3.26 \(\int x^6 (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=117 \[ \frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{1}{7} a^5 A x^7+\frac{1}{22} b^4 x^{22} (5 a B+A b)+\frac{5}{19} a b^3 x^{19} (2 a B+A b)+\frac{1}{25} b^5 B x^{25} \]

[Out]

(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (5*a^2*b^2*(A*b + a*B)*x^16)/8
+ (5*a*b^3*(A*b + 2*a*B)*x^19)/19 + (b^4*(A*b + 5*a*B)*x^22)/22 + (b^5*B*x^25)/25

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Rubi [A]  time = 0.0681405, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{1}{7} a^5 A x^7+\frac{1}{22} b^4 x^{22} (5 a B+A b)+\frac{5}{19} a b^3 x^{19} (2 a B+A b)+\frac{1}{25} b^5 B x^{25} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (5*a^2*b^2*(A*b + a*B)*x^16)/8
+ (5*a*b^3*(A*b + 2*a*B)*x^19)/19 + (b^4*(A*b + 5*a*B)*x^22)/22 + (b^5*B*x^25)/25

Rule 448

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

Rubi steps

\begin{align*} \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\int \left (a^5 A x^6+a^4 (5 A b+a B) x^9+5 a^3 b (2 A b+a B) x^{12}+10 a^2 b^2 (A b+a B) x^{15}+5 a b^3 (A b+2 a B) x^{18}+b^4 (A b+5 a B) x^{21}+b^5 B x^{24}\right ) \, dx\\ &=\frac{1}{7} a^5 A x^7+\frac{1}{10} a^4 (5 A b+a B) x^{10}+\frac{5}{13} a^3 b (2 A b+a B) x^{13}+\frac{5}{8} a^2 b^2 (A b+a B) x^{16}+\frac{5}{19} a b^3 (A b+2 a B) x^{19}+\frac{1}{22} b^4 (A b+5 a B) x^{22}+\frac{1}{25} b^5 B x^{25}\\ \end{align*}

Mathematica [A]  time = 0.0182178, size = 117, normalized size = 1. \[ \frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{1}{7} a^5 A x^7+\frac{1}{22} b^4 x^{22} (5 a B+A b)+\frac{5}{19} a b^3 x^{19} (2 a B+A b)+\frac{1}{25} b^5 B x^{25} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (5*a^2*b^2*(A*b + a*B)*x^16)/8
+ (5*a*b^3*(A*b + 2*a*B)*x^19)/19 + (b^4*(A*b + 5*a*B)*x^22)/22 + (b^5*B*x^25)/25

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Maple [A]  time = 0.002, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{25}}{25}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{22}}{22}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{19}}{19}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{16}}{16}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{13}}{13}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{10}}{10}}+{\frac{{a}^{5}A{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/25*b^5*B*x^25+1/22*(A*b^5+5*B*a*b^4)*x^22+1/19*(5*A*a*b^4+10*B*a^2*b^3)*x^19+1/16*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^16+1/13*(10*A*a^3*b^2+5*B*a^4*b)*x^13+1/10*(5*A*a^4*b+B*a^5)*x^10+1/7*a^5*A*x^7

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Maxima [A]  time = 1.08949, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{25} \, B b^{5} x^{25} + \frac{1}{22} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{22} + \frac{5}{19} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{19} + \frac{5}{8} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac{5}{13} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac{1}{7} \, A a^{5} x^{7} + \frac{1}{10} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")

[Out]

1/25*B*b^5*x^25 + 1/22*(5*B*a*b^4 + A*b^5)*x^22 + 5/19*(2*B*a^2*b^3 + A*a*b^4)*x^19 + 5/8*(B*a^3*b^2 + A*a^2*b
^3)*x^16 + 5/13*(B*a^4*b + 2*A*a^3*b^2)*x^13 + 1/7*A*a^5*x^7 + 1/10*(B*a^5 + 5*A*a^4*b)*x^10

