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

Optimal. Leaf size=70 \[ \frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4} \]

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Rubi [A]  time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x)^(4/3),x]

[Out]

(3*a^3)/(b^4*(a + b*x)^(1/3)) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) - (9*a*(a + b*x)^(5/3))/(5*b^4) + (3*(a + b*x)
^(8/3))/(8*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 \frac {x^3}{(a+b x)^{4/3}} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^{4/3}}+\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {(a+b x)^{5/3}}{b^3}\right ) \, dx\\ &=\frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 46, normalized size = 0.66 \begin {gather*} \frac {3 \left (81 a^3+27 a^2 b x-9 a b^2 x^2+5 b^3 x^3\right )}{40 b^4 \sqrt [3]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^(4/3),x]

[Out]

(3*(81*a^3 + 27*a^2*b*x - 9*a*b^2*x^2 + 5*b^3*x^3))/(40*b^4*(a + b*x)^(1/3))

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IntegrateAlgebraic [A]  time = 0.03, size = 51, normalized size = 0.73 \begin {gather*} \frac {3 \left (40 a^3+60 a^2 (a+b x)-24 a (a+b x)^2+5 (a+b x)^3\right )}{40 b^4 \sqrt [3]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/(a + b*x)^(4/3),x]

[Out]

(3*(40*a^3 + 60*a^2*(a + b*x) - 24*a*(a + b*x)^2 + 5*(a + b*x)^3))/(40*b^4*(a + b*x)^(1/3))

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fricas [A]  time = 0.86, size = 52, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (5 \, b^{3} x^{3} - 9 \, a b^{2} x^{2} + 27 \, a^{2} b x + 81 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{40 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(5*b^3*x^3 - 9*a*b^2*x^2 + 27*a^2*b*x + 81*a^3)*(b*x + a)^(2/3)/(b^5*x + a*b^4)

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giac [A]  time = 1.10, size = 62, normalized size = 0.89 \begin {gather*} \frac {3 \, a^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{4}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {8}{3}} b^{28} - 24 \, {\left (b x + a\right )}^{\frac {5}{3}} a b^{28} + 60 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2} b^{28}\right )}}{40 \, b^{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^3/((b*x + a)^(1/3)*b^4) + 3/40*(5*(b*x + a)^(8/3)*b^28 - 24*(b*x + a)^(5/3)*a*b^28 + 60*(b*x + a)^(2/3)*a^
2*b^28)/b^32

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maple [A]  time = 0.00, size = 43, normalized size = 0.61 \begin {gather*} \frac {\frac {3}{8} b^{3} x^{3}-\frac {27}{40} a \,b^{2} x^{2}+\frac {81}{40} a^{2} b x +\frac {243}{40} a^{3}}{\left (b x +a \right )^{\frac {1}{3}} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40/(b*x+a)^(1/3)*(5*b^3*x^3-9*a*b^2*x^2+27*a^2*b*x+81*a^3)/b^4

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maxima [A]  time = 1.36, size = 56, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}}}{8 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {5}{3}} a}{5 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2}}{2 \, b^{4}} + \frac {3 \, a^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x + a)^(8/3)/b^4 - 9/5*(b*x + a)^(5/3)*a/b^4 + 9/2*(b*x + a)^(2/3)*a^2/b^4 + 3*a^3/((b*x + a)^(1/3)*b^4
)

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mupad [B]  time = 0.05, size = 56, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{8/3}}{8\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{2/3}}{2\,b^4}+\frac {3\,a^3}{b^4\,{\left (a+b\,x\right )}^{1/3}}-\frac {9\,a\,{\left (a+b\,x\right )}^{5/3}}{5\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x)^(4/3),x)

[Out]

(3*(a + b*x)^(8/3))/(8*b^4) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) + (3*a^3)/(b^4*(a + b*x)^(1/3)) - (9*a*(a + b*x)
^(5/3))/(5*b^4)

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sympy [B]  time = 2.89, size = 1538, normalized size = 21.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x+a)**(4/3),x)

[Out]

243*a**(68/3)*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3
+ 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 243*a**(68/3)/(40*a**20*b**4 + 240*a**19
*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**
10*x**6) + 1296*a**(65/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800
*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(65/3)*b*x/(40*a
**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**
9*x**5 + 40*a**14*b**10*x**6) + 2808*a**(62/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x
+ 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6)
- 3645*a**(62/3)*b**2*x**2/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 60
0*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 3120*a**(59/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(
40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15
*b**9*x**5 + 40*a**14*b**10*x**6) - 4860*a**(59/3)*b**3*x**3/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**
6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 1830*a**(56/
3)*b**4*x**4*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3
+ 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 3645*a**(56/3)*b**4*x**4/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6) + 528*a**(53/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*
b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(
53/3)*b**5*x**5/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8
*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 96*a**(50/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(40*a**20*b**4
+ 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 4
0*a**14*b**10*x**6) - 243*a**(50/3)*b**6*x**6/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a*
*17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 48*a**(47/3)*b**7*x**7*(1 +
b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*
x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 15*a**(44/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6)

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