3.602 \(\int x^{11} (a+b x^3)^p \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^3 \left (a+b x^3\right )^{p+1}}{3 b^4 (p+1)}+\frac{a^2 \left (a+b x^3\right )^{p+2}}{b^4 (p+2)}-\frac{a \left (a+b x^3\right )^{p+3}}{b^4 (p+3)}+\frac{\left (a+b x^3\right )^{p+4}}{3 b^4 (p+4)} \]

[Out]

-(a^3*(a + b*x^3)^(1 + p))/(3*b^4*(1 + p)) + (a^2*(a + b*x^3)^(2 + p))/(b^4*(2 + p)) - (a*(a + b*x^3)^(3 + p))
/(b^4*(3 + p)) + (a + b*x^3)^(4 + p)/(3*b^4*(4 + p))

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Rubi [A]  time = 0.0595246, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^3 \left (a+b x^3\right )^{p+1}}{3 b^4 (p+1)}+\frac{a^2 \left (a+b x^3\right )^{p+2}}{b^4 (p+2)}-\frac{a \left (a+b x^3\right )^{p+3}}{b^4 (p+3)}+\frac{\left (a+b x^3\right )^{p+4}}{3 b^4 (p+4)} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^p,x]

[Out]

-(a^3*(a + b*x^3)^(1 + p))/(3*b^4*(1 + p)) + (a^2*(a + b*x^3)^(2 + p))/(b^4*(2 + p)) - (a*(a + b*x^3)^(3 + p))
/(b^4*(3 + p)) + (a + b*x^3)^(4 + p)/(3*b^4*(4 + p))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^{11} \left (a+b x^3\right )^p \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 (a+b x)^p \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^p}{b^3}+\frac{3 a^2 (a+b x)^{1+p}}{b^3}-\frac{3 a (a+b x)^{2+p}}{b^3}+\frac{(a+b x)^{3+p}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^{1+p}}{3 b^4 (1+p)}+\frac{a^2 \left (a+b x^3\right )^{2+p}}{b^4 (2+p)}-\frac{a \left (a+b x^3\right )^{3+p}}{b^4 (3+p)}+\frac{\left (a+b x^3\right )^{4+p}}{3 b^4 (4+p)}\\ \end{align*}

Mathematica [A]  time = 0.0422237, size = 95, normalized size = 1. \[ -\frac{a^3 \left (a+b x^3\right )^{p+1}}{3 b^4 (p+1)}+\frac{a^2 \left (a+b x^3\right )^{p+2}}{b^4 (p+2)}-\frac{a \left (a+b x^3\right )^{p+3}}{b^4 (p+3)}+\frac{\left (a+b x^3\right )^{p+4}}{3 b^4 (p+4)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^3)^p,x]

[Out]

-(a^3*(a + b*x^3)^(1 + p))/(3*b^4*(1 + p)) + (a^2*(a + b*x^3)^(2 + p))/(b^4*(2 + p)) - (a*(a + b*x^3)^(3 + p))
/(b^4*(3 + p)) + (a + b*x^3)^(4 + p)/(3*b^4*(4 + p))

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Maple [A]  time = 0.006, size = 132, normalized size = 1.4 \begin{align*} -{\frac{ \left ( b{x}^{3}+a \right ) ^{1+p} \left ( -{b}^{3}{p}^{3}{x}^{9}-6\,{b}^{3}{p}^{2}{x}^{9}-11\,{b}^{3}p{x}^{9}-6\,{b}^{3}{x}^{9}+3\,a{b}^{2}{p}^{2}{x}^{6}+9\,a{b}^{2}p{x}^{6}+6\,a{b}^{2}{x}^{6}-6\,{a}^{2}bp{x}^{3}-6\,{a}^{2}b{x}^{3}+6\,{a}^{3} \right ) }{3\,{b}^{4} \left ({p}^{4}+10\,{p}^{3}+35\,{p}^{2}+50\,p+24 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^3+a)^p,x)

[Out]

-1/3*(b*x^3+a)^(1+p)*(-b^3*p^3*x^9-6*b^3*p^2*x^9-11*b^3*p*x^9-6*b^3*x^9+3*a*b^2*p^2*x^6+9*a*b^2*p*x^6+6*a*b^2*
x^6-6*a^2*b*p*x^3-6*a^2*b*x^3+6*a^3)/b^4/(p^4+10*p^3+35*p^2+50*p+24)

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Maxima [A]  time = 1.00014, size = 143, normalized size = 1.51 \begin{align*} \frac{{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} x^{12} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} x^{9} - 3 \,{\left (p^{2} + p\right )} a^{2} b^{2} x^{6} + 6 \, a^{3} b p x^{3} - 6 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^p,x, algorithm="maxima")

[Out]

