3.406 \(\int \frac{x^3}{(a+b x)^{2/3}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac{9 a^2 (a+b x)^{4/3}}{4 b^4}+\frac{3 (a+b x)^{10/3}}{10 b^4}-\frac{9 a (a+b x)^{7/3}}{7 b^4} \]

[Out]

(-3*a^3*(a + b*x)^(1/3))/b^4 + (9*a^2*(a + b*x)^(4/3))/(4*b^4) - (9*a*(a + b*x)^
(7/3))/(7*b^4) + (3*(a + b*x)^(10/3))/(10*b^4)

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Rubi [A]  time = 0.0524753, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac{9 a^2 (a+b x)^{4/3}}{4 b^4}+\frac{3 (a+b x)^{10/3}}{10 b^4}-\frac{9 a (a+b x)^{7/3}}{7 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x)^(2/3),x]

[Out]

(-3*a^3*(a + b*x)^(1/3))/b^4 + (9*a^2*(a + b*x)^(4/3))/(4*b^4) - (9*a*(a + b*x)^
(7/3))/(7*b^4) + (3*(a + b*x)^(10/3))/(10*b^4)

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Rubi in Sympy [A]  time = 11.0077, size = 66, normalized size = 0.94 \[ - \frac{3 a^{3} \sqrt [3]{a + b x}}{b^{4}} + \frac{9 a^{2} \left (a + b x\right )^{\frac{4}{3}}}{4 b^{4}} - \frac{9 a \left (a + b x\right )^{\frac{7}{3}}}{7 b^{4}} + \frac{3 \left (a + b x\right )^{\frac{10}{3}}}{10 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(2/3),x)

[Out]

-3*a**3*(a + b*x)**(1/3)/b**4 + 9*a**2*(a + b*x)**(4/3)/(4*b**4) - 9*a*(a + b*x)
**(7/3)/(7*b**4) + 3*(a + b*x)**(10/3)/(10*b**4)

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Mathematica [A]  time = 0.0229735, size = 46, normalized size = 0.66 \[ \frac{3 \sqrt [3]{a+b x} \left (-81 a^3+27 a^2 b x-18 a b^2 x^2+14 b^3 x^3\right )}{140 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(1/3)*(-81*a^3 + 27*a^2*b*x - 18*a*b^2*x^2 + 14*b^3*x^3))/(140*b^4)

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Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[ -{\frac{-42\,{b}^{3}{x}^{3}+54\,a{b}^{2}{x}^{2}-81\,{a}^{2}bx+243\,{a}^{3}}{140\,{b}^{4}}\sqrt [3]{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(2/3),x)

[Out]

-3/140*(b*x+a)^(1/3)*(-14*b^3*x^3+18*a*b^2*x^2-27*a^2*b*x+81*a^3)/b^4

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Maxima [A]  time = 1.34226, size = 76, normalized size = 1.09 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{10}{3}}}{10 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{7}{3}} a}{7 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{2}}{4 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(2/3),x, algorithm="maxima")

[Out]

3/10*(b*x + a)^(10/3)/b^4 - 9/7*(b*x + a)^(7/3)*a/b^4 + 9/4*(b*x + a)^(4/3)*a^2/
b^4 - 3*(b*x + a)^(1/3)*a^3/b^4

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Fricas [A]  time = 0.206901, size = 57, normalized size = 0.81 \[ \frac{3 \,{\left (14 \, b^{3} x^{3} - 18 \, a b^{2} x^{2} + 27 \, a^{2} b x - 81 \, a^{3}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{140 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(2/3),x, algorithm="fricas")

[Out]

3/140*(14*b^3*x^3 - 18*a*b^2*x^2 + 27*a^2*b*x - 81*a^3)*(b*x + a)^(1/3)/b^4

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Sympy [A]  time = 8.24214, size = 1640, normalized size = 23.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**(2/3),x)

[Out]

-243*a**(70/3)*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**1
8*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5
+ 140*a**14*b**10*x**6) + 243*a**(70/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 210
0*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9
*x**5 + 140*a**14*b**10*x**6) - 1377*a**(67/3)*b*x*(1 + b*x/a)**(1/3)/(140*a**20
*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a*
*16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 1458*a**(67/3)*b*x
/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**
3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) - 3213*a*
*(64/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a
**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x*
*5 + 140*a**14*b**10*x**6) + 3645*a**(64/3)*b**2*x**2/(140*a**20*b**4 + 840*a**1
9*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 +
840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) - 3927*a**(61/3)*b**3*x**3*(1 + b*x/
a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17
*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6)
+ 4860*a**(61/3)*b**3*x**3/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*
x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a
**14*b**10*x**6) - 2583*a**(58/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(140*a**20*b**4 +
840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**
8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 3645*a**(58/3)*b**4*x**4/
(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3
+ 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) - 693*a**(
55/3)*b**5*x**5*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**
18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5
+ 140*a**14*b**10*x**6) + 1458*a**(55/3)*b**5*x**5/(140*a**20*b**4 + 840*a**19*
b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 84
0*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 273*a**(52/3)*b**6*x**6*(1 + b*x/a)*
*(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b*
*7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 2
43*a**(52/3)*b**6*x**6/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2
+ 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14
*b**10*x**6) + 387*a**(49/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*
a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**
4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**6) + 198*a**(46/3)*b**8*x**8*(1 + b
*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*x + 2100*a**18*b**6*x**2 + 2800*a*
*17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**15*b**9*x**5 + 140*a**14*b**10*x**
6) + 42*a**(43/3)*b**9*x**9*(1 + b*x/a)**(1/3)/(140*a**20*b**4 + 840*a**19*b**5*
x + 2100*a**18*b**6*x**2 + 2800*a**17*b**7*x**3 + 2100*a**16*b**8*x**4 + 840*a**
15*b**9*x**5 + 140*a**14*b**10*x**6)

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GIAC/XCAS [A]  time = 0.204861, size = 82, normalized size = 1.17 \[ \frac{3 \,{\left (14 \,{\left (b x + a\right )}^{\frac{10}{3}} b^{27} - 60 \,{\left (b x + a\right )}^{\frac{7}{3}} a b^{27} + 105 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{2} b^{27} - 140 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{3} b^{27}\right )}}{140 \, b^{31}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^(2/3),x, algorithm="giac")

[Out]

3/140*(14*(b*x + a)^(10/3)*b^27 - 60*(b*x + a)^(7/3)*a*b^27 + 105*(b*x + a)^(4/3
)*a^2*b^27 - 140*(b*x + a)^(1/3)*a^3*b^27)/b^31