3.276 \(\int x^3 (a+b x^3)^5 \, dx\)

Optimal. Leaf size=66 \[ \frac{10}{13} a^2 b^3 x^{13}+a^3 b^2 x^{10}+\frac{5}{7} a^4 b x^7+\frac{a^5 x^4}{4}+\frac{5}{16} a b^4 x^{16}+\frac{b^5 x^{19}}{19} \]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^7)/7 + a^3*b^2*x^10 + (10*a^2*b^3*x^13)/13 + (5*a*b^4*x^16)/16 + (b^5*x^19)/19

________________________________________________________________________________________

Rubi [A]  time = 0.0221241, antiderivative size = 66, 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 = {270} \[ \frac{10}{13} a^2 b^3 x^{13}+a^3 b^2 x^{10}+\frac{5}{7} a^4 b x^7+\frac{a^5 x^4}{4}+\frac{5}{16} a b^4 x^{16}+\frac{b^5 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^3)^5,x]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^7)/7 + a^3*b^2*x^10 + (10*a^2*b^3*x^13)/13 + (5*a*b^4*x^16)/16 + (b^5*x^19)/19

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^3 \left (a+b x^3\right )^5 \, dx &=\int \left (a^5 x^3+5 a^4 b x^6+10 a^3 b^2 x^9+10 a^2 b^3 x^{12}+5 a b^4 x^{15}+b^5 x^{18}\right ) \, dx\\ &=\frac{a^5 x^4}{4}+\frac{5}{7} a^4 b x^7+a^3 b^2 x^{10}+\frac{10}{13} a^2 b^3 x^{13}+\frac{5}{16} a b^4 x^{16}+\frac{b^5 x^{19}}{19}\\ \end{align*}

Mathematica [A]  time = 0.002395, size = 66, normalized size = 1. \[ \frac{10}{13} a^2 b^3 x^{13}+a^3 b^2 x^{10}+\frac{5}{7} a^4 b x^7+\frac{a^5 x^4}{4}+\frac{5}{16} a b^4 x^{16}+\frac{b^5 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^3)^5,x]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^7)/7 + a^3*b^2*x^10 + (10*a^2*b^3*x^13)/13 + (5*a*b^4*x^16)/16 + (b^5*x^19)/19

________________________________________________________________________________________

Maple [A]  time = 0., size = 57, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{x}^{4}}{4}}+{\frac{5\,{a}^{4}b{x}^{7}}{7}}+{a}^{3}{b}^{2}{x}^{10}+{\frac{10\,{a}^{2}{b}^{3}{x}^{13}}{13}}+{\frac{5\,a{b}^{4}{x}^{16}}{16}}+{\frac{{b}^{5}{x}^{19}}{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^3+a)^5,x)

[Out]

1/4*a^5*x^4+5/7*a^4*b*x^7+a^3*b^2*x^10+10/13*a^2*b^3*x^13+5/16*a*b^4*x^16+1/19*b^5*x^19

________________________________________________________________________________________

Maxima [A]  time = 0.968489, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{19} \, b^{5} x^{19} + \frac{5}{16} \, a b^{4} x^{16} + \frac{10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4

________________________________________________________________________________________

Fricas [A]  time = 1.51353, size = 134, normalized size = 2.03 \begin{align*} \frac{1}{19} x^{19} b^{5} + \frac{5}{16} x^{16} b^{4} a + \frac{10}{13} x^{13} b^{3} a^{2} + x^{10} b^{2} a^{3} + \frac{5}{7} x^{7} b a^{4} + \frac{1}{4} x^{4} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/19*x^19*b^5 + 5/16*x^16*b^4*a + 10/13*x^13*b^3*a^2 + x^10*b^2*a^3 + 5/7*x^7*b*a^4 + 1/4*x^4*a^5

________________________________________________________________________________________

Sympy [A]  time = 0.094444, size = 63, normalized size = 0.95 \begin{align*} \frac{a^{5} x^{4}}{4} + \frac{5 a^{4} b x^{7}}{7} + a^{3} b^{2} x^{10} + \frac{10 a^{2} b^{3} x^{13}}{13} + \frac{5 a b^{4} x^{16}}{16} + \frac{b^{5} x^{19}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**3+a)**5,x)

[Out]

a**5*x**4/4 + 5*a**4*b*x**7/7 + a**3*b**2*x**10 + 10*a**2*b**3*x**13/13 + 5*a*b**4*x**16/16 + b**5*x**19/19

________________________________________________________________________________________

Giac [A]  time = 1.10006, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{19} \, b^{5} x^{19} + \frac{5}{16} \, a b^{4} x^{16} + \frac{10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4