3.385 \(\int x^3 (a+b x)^{4/3} \, dx\)

Optimal. Leaf size=72 \[ \frac{9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac{3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac{3 (a+b x)^{16/3}}{16 b^4}-\frac{9 a (a+b x)^{13/3}}{13 b^4} \]

[Out]

(-3*a^3*(a + b*x)^(7/3))/(7*b^4) + (9*a^2*(a + b*x)^(10/3))/(10*b^4) - (9*a*(a + b*x)^(13/3))/(13*b^4) + (3*(a
+ b*x)^(16/3))/(16*b^4)

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Rubi [A]  time = 0.0179075, 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} \[ \frac{9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac{3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac{3 (a+b x)^{16/3}}{16 b^4}-\frac{9 a (a+b x)^{13/3}}{13 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(4/3),x]

[Out]

(-3*a^3*(a + b*x)^(7/3))/(7*b^4) + (9*a^2*(a + b*x)^(10/3))/(10*b^4) - (9*a*(a + b*x)^(13/3))/(13*b^4) + (3*(a
+ b*x)^(16/3))/(16*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 (a+b x)^{4/3} \, dx &=\int \left (-\frac{a^3 (a+b x)^{4/3}}{b^3}+\frac{3 a^2 (a+b x)^{7/3}}{b^3}-\frac{3 a (a+b x)^{10/3}}{b^3}+\frac{(a+b x)^{13/3}}{b^3}\right ) \, dx\\ &=-\frac{3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac{9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac{9 a (a+b x)^{13/3}}{13 b^4}+\frac{3 (a+b x)^{16/3}}{16 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0758572, size = 46, normalized size = 0.64 \[ \frac{3 (a+b x)^{7/3} \left (189 a^2 b x-81 a^3-315 a b^2 x^2+455 b^3 x^3\right )}{7280 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3)*(-81*a^3 + 189*a^2*b*x - 315*a*b^2*x^2 + 455*b^3*x^3))/(7280*b^4)

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Maple [A]  time = 0.004, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-1365\,{b}^{3}{x}^{3}+945\,a{b}^{2}{x}^{2}-567\,{a}^{2}bx+243\,{a}^{3}}{7280\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(4/3),x)

[Out]

-3/7280*(b*x+a)^(7/3)*(-455*b^3*x^3+315*a*b^2*x^2-189*a^2*b*x+81*a^3)/b^4

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Maxima [A]  time = 1.04098, size = 76, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{16}{3}}}{16 \, b^{4}} - \frac{9 \,{\left (b x + a\right )}^{\frac{13}{3}} a}{13 \, b^{4}} + \frac{9 \,{\left (b x + a\right )}^{\frac{10}{3}} a^{2}}{10 \, b^{4}} - \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{3}}{7 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(4/3),x, algorithm="maxima")

[Out]

3/16*(b*x + a)^(16/3)/b^4 - 9/13*(b*x + a)^(13/3)*a/b^4 + 9/10*(b*x + a)^(10/3)*a^2/b^4 - 3/7*(b*x + a)^(7/3)*
a^3/b^4

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Fricas [A]  time = 1.47519, size = 154, normalized size = 2.14 \begin{align*} \frac{3 \,{\left (455 \, b^{5} x^{5} + 595 \, a b^{4} x^{4} + 14 \, a^{2} b^{3} x^{3} - 18 \, a^{3} b^{2} x^{2} + 27 \, a^{4} b x - 81 \, a^{5}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{7280 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(4/3),x, algorithm="fricas")

[Out]

3/7280*(455*b^5*x^5 + 595*a*b^4*x^4 + 14*a^2*b^3*x^3 - 18*a^3*b^2*x^2 + 27*a^4*b*x - 81*a^5)*(b*x + a)^(1/3)/b
^4

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Sympy [B]  time = 4.66662, size = 1844, normalized size = 25.61 \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)**(4/3),x)

[Out]

-243*a**(76/3)*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**1
7*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 243*a**(76/3)/(7280*a*
*20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 436
80*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) - 1377*a**(73/3)*b*x*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a
**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5
+ 7280*a**14*b**10*x**6) + 1458*a**(73/3)*b*x/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2
+ 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) - 3213*a**(
70/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**
17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 3645*a**(70/3)*b**2*x
**2/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**
8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) - 3927*a**(67/3)*b**3*x**3*(1 + b*x/a)**(1/3)/(7280*a*
*20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 436
80*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 4860*a**(67/3)*b**3*x**3/(7280*a**20*b**4 + 43680*a**19*b**5*x +
109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14
*b**10*x**6) - 798*a**(64/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18
*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6)
+ 3645*a**(64/3)*b**4*x**4/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*
x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 11382*a**(61/3)*b**5*x**5*(1
+ b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 1092
00*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 1458*a**(61/3)*b**5*x**5/(7280*a**20*b**
4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**1
5*b**9*x**5 + 7280*a**14*b**10*x**6) + 35238*a**(58/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a
**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5
+ 7280*a**14*b**10*x**6) + 243*a**(58/3)*b**6*x**6/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*
x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 5656
2*a**(55/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 1456
00*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 54273*a**(52/3)
*b**8*x**8*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b*
*7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 31227*a**(49/3)*b**9*x**9*
(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 1
09200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 9975*a**(46/3)*b**10*x**10*(1 + b*x/a
)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**1
6*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 1365*a**(43/3)*b**11*x**11*(1 + b*x/a)**(1/3)/(
7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**
4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6)

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Giac [B]  time = 1.18533, size = 157, normalized size = 2.18 \begin{align*} \frac{3 \,{\left (\frac{4 \,{\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 )} a}{b^{3}} + \frac{455 \,{\left (b x + a\right )}^{\frac{16}{3}} - 2240 \,{\left (b x + a\right )}^{\frac{13}{3}} a + 4368 \,{\left (b x + a\right )}^{\frac{10}{3}} a^{2} - 4160 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{3} + 1820 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{4}}{b^{3}}\right )}}{7280 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(4/3),x, algorithm="giac")

[Out]

3/7280*(4*(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)*
a/b^3 + (455*(b*x + a)^(16/3) - 2240*(b*x + a)^(13/3)*a + 4368*(b*x + a)^(10/3)*a^2 - 4160*(b*x + a)^(7/3)*a^3
+ 1820*(b*x + a)^(4/3)*a^4)/b^3)/b