*x^2+180*a^2*b*d^3*x+72*a*b^
2*c*d^2*x-9*b^3*c^2*d*x+40*a^3*d^3+60*a^2*b*c*d^2-24*a*b^2*c^2*d+5*b^3*c^3)/(b*x
+a)^(8/3)/(d*x+c)^(1/3)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b
^4*c^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{11}{3}}{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/3)*(d*x + c)^(4/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(11/3)*(d*x + c)^(4/3)), x)

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Fricas [A]  time = 0.217988, size = 392, normalized size = 2.88 \[ -\frac{3 \,{\left (81 \, b^{3} d^{3} x^{3} + 5 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} + 40 \, a^{3} d^{3} + 27 \,{\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 9 \,{\left (b^{3} c^{2} d - 8 \, a b^{2} c d^{2} - 20 \, a^{2} b d^{3}\right )} x\right )}}{40 \,{\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} +{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/3)*(d*x + c)^(4/3)),x, algorithm="fricas")

[Out]

-3/40*(81*b^3*d^3*x^3 + 5*b^3*c^3 - 24*a*b^2*c^2*d + 60*a^2*b*c*d^2 + 40*a^3*d^3
+ 27*(b^3*c*d^2 + 8*a*b^2*d^3)*x^2 - 9*(b^3*c^2*d - 8*a*b^2*c*d^2 - 20*a^2*b*d^
3)*x)/((a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*
d^4 + (b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d
^4)*x^2 + 2*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 +
a^5*b*d^4)*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)**(4/3),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/3)*(d*x + c)^(4/3)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError