3.32 \(\int (a+b x^2)^5 (A+B x^2) \, dx\)

Optimal. Leaf size=109 \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+a^3 b x^5 (a B+2 A b)+\frac{1}{3} a^4 x^3 (a B+5 A b)+a^5 A x+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{13} b^5 B x^{13} \]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^3)/3 + a^3*b*(2*A*b + a*B)*x^5 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (5*a*b^3*(A*b
+ 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^13)/13

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Rubi [A]  time = 0.0600984, 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} \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+a^3 b x^5 (a B+2 A b)+\frac{1}{3} a^4 x^3 (a B+5 A b)+a^5 A x+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{13} b^5 B x^{13} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5*(A + B*x^2),x]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^3)/3 + a^3*b*(2*A*b + a*B)*x^5 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (5*a*b^3*(A*b
+ 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^13)/13

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^2\right )^5 \left (A+B x^2\right ) \, dx &=\int \left (a^5 A+a^4 (5 A b+a B) x^2+5 a^3 b (2 A b+a B) x^4+10 a^2 b^2 (A b+a B) x^6+5 a b^3 (A b+2 a B) x^8+b^4 (A b+5 a B) x^{10}+b^5 B x^{12}\right ) \, dx\\ &=a^5 A x+\frac{1}{3} a^4 (5 A b+a B) x^3+a^3 b (2 A b+a B) x^5+\frac{10}{7} a^2 b^2 (A b+a B) x^7+\frac{5}{9} a b^3 (A b+2 a B) x^9+\frac{1}{11} b^4 (A b+5 a B) x^{11}+\frac{1}{13} b^5 B x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0167549, size = 109, normalized size = 1. \[ \frac{10}{7} a^2 b^2 x^7 (a B+A b)+a^3 b x^5 (a B+2 A b)+\frac{1}{3} a^4 x^3 (a B+5 A b)+a^5 A x+\frac{1}{11} b^4 x^{11} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{13} b^5 B x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5*(A + B*x^2),x]

[Out]

a^5*A*x + (a^4*(5*A*b + a*B)*x^3)/3 + a^3*b*(2*A*b + a*B)*x^5 + (10*a^2*b^2*(A*b + a*B)*x^7)/7 + (5*a*b^3*(A*b
+ 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^13)/13

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Maple [A]  time = 0.001, size = 121, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{13}}{13}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{11}}{11}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{9}}{9}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{5}}{5}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{3}}{3}}+{a}^{5}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A),x)

[Out]

1/13*b^5*B*x^13+1/11*(A*b^5+5*B*a*b^4)*x^11+1/9*(5*A*a*b^4+10*B*a^2*b^3)*x^9+1/7*(10*A*a^2*b^3+10*B*a^3*b^2)*x
^7+1/5*(10*A*a^3*b^2+5*B*a^4*b)*x^5+1/3*(5*A*a^4*b+B*a^5)*x^3+a^5*A*x

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Maxima [A]  time = 0.975597, size = 155, normalized size = 1.42 \begin{align*} \frac{1}{13} \, B b^{5} x^{13} + \frac{1}{11} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac{5}{9} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac{10}{7} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + A a^{5} x +{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")

[Out]

1/13*B*b^5*x^13 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 5/9*(2*B*a^2*b^3 + A*a*b^4)*x^9 + 10/7*(B*a^3*b^2 + A*a^2*b^
3)*x^7 + A*a^5*x + (B*a^4*b + 2*A*a^3*b^2)*x^5 + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

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Fricas [A]  time = 1.28871, size = 286, normalized size = 2.62 \begin{align*} \frac{1}{13} x^{13} b^{5} B + \frac{5}{11} x^{11} b^{4} a B + \frac{1}{11} x^{11} b^{5} A + \frac{10}{9} x^{9} b^{3} a^{2} B + \frac{5}{9} x^{9} b^{4} a A + \frac{10}{7} x^{7} b^{2} a^{3} B + \frac{10}{7} x^{7} b^{3} a^{2} A + x^{5} b a^{4} B + 2 x^{5} b^{2} a^{3} A + \frac{1}{3} x^{3} a^{5} B + \frac{5}{3} x^{3} b a^{4} A + x a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A),x, algorithm="fricas")

[Out]

1/13*x^13*b^5*B + 5/11*x^11*b^4*a*B + 1/11*x^11*b^5*A + 10/9*x^9*b^3*a^2*B + 5/9*x^9*b^4*a*A + 10/7*x^7*b^2*a^
3*B + 10/7*x^7*b^3*a^2*A + x^5*b*a^4*B + 2*x^5*b^2*a^3*A + 1/3*x^3*a^5*B + 5/3*x^3*b*a^4*A + x*a^5*A

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Sympy [A]  time = 0.079743, size = 129, normalized size = 1.18 \begin{align*} A a^{5} x + \frac{B b^{5} x^{13}}{13} + x^{11} \left (\frac{A b^{5}}{11} + \frac{5 B a b^{4}}{11}\right ) + x^{9} \left (\frac{5 A a b^{4}}{9} + \frac{10 B a^{2} b^{3}}{9}\right ) + x^{7} \left (\frac{10 A a^{2} b^{3}}{7} + \frac{10 B a^{3} b^{2}}{7}\right ) + x^{5} \left (2 A a^{3} b^{2} + B a^{4} b\right ) + x^{3} \left (\frac{5 A a^{4} b}{3} + \frac{B a^{5}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x + B*b**5*x**13/13 + x**11*(A*b**5/11 + 5*B*a*b**4/11) + x**9*(5*A*a*b**4/9 + 10*B*a**2*b**3/9) + x**7
*(10*A*a**2*b**3/7 + 10*B*a**3*b**2/7) + x**5*(2*A*a**3*b**2 + B*a**4*b) + x**3*(5*A*a**4*b/3 + B*a**5/3)

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Giac [A]  time = 1.15942, size = 163, normalized size = 1.5 \begin{align*} \frac{1}{13} \, B b^{5} x^{13} + \frac{5}{11} \, B a b^{4} x^{11} + \frac{1}{11} \, A b^{5} x^{11} + \frac{10}{9} \, B a^{2} b^{3} x^{9} + \frac{5}{9} \, A a b^{4} x^{9} + \frac{10}{7} \, B a^{3} b^{2} x^{7} + \frac{10}{7} \, A a^{2} b^{3} x^{7} + B a^{4} b x^{5} + 2 \, A a^{3} b^{2} x^{5} + \frac{1}{3} \, B a^{5} x^{3} + \frac{5}{3} \, A a^{4} b x^{3} + A a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A),x, algorithm="giac")

[Out]

1/13*B*b^5*x^13 + 5/11*B*a*b^4*x^11 + 1/11*A*b^5*x^11 + 10/9*B*a^2*b^3*x^9 + 5/9*A*a*b^4*x^9 + 10/7*B*a^3*b^2*
x^7 + 10/7*A*a^2*b^3*x^7 + B*a^4*b*x^5 + 2*A*a^3*b^2*x^5 + 1/3*B*a^5*x^3 + 5/3*A*a^4*b*x^3 + A*a^5*x