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

Optimal. Leaf size=34 \[ \frac{\left (a+b x^3\right )^7}{21 b^2}-\frac{a \left (a+b x^3\right )^6}{18 b^2} \]

[Out]

-(a*(a + b*x^3)^6)/(18*b^2) + (a + b*x^3)^7/(21*b^2)

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Rubi [A]  time = 0.0363607, antiderivative size = 34, 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} \[ \frac{\left (a+b x^3\right )^7}{21 b^2}-\frac{a \left (a+b x^3\right )^6}{18 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^3)^5,x]

[Out]

-(a*(a + b*x^3)^6)/(18*b^2) + (a + b*x^3)^7/(21*b^2)

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^5 \left (a+b x^3\right )^5 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^5}{b}+\frac{(a+b x)^6}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (a+b x^3\right )^6}{18 b^2}+\frac{\left (a+b x^3\right )^7}{21 b^2}\\ \end{align*}

Mathematica [B]  time = 0.0022518, size = 69, normalized size = 2.03 \[ \frac{2}{3} a^2 b^3 x^{15}+\frac{5}{6} a^3 b^2 x^{12}+\frac{5}{9} a^4 b x^9+\frac{a^5 x^6}{6}+\frac{5}{18} a b^4 x^{18}+\frac{b^5 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^3)^5,x]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^9)/9 + (5*a^3*b^2*x^12)/6 + (2*a^2*b^3*x^15)/3 + (5*a*b^4*x^18)/18 + (b^5*x^21)/21

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Maple [A]  time = 0.001, size = 58, normalized size = 1.7 \begin{align*}{\frac{{b}^{5}{x}^{21}}{21}}+{\frac{5\,a{b}^{4}{x}^{18}}{18}}+{\frac{2\,{a}^{2}{b}^{3}{x}^{15}}{3}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{12}}{6}}+{\frac{5\,{a}^{4}b{x}^{9}}{9}}+{\frac{{a}^{5}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^5*x^21+5/18*a*b^4*x^18+2/3*a^2*b^3*x^15+5/6*a^3*b^2*x^12+5/9*a^4*b*x^9+1/6*a^5*x^6

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Maxima [A]  time = 1.0309, size = 77, normalized size = 2.26 \begin{align*} \frac{1}{21} \, b^{5} x^{21} + \frac{5}{18} \, a b^{4} x^{18} + \frac{2}{3} \, a^{2} b^{3} x^{15} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{6} \, a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^5*x^21 + 5/18*a*b^4*x^18 + 2/3*a^2*b^3*x^15 + 5/6*a^3*b^2*x^12 + 5/9*a^4*b*x^9 + 1/6*a^5*x^6

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Fricas [A]  time = 1.46043, size = 136, normalized size = 4. \begin{align*} \frac{1}{21} x^{21} b^{5} + \frac{5}{18} x^{18} b^{4} a + \frac{2}{3} x^{15} b^{3} a^{2} + \frac{5}{6} x^{12} b^{2} a^{3} + \frac{5}{9} x^{9} b a^{4} + \frac{1}{6} x^{6} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*x^21*b^5 + 5/18*x^18*b^4*a + 2/3*x^15*b^3*a^2 + 5/6*x^12*b^2*a^3 + 5/9*x^9*b*a^4 + 1/6*x^6*a^5

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Sympy [B]  time = 0.101785, size = 66, normalized size = 1.94 \begin{align*} \frac{a^{5} x^{6}}{6} + \frac{5 a^{4} b x^{9}}{9} + \frac{5 a^{3} b^{2} x^{12}}{6} + \frac{2 a^{2} b^{3} x^{15}}{3} + \frac{5 a b^{4} x^{18}}{18} + \frac{b^{5} x^{21}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x**6/6 + 5*a**4*b*x**9/9 + 5*a**3*b**2*x**12/6 + 2*a**2*b**3*x**15/3 + 5*a*b**4*x**18/18 + b**5*x**21/21

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Giac [A]  time = 1.11928, size = 77, normalized size = 2.26 \begin{align*} \frac{1}{21} \, b^{5} x^{21} + \frac{5}{18} \, a b^{4} x^{18} + \frac{2}{3} \, a^{2} b^{3} x^{15} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{6} \, a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^5*x^21 + 5/18*a*b^4*x^18 + 2/3*a^2*b^3*x^15 + 5/6*a^3*b^2*x^12 + 5/9*a^4*b*x^9 + 1/6*a^5*x^6