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Fricas [A]  time = 1.22717, size = 319, normalized size = 2.73 \begin{align*} \frac{1}{25} x^{25} b^{5} B + \frac{5}{22} x^{22} b^{4} a B + \frac{1}{22} x^{22} b^{5} A + \frac{10}{19} x^{19} b^{3} a^{2} B + \frac{5}{19} x^{19} b^{4} a A + \frac{5}{8} x^{16} b^{2} a^{3} B + \frac{5}{8} x^{16} b^{3} a^{2} A + \frac{5}{13} x^{13} b a^{4} B + \frac{10}{13} x^{13} b^{2} a^{3} A + \frac{1}{10} x^{10} a^{5} B + \frac{1}{2} x^{10} b a^{4} A + \frac{1}{7} x^{7} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")

[Out]

1/25*x^25*b^5*B + 5/22*x^22*b^4*a*B + 1/22*x^22*b^5*A + 10/19*x^19*b^3*a^2*B + 5/19*x^19*b^4*a*A + 5/8*x^16*b^
2*a^3*B + 5/8*x^16*b^3*a^2*A + 5/13*x^13*b*a^4*B + 10/13*x^13*b^2*a^3*A + 1/10*x^10*a^5*B + 1/2*x^10*b*a^4*A +
1/7*x^7*a^5*A

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Sympy [A]  time = 0.084589, size = 136, normalized size = 1.16 \begin{align*} \frac{A a^{5} x^{7}}{7} + \frac{B b^{5} x^{25}}{25} + x^{22} \left (\frac{A b^{5}}{22} + \frac{5 B a b^{4}}{22}\right ) + x^{19} \left (\frac{5 A a b^{4}}{19} + \frac{10 B a^{2} b^{3}}{19}\right ) + x^{16} \left (\frac{5 A a^{2} b^{3}}{8} + \frac{5 B a^{3} b^{2}}{8}\right ) + x^{13} \left (\frac{10 A a^{3} b^{2}}{13} + \frac{5 B a^{4} b}{13}\right ) + x^{10} \left (\frac{A a^{4} b}{2} + \frac{B a^{5}}{10}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**7/7 + B*b**5*x**25/25 + x**22*(A*b**5/22 + 5*B*a*b**4/22) + x**19*(5*A*a*b**4/19 + 10*B*a**2*b**3/19
) + x**16*(5*A*a**2*b**3/8 + 5*B*a**3*b**2/8) + x**13*(10*A*a**3*b**2/13 + 5*B*a**4*b/13) + x**10*(A*a**4*b/2
+ B*a**5/10)

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Giac [A]  time = 1.44056, size = 169, normalized size = 1.44 \begin{align*} \frac{1}{25} \, B b^{5} x^{25} + \frac{5}{22} \, B a b^{4} x^{22} + \frac{1}{22} \, A b^{5} x^{22} + \frac{10}{19} \, B a^{2} b^{3} x^{19} + \frac{5}{19} \, A a b^{4} x^{19} + \frac{5}{8} \, B a^{3} b^{2} x^{16} + \frac{5}{8} \, A a^{2} b^{3} x^{16} + \frac{5}{13} \, B a^{4} b x^{13} + \frac{10}{13} \, A a^{3} b^{2} x^{13} + \frac{1}{10} \, B a^{5} x^{10} + \frac{1}{2} \, A a^{4} b x^{10} + \frac{1}{7} \, A a^{5} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")

[Out]

1/25*B*b^5*x^25 + 5/22*B*a*b^4*x^22 + 1/22*A*b^5*x^22 + 10/19*B*a^2*b^3*x^19 + 5/19*A*a*b^4*x^19 + 5/8*B*a^3*b
^2*x^16 + 5/8*A*a^2*b^3*x^16 + 5/13*B*a^4*b*x^13 + 10/13*A*a^3*b^2*x^13 + 1/10*B*a^5*x^10 + 1/2*A*a^4*b*x^10 +
1/7*A*a^5*x^7