3.25 \(\int x^7 (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=117 \[ \frac{10}{17} a^2 b^2 x^{17} (a B+A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{1}{8} a^5 A x^8+\frac{1}{23} b^4 x^{23} (5 a B+A b)+\frac{1}{4} a b^3 x^{20} (2 a B+A b)+\frac{1}{26} b^5 B x^{26} \]

[Out]

(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (10*a^2*b^2*(A*b + a*B)*x^17)/
17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26

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Rubi [A]  time = 0.0749274, 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} \[ \frac{10}{17} a^2 b^2 x^{17} (a B+A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{1}{8} a^5 A x^8+\frac{1}{23} b^4 x^{23} (5 a B+A b)+\frac{1}{4} a b^3 x^{20} (2 a B+A b)+\frac{1}{26} b^5 B x^{26} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (10*a^2*b^2*(A*b + a*B)*x^17)/
17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26

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^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\int \left (a^5 A x^7+a^4 (5 A b+a B) x^{10}+5 a^3 b (2 A b+a B) x^{13}+10 a^2 b^2 (A b+a B) x^{16}+5 a b^3 (A b+2 a B) x^{19}+b^4 (A b+5 a B) x^{22}+b^5 B x^{25}\right ) \, dx\\ &=\frac{1}{8} a^5 A x^8+\frac{1}{11} a^4 (5 A b+a B) x^{11}+\frac{5}{14} a^3 b (2 A b+a B) x^{14}+\frac{10}{17} a^2 b^2 (A b+a B) x^{17}+\frac{1}{4} a b^3 (A b+2 a B) x^{20}+\frac{1}{23} b^4 (A b+5 a B) x^{23}+\frac{1}{26} b^5 B x^{26}\\ \end{align*}

Mathematica [A]  time = 0.0180576, size = 117, normalized size = 1. \[ \frac{10}{17} a^2 b^2 x^{17} (a B+A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{1}{8} a^5 A x^8+\frac{1}{23} b^4 x^{23} (5 a B+A b)+\frac{1}{4} a b^3 x^{20} (2 a B+A b)+\frac{1}{26} b^5 B x^{26} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (10*a^2*b^2*(A*b + a*B)*x^17)/
17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26

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Maple [A]  time = 0.002, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{26}}{26}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{23}}{23}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{20}}{20}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{17}}{17}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{14}}{14}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{11}}{11}}+{\frac{{a}^{5}A{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/26*b^5*B*x^26+1/23*(A*b^5+5*B*a*b^4)*x^23+1/20*(5*A*a*b^4+10*B*a^2*b^3)*x^20+1/17*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^17+1/14*(10*A*a^3*b^2+5*B*a^4*b)*x^14+1/11*(5*A*a^4*b+B*a^5)*x^11+1/8*a^5*A*x^8

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Maxima [A]  time = 1.19773, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{26} \, B b^{5} x^{26} + \frac{1}{23} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{23} + \frac{1}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{20} + \frac{10}{17} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{17} + \frac{5}{14} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac{1}{8} \, A a^{5} x^{8} + \frac{1}{11} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")

[Out]

1/26*B*b^5*x^26 + 1/23*(5*B*a*b^4 + A*b^5)*x^23 + 1/4*(2*B*a^2*b^3 + A*a*b^4)*x^20 + 10/17*(B*a^3*b^2 + A*a^2*
b^3)*x^17 + 5/14*(B*a^4*b + 2*A*a^3*b^2)*x^14 + 1/8*A*a^5*x^8 + 1/11*(B*a^5 + 5*A*a^4*b)*x^11

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Fricas [A]  time = 1.27853, size = 319, normalized size = 2.73 \begin{align*} \frac{1}{26} x^{26} b^{5} B + \frac{5}{23} x^{23} b^{4} a B + \frac{1}{23} x^{23} b^{5} A + \frac{1}{2} x^{20} b^{3} a^{2} B + \frac{1}{4} x^{20} b^{4} a A + \frac{10}{17} x^{17} b^{2} a^{3} B + \frac{10}{17} x^{17} b^{3} a^{2} A + \frac{5}{14} x^{14} b a^{4} B + \frac{5}{7} x^{14} b^{2} a^{3} A + \frac{1}{11} x^{11} a^{5} B + \frac{5}{11} x^{11} b a^{4} A + \frac{1}{8} x^{8} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")

[Out]

1/26*x^26*b^5*B + 5/23*x^23*b^4*a*B + 1/23*x^23*b^5*A + 1/2*x^20*b^3*a^2*B + 1/4*x^20*b^4*a*A + 10/17*x^17*b^2
*a^3*B + 10/17*x^17*b^3*a^2*A + 5/14*x^14*b*a^4*B + 5/7*x^14*b^2*a^3*A + 1/11*x^11*a^5*B + 5/11*x^11*b*a^4*A +
1/8*x^8*a^5*A

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Sympy [A]  time = 0.085915, size = 134, normalized size = 1.15 \begin{align*} \frac{A a^{5} x^{8}}{8} + \frac{B b^{5} x^{26}}{26} + x^{23} \left (\frac{A b^{5}}{23} + \frac{5 B a b^{4}}{23}\right ) + x^{20} \left (\frac{A a b^{4}}{4} + \frac{B a^{2} b^{3}}{2}\right ) + x^{17} \left (\frac{10 A a^{2} b^{3}}{17} + \frac{10 B a^{3} b^{2}}{17}\right ) + x^{14} \left (\frac{5 A a^{3} b^{2}}{7} + \frac{5 B a^{4} b}{14}\right ) + x^{11} \left (\frac{5 A a^{4} b}{11} + \frac{B a^{5}}{11}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**8/8 + B*b**5*x**26/26 + x**23*(A*b**5/23 + 5*B*a*b**4/23) + x**20*(A*a*b**4/4 + B*a**2*b**3/2) + x**
17*(10*A*a**2*b**3/17 + 10*B*a**3*b**2/17) + x**14*(5*A*a**3*b**2/7 + 5*B*a**4*b/14) + x**11*(5*A*a**4*b/11 +
B*a**5/11)

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Giac [A]  time = 1.20776, size = 169, normalized size = 1.44 \begin{align*} \frac{1}{26} \, B b^{5} x^{26} + \frac{5}{23} \, B a b^{4} x^{23} + \frac{1}{23} \, A b^{5} x^{23} + \frac{1}{2} \, B a^{2} b^{3} x^{20} + \frac{1}{4} \, A a b^{4} x^{20} + \frac{10}{17} \, B a^{3} b^{2} x^{17} + \frac{10}{17} \, A a^{2} b^{3} x^{17} + \frac{5}{14} \, B a^{4} b x^{14} + \frac{5}{7} \, A a^{3} b^{2} x^{14} + \frac{1}{11} \, B a^{5} x^{11} + \frac{5}{11} \, A a^{4} b x^{11} + \frac{1}{8} \, A a^{5} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")

[Out]

1/26*B*b^5*x^26 + 5/23*B*a*b^4*x^23 + 1/23*A*b^5*x^23 + 1/2*B*a^2*b^3*x^20 + 1/4*A*a*b^4*x^20 + 10/17*B*a^3*b^
2*x^17 + 10/17*A*a^2*b^3*x^17 + 5/14*B*a^4*b*x^14 + 5/7*A*a^3*b^2*x^14 + 1/11*B*a^5*x^11 + 5/11*A*a^4*b*x^11 +
1/8*A*a^5*x^8