3.27 \(\int x^5 (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=67 \[ \frac{\left (a+b x^3\right )^7 (A b-2 a B)}{21 b^3}-\frac{a \left (a+b x^3\right )^6 (A b-a B)}{18 b^3}+\frac{B \left (a+b x^3\right )^8}{24 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x^3)^6)/(18*b^3) + ((A*b - 2*a*B)*(a + b*x^3)^7)/(21*b^3) + (B*(a + b*x^3)^8)/(24*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.148173, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ \frac{\left (a+b x^3\right )^7 (A b-2 a B)}{21 b^3}-\frac{a \left (a+b x^3\right )^6 (A b-a B)}{18 b^3}+\frac{B \left (a+b x^3\right )^8}{24 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

-(a*(A*b - a*B)*(a + b*x^3)^6)/(18*b^3) + ((A*b - 2*a*B)*(a + b*x^3)^7)/(21*b^3) + (B*(a + b*x^3)^8)/(24*b^3)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^5 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^5 (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a (-A b+a B) (a+b x)^5}{b^2}+\frac{(A b-2 a B) (a+b x)^6}{b^2}+\frac{B (a+b x)^7}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac{a (A b-a B) \left (a+b x^3\right )^6}{18 b^3}+\frac{(A b-2 a B) \left (a+b x^3\right )^7}{21 b^3}+\frac{B \left (a+b x^3\right )^8}{24 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0240713, size = 107, normalized size = 1.6 \[ \frac{1}{504} x^6 \left (336 a^2 b^2 x^9 (a B+A b)+210 a^3 b x^6 (a B+2 A b)+56 a^4 x^3 (a B+5 A b)+84 a^5 A+24 b^4 x^{15} (5 a B+A b)+140 a b^3 x^{12} (2 a B+A b)+21 b^5 B x^{18}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(x^6*(84*a^5*A + 56*a^4*(5*A*b + a*B)*x^3 + 210*a^3*b*(2*A*b + a*B)*x^6 + 336*a^2*b^2*(A*b + a*B)*x^9 + 140*a*
b^3*(A*b + 2*a*B)*x^12 + 24*b^4*(A*b + 5*a*B)*x^15 + 21*b^5*B*x^18))/504

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 124, normalized size = 1.9 \begin{align*}{\frac{{b}^{5}B{x}^{24}}{24}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{21}}{21}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{12}}{12}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{9}}{9}}+{\frac{{a}^{5}A{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b^5*B*x^24+1/21*(A*b^5+5*B*a*b^4)*x^21+1/18*(5*A*a*b^4+10*B*a^2*b^3)*x^18+1/15*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^15+1/12*(10*A*a^3*b^2+5*B*a^4*b)*x^12+1/9*(5*A*a^4*b+B*a^5)*x^9+1/6*a^5*A*x^6

________________________________________________________________________________________

Maxima [A]  time = 1.26554, size = 161, normalized size = 2.4 \begin{align*} \frac{1}{24} \, B b^{5} x^{24} + \frac{1}{21} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{21} + \frac{5}{18} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac{2}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac{5}{12} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{12} + \frac{1}{6} \, A a^{5} x^{6} + \frac{1}{9} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*B*b^5*x^24 + 1/21*(5*B*a*b^4 + A*b^5)*x^21 + 5/18*(2*B*a^2*b^3 + A*a*b^4)*x^18 + 2/3*(B*a^3*b^2 + A*a^2*b
^3)*x^15 + 5/12*(B*a^4*b + 2*A*a^3*b^2)*x^12 + 1/6*A*a^5*x^6 + 1/9*(B*a^5 + 5*A*a^4*b)*x^9

________________________________________________________________________________________

Fricas [B]  time = 1.25691, size = 309, normalized size = 4.61 \begin{align*} \frac{1}{24} x^{24} b^{5} B + \frac{5}{21} x^{21} b^{4} a B + \frac{1}{21} x^{21} b^{5} A + \frac{5}{9} x^{18} b^{3} a^{2} B + \frac{5}{18} x^{18} b^{4} a A + \frac{2}{3} x^{15} b^{2} a^{3} B + \frac{2}{3} x^{15} b^{3} a^{2} A + \frac{5}{12} x^{12} b a^{4} B + \frac{5}{6} x^{12} b^{2} a^{3} A + \frac{1}{9} x^{9} a^{5} B + \frac{5}{9} x^{9} b a^{4} A + \frac{1}{6} x^{6} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24*b^5*B + 5/21*x^21*b^4*a*B + 1/21*x^21*b^5*A + 5/9*x^18*b^3*a^2*B + 5/18*x^18*b^4*a*A + 2/3*x^15*b^2*
a^3*B + 2/3*x^15*b^3*a^2*A + 5/12*x^12*b*a^4*B + 5/6*x^12*b^2*a^3*A + 1/9*x^9*a^5*B + 5/9*x^9*b*a^4*A + 1/6*x^
6*a^5*A

________________________________________________________________________________________

Sympy [B]  time = 0.086819, size = 138, normalized size = 2.06 \begin{align*} \frac{A a^{5} x^{6}}{6} + \frac{B b^{5} x^{24}}{24} + x^{21} \left (\frac{A b^{5}}{21} + \frac{5 B a b^{4}}{21}\right ) + x^{18} \left (\frac{5 A a b^{4}}{18} + \frac{5 B a^{2} b^{3}}{9}\right ) + x^{15} \left (\frac{2 A a^{2} b^{3}}{3} + \frac{2 B a^{3} b^{2}}{3}\right ) + x^{12} \left (\frac{5 A a^{3} b^{2}}{6} + \frac{5 B a^{4} b}{12}\right ) + x^{9} \left (\frac{5 A a^{4} b}{9} + \frac{B a^{5}}{9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**5*x**6/6 + B*b**5*x**24/24 + x**21*(A*b**5/21 + 5*B*a*b**4/21) + x**18*(5*A*a*b**4/18 + 5*B*a**2*b**3/9)
+ x**15*(2*A*a**2*b**3/3 + 2*B*a**3*b**2/3) + x**12*(5*A*a**3*b**2/6 + 5*B*a**4*b/12) + x**9*(5*A*a**4*b/9 + B
*a**5/9)

________________________________________________________________________________________

Giac [B]  time = 1.18118, size = 169, normalized size = 2.52 \begin{align*} \frac{1}{24} \, B b^{5} x^{24} + \frac{5}{21} \, B a b^{4} x^{21} + \frac{1}{21} \, A b^{5} x^{21} + \frac{5}{9} \, B a^{2} b^{3} x^{18} + \frac{5}{18} \, A a b^{4} x^{18} + \frac{2}{3} \, B a^{3} b^{2} x^{15} + \frac{2}{3} \, A a^{2} b^{3} x^{15} + \frac{5}{12} \, B a^{4} b x^{12} + \frac{5}{6} \, A a^{3} b^{2} x^{12} + \frac{1}{9} \, B a^{5} x^{9} + \frac{5}{9} \, A a^{4} b x^{9} + \frac{1}{6} \, A 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*(B*x^3+A),x, algorithm="giac")

[Out]

1/24*B*b^5*x^24 + 5/21*B*a*b^4*x^21 + 1/21*A*b^5*x^21 + 5/9*B*a^2*b^3*x^18 + 5/18*A*a*b^4*x^18 + 2/3*B*a^3*b^2
*x^15 + 2/3*A*a^2*b^3*x^15 + 5/12*B*a^4*b*x^12 + 5/6*A*a^3*b^2*x^12 + 1/9*B*a^5*x^9 + 5/9*A*a^4*b*x^9 + 1/6*A*
a^5*x^6