1/3*((p^3 + 6*p^2 + 11*p + 6)*b^4*x^12 + (p^3 + 3*p^2 + 2*p)*a*b^3*x^9 - 3*(p^2 + p)*a^2*b^2*x^6 + 6*a^3*b*p*x
^3 - 6*a^4)*(b*x^3 + a)^p/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^4)

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Fricas [A]  time = 1.50315, size = 304, normalized size = 3.2 \begin{align*} \frac{{\left ({\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} x^{12} +{\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} x^{9} + 6 \, a^{3} b p x^{3} - 3 \,{\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{6} - 6 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (b^{4} p^{4} + 10 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 50 \, b^{4} p + 24 \, b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^p,x, algorithm="fricas")

[Out]

1/3*((b^4*p^3 + 6*b^4*p^2 + 11*b^4*p + 6*b^4)*x^12 + (a*b^3*p^3 + 3*a*b^3*p^2 + 2*a*b^3*p)*x^9 + 6*a^3*b*p*x^3
- 3*(a^2*b^2*p^2 + a^2*b^2*p)*x^6 - 6*a^4)*(b*x^3 + a)^p/(b^4*p^4 + 10*b^4*p^3 + 35*b^4*p^2 + 50*b^4*p + 24*b
^4)

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Sympy [A]  time = 77.5925, size = 2771, normalized size = 29.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**3+a)**p,x)

[Out]

Piecewise((a**p*x**12/12, Eq(b, 0)), (6*a**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(18*a**3*b**4 + 54*a*
*2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) + 6*a**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)
*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) - 12*a**
3*log(2)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) + 2*a**3/(18*a**3*b**4 + 54*a**2*b
**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) + 18*a**2*b*x**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(18*a**
3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) + 18*a**2*b*x**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)*
*(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 +
18*b**7*x**9) - 36*a**2*b*x**3*log(2)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) + 18*
a*b**2*x**6*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 1
8*b**7*x**9) + 18*a*b**2*x**6*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3)
+ 4*x**2)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) - 36*a*b**2*x**6*log(2)/(18*a**3*
b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) - 9*a*b**2*x**6/(18*a**3*b**4 + 54*a**2*b**5*x**3 +
54*a*b**6*x**6 + 18*b**7*x**9) + 6*b**3*x**9*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(18*a**3*b**4 + 54*a*
*2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) + 6*b**3*x**9*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**
(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) - 9
*b**3*x**9/(18*a**3*b**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9) - 12*b**3*x**9*log(2)/(18*a**3*b
**4 + 54*a**2*b**5*x**3 + 54*a*b**6*x**6 + 18*b**7*x**9), Eq(p, -4)), (-6*a**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)
**(1/3) + x)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) - 6*a**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4
*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) - 3*a**3/(6*a**2*b
**4 + 12*a*b**5*x**3 + 6*b**6*x**6) + 12*a**3*log(2)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) - 12*a**2*b*
x**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) - 12*a**2*b*x**3
*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a**2*b**4 + 12*a
*b**5*x**3 + 6*b**6*x**6) + 24*a**2*b*x**3*log(2)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) - 6*a*b**2*x**6
*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) - 6*a*b**2*x**6*log(
4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a**2*b**4 + 12*a*b**5
*x**3 + 6*b**6*x**6) + 6*a*b**2*x**6/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) + 12*a*b**2*x**6*log(2)/(6*a
**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) + 2*b**3*x**9/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6), Eq(p, -3)
), (6*a**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(6*a*b**4 + 6*b**5*x**3) + 6*a**3*log(4*(-1)**(2/3)*a**
(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a*b**4 + 6*b**5*x**3) - 12*a**3*log(2)
/(6*a*b**4 + 6*b**5*x**3) + 6*a**3/(6*a*b**4 + 6*b**5*x**3) + 6*a**2*b*x**3*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(
1/3) + x)/(6*a*b**4 + 6*b**5*x**3) + 6*a**2*b*x**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**
(1/3)*x*(1/b)**(1/3) + 4*x**2)/(6*a*b**4 + 6*b**5*x**3) - 12*a**2*b*x**3*log(2)/(6*a*b**4 + 6*b**5*x**3) - 3*a
*b**2*x**6/(6*a*b**4 + 6*b**5*x**3) + b**3*x**9/(6*a*b**4 + 6*b**5*x**3), Eq(p, -2)), (-a**3*log(-(-1)**(1/3)*
a**(1/3)*(1/b)**(1/3) + x)/(3*b**4) - a**3*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*
(1/b)**(1/3) + 4*x**2)/(3*b**4) + a**2*x**3/(3*b**3) - a*x**6/(6*b**2) + x**9/(9*b), Eq(p, -1)), (-6*a**4*(a +
b*x**3)**p/(3*b**4*p**4 + 30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) + 6*a**3*b*p*x**3*(a + b*x**3)
**p/(3*b**4*p**4 + 30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) - 3*a**2*b**2*p**2*x**6*(a + b*x**3)**
p/(3*b**4*p**4 + 30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) - 3*a**2*b**2*p*x**6*(a + b*x**3)**p/(3*
b**4*p**4 + 30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) + a*b**3*p**3*x**9*(a + b*x**3)**p/(3*b**4*p*
*4 + 30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) + 3*a*b**3*p**2*x**9*(a + b*x**3)**p/(3*b**4*p**4 +
30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) + 2*a*b**3*p*x**9*(a + b*x**3)**p/(3*b**4*p**4 + 30*b**4*
p**3 + 105*b**4*p**2 + 150*b**4*p + 72*b**4) + b**4*p**3*x**12*(a + b*x**3)**p/(3*b**4*p**4 + 30*b**4*p**3 + 1
05*b**4*p**2 + 150*b**4*p + 72*b**4) + 6*b**4*p**2*x**12*(a + b*x**3)**p/(3*b**4*p**4 + 30*b**4*p**3 + 105*b**
4*p**2 + 150*b**4*p + 72*b**4) + 11*b**4*p*x**12*(a + b*x**3)**p/(3*b**4*p**4 + 30*b**4*p**3 + 105*b**4*p**2 +
150*b**4*p + 72*b**4) + 6*b**4*x**12*(a + b*x**3)**p/(3*b**4*p**4 + 30*b**4*p**3 + 105*b**4*p**2 + 150*b**4*p
+ 72*b**4), True))

