3.371 \(\int x^3 \sqrt [3]{a+b x} \, dx\)
Optimal. Leaf size=72 \[ \frac{9 a^2 (a+b x)^{7/3}}{7 b^4}-\frac{3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac{3 (a+b x)^{13/3}}{13 b^4}-\frac{9 a (a+b x)^{10/3}}{10 b^4} \]
[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 = 0.0186468, 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, Rules used =
{43} \[ \frac{9 a^2 (a+b x)^{7/3}}{7 b^4}-\frac{3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac{3 (a+b x)^{13/3}}{13 b^4}-\frac{9 a (a+b x)^{10/3}}{10 b^4} \]
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)
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int x^3 \sqrt [3]{a+b x} \, dx &=\int \left (-\frac{a^3 \sqrt [3]{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{4/3}}{b^3}-\frac{3 a (a+b x)^{7/3}}{b^3}+\frac{(a+b x)^{10/3}}{b^3}\right ) \, dx\\ &=-\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{9 a (a+b x)^{10/3}}{10 b^4}+\frac{3 (a+b x)^{13/3}}{13 b^4}\\ \end{align*}
Mathematica [A] time = 0.052127, size = 46, normalized size = 0.64 \[ \frac{3 (a+b x)^{4/3} \left (108 a^2 b x-81 a^3-126 a b^2 x^2+140 b^3 x^3\right )}{1820 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*(a + b*x)^(1/3),x]
[Out]
(3*(a + b*x)^(4/3)*(-81*a^3 + 108*a^2*b*x - 126*a*b^2*x^2 + 140*b^3*x^3))/(1820*b^4)
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Maple [A] time = 0.004, size = 43, normalized size = 0.6 \begin{align*} -{\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}}}} \end{align*}
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 = 1.07404, size = 76, normalized size = 1.06 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(b*x+a)^(1/3),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 = 1.65421, size = 130, normalized size = 1.81 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(b*x+a)^(1/3),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 [B] time = 3.53965, size = 1742, normalized size = 24.19 \begin{align*} \text{result too large to display} \end{align*}
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 + 10920*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 + 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**(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)/(18
20*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 + 1
0920*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 + 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) + 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 [A] time = 1.1697, size = 66, normalized size = 0.92 \begin{align*} \frac{3 \,{\left (140 \,{\left (b x + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{3}\right )}}{1820 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(b*x+a)^(1/3),x, algorithm="giac")
[Out]
3/1820*(140*(b*x + a)^(13/3) - 546*(b*x + a)^(10/3)*a + 780*(b*x + a)^(7/3)*a^2 - 455*(b*x + a)^(4/3)*a^3)/b^4