3.28 \(\int x^4 (a+b x^2)^5 (A+B x^2) \, dx\)

Optimal. Leaf size=117 \[ \frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{5}{9} a^3 b x^9 (a B+2 A b)+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{5} a^5 A x^5+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{17} b^5 B x^{17} \]

[Out]

(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^9)/9 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 +
(5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^17)/17

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Rubi [A]  time = 0.0701615, 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} \[ \frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{5}{9} a^3 b x^9 (a B+2 A b)+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{5} a^5 A x^5+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{17} b^5 B x^{17} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^9)/9 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 +
(5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^17)/17

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^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\int \left (a^5 A x^4+a^4 (5 A b+a B) x^6+5 a^3 b (2 A b+a B) x^8+10 a^2 b^2 (A b+a B) x^{10}+5 a b^3 (A b+2 a B) x^{12}+b^4 (A b+5 a B) x^{14}+b^5 B x^{16}\right ) \, dx\\ &=\frac{1}{5} a^5 A x^5+\frac{1}{7} a^4 (5 A b+a B) x^7+\frac{5}{9} a^3 b (2 A b+a B) x^9+\frac{10}{11} a^2 b^2 (A b+a B) x^{11}+\frac{5}{13} a b^3 (A b+2 a B) x^{13}+\frac{1}{15} b^4 (A b+5 a B) x^{15}+\frac{1}{17} b^5 B x^{17}\\ \end{align*}

Mathematica [A]  time = 0.0152669, size = 117, normalized size = 1. \[ \frac{10}{11} a^2 b^2 x^{11} (a B+A b)+\frac{5}{9} a^3 b x^9 (a B+2 A b)+\frac{1}{7} a^4 x^7 (a B+5 A b)+\frac{1}{5} a^5 A x^5+\frac{1}{15} b^4 x^{15} (5 a B+A b)+\frac{5}{13} a b^3 x^{13} (2 a B+A b)+\frac{1}{17} b^5 B x^{17} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^9)/9 + (10*a^2*b^2*(A*b + a*B)*x^11)/11 +
(5*a*b^3*(A*b + 2*a*B)*x^13)/13 + (b^4*(A*b + 5*a*B)*x^15)/15 + (b^5*B*x^17)/17

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Maple [A]  time = 0.001, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{17}}{17}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{15}}{15}}+{\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}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{9}}{9}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{7}}{7}}+{\frac{{a}^{5}A{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^5*(B*x^2+A),x)

[Out]

1/17*b^5*B*x^17+1/15*(A*b^5+5*B*a*b^4)*x^15+1/13*(5*A*a*b^4+10*B*a^2*b^3)*x^13+1/11*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^11+1/9*(10*A*a^3*b^2+5*B*a^4*b)*x^9+1/7*(5*A*a^4*b+B*a^5)*x^7+1/5*a^5*A*x^5

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Maxima [A]  time = 1.01344, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{17} \, B b^{5} x^{17} + \frac{1}{15} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac{5}{13} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{13} + \frac{10}{11} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{11} + \frac{1}{5} \, A a^{5} x^{5} + \frac{5}{9} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{9} + \frac{1}{7} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")

[Out]

1/17*B*b^5*x^17 + 1/15*(5*B*a*b^4 + A*b^5)*x^15 + 5/13*(2*B*a^2*b^3 + A*a*b^4)*x^13 + 10/11*(B*a^3*b^2 + A*a^2
*b^3)*x^11 + 1/5*A*a^5*x^5 + 5/9*(B*a^4*b + 2*A*a^3*b^2)*x^9 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7

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Fricas [A]  time = 1.28002, size = 313, normalized size = 2.68 \begin{align*} \frac{1}{17} x^{17} b^{5} B + \frac{1}{3} x^{15} b^{4} a B + \frac{1}{15} x^{15} b^{5} A + \frac{10}{13} x^{13} b^{3} a^{2} B + \frac{5}{13} x^{13} b^{4} a A + \frac{10}{11} x^{11} b^{2} a^{3} B + \frac{10}{11} x^{11} b^{3} a^{2} A + \frac{5}{9} x^{9} b a^{4} B + \frac{10}{9} x^{9} b^{2} a^{3} A + \frac{1}{7} x^{7} a^{5} B + \frac{5}{7} x^{7} b a^{4} A + \frac{1}{5} x^{5} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^5*(B*x^2+A),x, algorithm="fricas")

[Out]

1/17*x^17*b^5*B + 1/3*x^15*b^4*a*B + 1/15*x^15*b^5*A + 10/13*x^13*b^3*a^2*B + 5/13*x^13*b^4*a*A + 10/11*x^11*b
^2*a^3*B + 10/11*x^11*b^3*a^2*A + 5/9*x^9*b*a^4*B + 10/9*x^9*b^2*a^3*A + 1/7*x^7*a^5*B + 5/7*x^7*b*a^4*A + 1/5
*x^5*a^5*A

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Sympy [A]  time = 0.083354, size = 136, normalized size = 1.16 \begin{align*} \frac{A a^{5} x^{5}}{5} + \frac{B b^{5} x^{17}}{17} + x^{15} \left (\frac{A b^{5}}{15} + \frac{B a b^{4}}{3}\right ) + x^{13} \left (\frac{5 A a b^{4}}{13} + \frac{10 B a^{2} b^{3}}{13}\right ) + x^{11} \left (\frac{10 A a^{2} b^{3}}{11} + \frac{10 B a^{3} b^{2}}{11}\right ) + x^{9} \left (\frac{10 A a^{3} b^{2}}{9} + \frac{5 B a^{4} b}{9}\right ) + x^{7} \left (\frac{5 A a^{4} b}{7} + \frac{B a^{5}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**5/5 + B*b**5*x**17/17 + x**15*(A*b**5/15 + B*a*b**4/3) + x**13*(5*A*a*b**4/13 + 10*B*a**2*b**3/13) +
x**11*(10*A*a**2*b**3/11 + 10*B*a**3*b**2/11) + x**9*(10*A*a**3*b**2/9 + 5*B*a**4*b/9) + x**7*(5*A*a**4*b/7 +
B*a**5/7)

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Giac [A]  time = 1.13461, size = 169, normalized size = 1.44 \begin{align*} \frac{1}{17} \, B b^{5} x^{17} + \frac{1}{3} \, B a b^{4} x^{15} + \frac{1}{15} \, A b^{5} x^{15} + \frac{10}{13} \, B a^{2} b^{3} x^{13} + \frac{5}{13} \, A a b^{4} x^{13} + \frac{10}{11} \, B a^{3} b^{2} x^{11} + \frac{10}{11} \, A a^{2} b^{3} x^{11} + \frac{5}{9} \, B a^{4} b x^{9} + \frac{10}{9} \, A a^{3} b^{2} x^{9} + \frac{1}{7} \, B a^{5} x^{7} + \frac{5}{7} \, A a^{4} b x^{7} + \frac{1}{5} \, A a^{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^5*(B*x^2+A),x, algorithm="giac")

[Out]

1/17*B*b^5*x^17 + 1/3*B*a*b^4*x^15 + 1/15*A*b^5*x^15 + 10/13*B*a^2*b^3*x^13 + 5/13*A*a*b^4*x^13 + 10/11*B*a^3*
b^2*x^11 + 10/11*A*a^2*b^3*x^11 + 5/9*B*a^4*b*x^9 + 10/9*A*a^3*b^2*x^9 + 1/7*B*a^5*x^7 + 5/7*A*a^4*b*x^7 + 1/5
*A*a^5*x^5