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

Optimal. Leaf size=109 \[ a^2 b^2 x^{10} (a B+A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{1}{4} a^4 x^4 (a B+5 A b)+a^5 A x+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

[Out]

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

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Rubi [A]  time = 0.0554363, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {373} \[ a^2 b^2 x^{10} (a B+A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{1}{4} a^4 x^4 (a B+5 A b)+a^5 A x+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5*(A + B*x^3),x]

[Out]

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

Rule 373

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

Rubi steps

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

Mathematica [A]  time = 0.0148256, size = 109, normalized size = 1. \[ a^2 b^2 x^{10} (a B+A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{1}{4} a^4 x^4 (a B+5 A b)+a^5 A x+\frac{1}{16} b^4 x^{16} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 121, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{19}}{19}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{16}}{16}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{13}}{13}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{4}}{4}}+{a}^{5}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.31041, size = 155, normalized size = 1.42 \begin{align*} \frac{1}{19} \, B b^{5} x^{19} + \frac{1}{16} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac{5}{13} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{13} +{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{10} + \frac{5}{7} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + A a^{5} x + \frac{1}{4} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28977, size = 290, normalized size = 2.66 \begin{align*} \frac{1}{19} x^{19} b^{5} B + \frac{5}{16} x^{16} b^{4} a B + \frac{1}{16} x^{16} b^{5} A + \frac{10}{13} x^{13} b^{3} a^{2} B + \frac{5}{13} x^{13} b^{4} a A + x^{10} b^{2} a^{3} B + x^{10} b^{3} a^{2} A + \frac{5}{7} x^{7} b a^{4} B + \frac{10}{7} x^{7} b^{2} a^{3} A + \frac{1}{4} x^{4} a^{5} B + \frac{5}{4} x^{4} b a^{4} A + x a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.082786, size = 128, normalized size = 1.17 \begin{align*} A a^{5} x + \frac{B b^{5} x^{19}}{19} + x^{16} \left (\frac{A b^{5}}{16} + \frac{5 B a b^{4}}{16}\right ) + x^{13} \left (\frac{5 A a b^{4}}{13} + \frac{10 B a^{2} b^{3}}{13}\right ) + x^{10} \left (A a^{2} b^{3} + B a^{3} b^{2}\right ) + x^{7} \left (\frac{10 A a^{3} b^{2}}{7} + \frac{5 B a^{4} b}{7}\right ) + x^{4} \left (\frac{5 A a^{4} b}{4} + \frac{B a^{5}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19373, size = 162, normalized size = 1.49 \begin{align*} \frac{1}{19} \, B b^{5} x^{19} + \frac{5}{16} \, B a b^{4} x^{16} + \frac{1}{16} \, A b^{5} x^{16} + \frac{10}{13} \, B a^{2} b^{3} x^{13} + \frac{5}{13} \, A a b^{4} x^{13} + B a^{3} b^{2} x^{10} + A a^{2} b^{3} x^{10} + \frac{5}{7} \, B a^{4} b x^{7} + \frac{10}{7} \, A a^{3} b^{2} x^{7} + \frac{1}{4} \, B a^{5} x^{4} + \frac{5}{4} \, A a^{4} b x^{4} + A a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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