3.371 \(\int x^3 \sqrt [3]{a+b x} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac{9 a^2 (a+b x)^{7/3}}{7 b^4}+\frac{3 (a+b x)^{13/3}}{13 b^4}-\frac{9 a (a+b x)^{10/3}}{10 b^4} \]

[Out]

(-3*a^3*(a + b*x)^(4/3))/(4*b^4) + (9*a^2*(a + b*x)^(7/3))/(7*b^4) - (9*a*(a + b
*x)^(10/3))/(10*b^4) + (3*(a + b*x)^(13/3))/(13*b^4)

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Rubi [A]  time = 0.0530509, antiderivative size = 72, 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 \[ -\frac{3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac{9 a^2 (a+b x)^{7/3}}{7 b^4}+\frac{3 (a+b x)^{13/3}}{13 b^4}-\frac{9 a (a+b x)^{10/3}}{10 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^(1/3),x]

[Out]

(-3*a^3*(a + b*x)^(4/3))/(4*b^4) + (9*a^2*(a + b*x)^(7/3))/(7*b^4) - (9*a*(a + b
*x)^(10/3))/(10*b^4) + (3*(a + b*x)^(13/3))/(13*b^4)

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Rubi in Sympy [A]  time = 11.048, size = 68, normalized size = 0.94 \[ - \frac{3 a^{3} \left (a + b x\right )^{\frac{4}{3}}}{4 b^{4}} + \frac{9 a^{2} \left (a + b x\right )^{\frac{7}{3}}}{7 b^{4}} - \frac{9 a \left (a + b x\right )^{\frac{10}{3}}}{10 b^{4}} + \frac{3 \left (a + b x\right )^{\frac{13}{3}}}{13 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**3*(a + b*x)**(4/3)/(4*b**4) + 9*a**2*(a + b*x)**(7/3)/(7*b**4) - 9*a*(a +
b*x)**(10/3)/(10*b**4) + 3*(a + b*x)**(13/3)/(13*b**4)

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Mathematica [A]  time = 0.021053, size = 57, normalized size = 0.79 \[ \frac{3 \sqrt [3]{a+b x} \left (-81 a^4+27 a^3 b x-18 a^2 b^2 x^2+14 a b^3 x^3+140 b^4 x^4\right )}{1820 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x)^(1/3)*(-81*a^4 + 27*a^3*b*x - 18*a^2*b^2*x^2 + 14*a*b^3*x^3 + 140*b
^4*x^4))/(1820*b^4)

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Maple [A]  time = 0.009, size = 43, normalized size = 0.6 \[ -{\frac{-420\,{b}^{3}{x}^{3}+378\,a{b}^{2}{x}^{2}-324\,{a}^{2}bx+243\,{a}^{3}}{1820\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(1/3),x)

[Out]

-3/1820*(b*x+a)^(4/3)*(-140*b^3*x^3+126*a*b^2*x^2-108*a^2*b*x+81*a^3)/b^4

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Maxima [A]  time = 1.34008, size = 76, normalized size = 1.06 \[ \frac{3 \,{\left (b x + a\right )}^{\frac{13}{3}}}{13 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{10}{3}} a}{10 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{2}}{7 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{3}}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/13*(b*x + a)^(13/3)/b^4 - 9/10*(b*x + a)^(10/3)*a/b^4 + 9/7*(b*x + a)^(7/3)*a^
2/b^4 - 3/4*(b*x + a)^(4/3)*a^3/b^4

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Fricas [A]  time = 0.20959, size = 72, normalized size = 1. \[ \frac{3 \,{\left (140 \, b^{4} x^{4} + 14 \, a b^{3} x^{3} - 18 \, a^{2} b^{2} x^{2} + 27 \, a^{3} b x - 81 \, a^{4}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{1820 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/1820*(140*b^4*x^4 + 14*a*b^3*x^3 - 18*a^2*b^2*x^2 + 27*a^3*b*x - 81*a^4)*(b*x
+ a)^(1/3)/b^4

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Sympy [A]  time = 8.37901, size = 1742, normalized size = 24.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243*a**(73/3)*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*
a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b*
*9*x**5 + 1820*a**14*b**10*x**6) + 243*a**(73/3)/(1820*a**20*b**4 + 10920*a**19*
b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 +
10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) - 1377*a**(70/3)*b*x*(1 + b*x/a)
**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a*
*17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10
*x**6) + 1458*a**(70/3)*b*x/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*
b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**
5 + 1820*a**14*b**10*x**6) - 3213*a**(67/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(1820*a
**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 +
27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 3645*a
**(67/3)*b**2*x**2/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2
+ 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*
a**14*b**10*x**6) - 3927*a**(64/3)*b**3*x**3*(1 + b*x/a)**(1/3)/(1820*a**20*b**4
+ 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a*
*16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 4860*a**(64/3)*
b**3*x**3/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*
a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**
10*x**6) - 2163*a**(61/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*
a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*
x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 3645*a**(61/3)*b**4*x**4
/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**
7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6)
+ 1827*a**(58/3)*b**5*x**5*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**
5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10
920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 1458*a**(58/3)*b**5*x**5/(1820*a*
*20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 +
27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 6573*a*
*(55/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 273
00*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15
*b**9*x**5 + 1820*a**14*b**10*x**6) + 243*a**(55/3)*b**6*x**6/(1820*a**20*b**4 +
10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**1
6*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 8787*a**(52/3)*b*
*7*x**7*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b
**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5
+ 1820*a**14*b**10*x**6) + 6498*a**(49/3)*b**8*x**8*(1 + b*x/a)**(1/3)/(1820*a*
*20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 +
27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 2562*a*
*(46/3)*b**9*x**9*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 273
00*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15
*b**9*x**5 + 1820*a**14*b**10*x**6) + 420*a**(43/3)*b**10*x**10*(1 + b*x/a)**(1/
3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b
**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6
)

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GIAC/XCAS [A]  time = 0.205932, size = 82, normalized size = 1.14 \[ \frac{3 \,{\left (140 \,{\left (b x + a\right )}^{\frac{13}{3}} b^{36} - 546 \,{\left (b x + a\right )}^{\frac{10}{3}} a b^{36} + 780 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{2} b^{36} - 455 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{3} b^{36}\right )}}{1820 \, b^{40}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/1820*(140*(b*x + a)^(13/3)*b^36 - 546*(b*x + a)^(10/3)*a*b^36 + 780*(b*x + a)^
(7/3)*a^2*b^36 - 455*(b*x + a)^(4/3)*a^3*b^36)/b^40