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Giac [B]  time = 1.11031, size = 554, normalized size = 5.83 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{4}{\left (b x^{3} + a\right )}^{p} p^{3} - 3 \,{\left (b x^{3} + a\right )}^{3}{\left (b x^{3} + a\right )}^{p} a p^{3} + 3 \,{\left (b x^{3} + a\right )}^{2}{\left (b x^{3} + a\right )}^{p} a^{2} p^{3} -{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{p} a^{3} p^{3} + 6 \,{\left (b x^{3} + a\right )}^{4}{\left (b x^{3} + a\right )}^{p} p^{2} - 21 \,{\left (b x^{3} + a\right )}^{3}{\left (b x^{3} + a\right )}^{p} a p^{2} + 24 \,{\left (b x^{3} + a\right )}^{2}{\left (b x^{3} + a\right )}^{p} a^{2} p^{2} - 9 \,{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{p} a^{3} p^{2} + 11 \,{\left (b x^{3} + a\right )}^{4}{\left (b x^{3} + a\right )}^{p} p - 42 \,{\left (b x^{3} + a\right )}^{3}{\left (b x^{3} + a\right )}^{p} a p + 57 \,{\left (b x^{3} + a\right )}^{2}{\left (b x^{3} + a\right )}^{p} a^{2} p - 26 \,{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{p} a^{3} p + 6 \,{\left (b x^{3} + a\right )}^{4}{\left (b x^{3} + a\right )}^{p} - 24 \,{\left (b x^{3} + a\right )}^{3}{\left (b x^{3} + a\right )}^{p} a + 36 \,{\left (b x^{3} + a\right )}^{2}{\left (b x^{3} + a\right )}^{p} a^{2} - 24 \,{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{p} a^{3}}{3 \,{\left (b^{3} p^{4} + 10 \, b^{3} p^{3} + 35 \, b^{3} p^{2} + 50 \, b^{3} p + 24 \, b^{3}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^p,x, algorithm="giac")

[Out]

1/3*((b*x^3 + a)^4*(b*x^3 + a)^p*p^3 - 3*(b*x^3 + a)^3*(b*x^3 + a)^p*a*p^3 + 3*(b*x^3 + a)^2*(b*x^3 + a)^p*a^2
*p^3 - (b*x^3 + a)*(b*x^3 + a)^p*a^3*p^3 + 6*(b*x^3 + a)^4*(b*x^3 + a)^p*p^2 - 21*(b*x^3 + a)^3*(b*x^3 + a)^p*
a*p^2 + 24*(b*x^3 + a)^2*(b*x^3 + a)^p*a^2*p^2 - 9*(b*x^3 + a)*(b*x^3 + a)^p*a^3*p^2 + 11*(b*x^3 + a)^4*(b*x^3
+ a)^p*p - 42*(b*x^3 + a)^3*(b*x^3 + a)^p*a*p + 57*(b*x^3 + a)^2*(b*x^3 + a)^p*a^2*p - 26*(b*x^3 + a)*(b*x^3
+ a)^p*a^3*p + 6*(b*x^3 + a)^4*(b*x^3 + a)^p - 24*(b*x^3 + a)^3*(b*x^3 + a)^p*a + 36*(b*x^3 + a)^2*(b*x^3 + a)
^p*a^2 - 24*(b*x^3 + a)*(b*x^3 + a)^p*a^3)/((b^3*p^4 + 10*b^3*p^3 + 35*b^3*p^2 + 50*b^3*p + 24*b^3)*b)