3.25 \(\int x^7 (a+b x^2)^5 (A+B x^2) \, dx\)

Optimal. Leaf size=122 \[ \frac{a^2 \left (a+b x^2\right )^7 (3 A b-4 a B)}{14 b^5}-\frac{a^3 \left (a+b x^2\right )^6 (A b-a B)}{12 b^5}+\frac{\left (a+b x^2\right )^9 (A b-4 a B)}{18 b^5}-\frac{3 a \left (a+b x^2\right )^8 (A b-2 a B)}{16 b^5}+\frac{B \left (a+b x^2\right )^{10}}{20 b^5} \]

[Out]

-(a^3*(A*b - a*B)*(a + b*x^2)^6)/(12*b^5) + (a^2*(3*A*b - 4*a*B)*(a + b*x^2)^7)/(14*b^5) - (3*a*(A*b - 2*a*B)*
(a + b*x^2)^8)/(16*b^5) + ((A*b - 4*a*B)*(a + b*x^2)^9)/(18*b^5) + (B*(a + b*x^2)^10)/(20*b^5)

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Rubi [A]  time = 0.280026, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ \frac{a^2 \left (a+b x^2\right )^7 (3 A b-4 a B)}{14 b^5}-\frac{a^3 \left (a+b x^2\right )^6 (A b-a B)}{12 b^5}+\frac{\left (a+b x^2\right )^9 (A b-4 a B)}{18 b^5}-\frac{3 a \left (a+b x^2\right )^8 (A b-2 a B)}{16 b^5}+\frac{B \left (a+b x^2\right )^{10}}{20 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

-(a^3*(A*b - a*B)*(a + b*x^2)^6)/(12*b^5) + (a^2*(3*A*b - 4*a*B)*(a + b*x^2)^7)/(14*b^5) - (3*a*(A*b - 2*a*B)*
(a + b*x^2)^8)/(16*b^5) + ((A*b - 4*a*B)*(a + b*x^2)^9)/(18*b^5) + (B*(a + b*x^2)^10)/(20*b^5)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^7 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^3 (-A b+a B) (a+b x)^5}{b^4}-\frac{a^2 (-3 A b+4 a B) (a+b x)^6}{b^4}+\frac{3 a (-A b+2 a B) (a+b x)^7}{b^4}+\frac{(A b-4 a B) (a+b x)^8}{b^4}+\frac{B (a+b x)^9}{b^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 (A b-a B) \left (a+b x^2\right )^6}{12 b^5}+\frac{a^2 (3 A b-4 a B) \left (a+b x^2\right )^7}{14 b^5}-\frac{3 a (A b-2 a B) \left (a+b x^2\right )^8}{16 b^5}+\frac{(A b-4 a B) \left (a+b x^2\right )^9}{18 b^5}+\frac{B \left (a+b x^2\right )^{10}}{20 b^5}\\ \end{align*}

Mathematica [A]  time = 0.014543, size = 117, normalized size = 0.96 \[ \frac{5}{7} a^2 b^2 x^{14} (a B+A b)+\frac{5}{12} a^3 b x^{12} (a B+2 A b)+\frac{1}{10} a^4 x^{10} (a B+5 A b)+\frac{1}{8} a^5 A x^8+\frac{1}{18} b^4 x^{18} (5 a B+A b)+\frac{5}{16} a b^3 x^{16} (2 a B+A b)+\frac{1}{20} b^5 B x^{20} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^12)/12 + (5*a^2*b^2*(A*b + a*B)*x^14)/7
+ (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4*(A*b + 5*a*B)*x^18)/18 + (b^5*B*x^20)/20

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Maple [A]  time = 0.001, size = 124, normalized size = 1. \begin{align*}{\frac{{b}^{5}B{x}^{20}}{20}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{18}}{18}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{16}}{16}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{14}}{14}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{12}}{12}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{10}}{10}}+{\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^2+a)^5*(B*x^2+A),x)

[Out]

1/20*b^5*B*x^20+1/18*(A*b^5+5*B*a*b^4)*x^18+1/16*(5*A*a*b^4+10*B*a^2*b^3)*x^16+1/14*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^14+1/12*(10*A*a^3*b^2+5*B*a^4*b)*x^12+1/10*(5*A*a^4*b+B*a^5)*x^10+1/8*a^5*A*x^8

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Maxima [A]  time = 0.993744, size = 161, normalized size = 1.32 \begin{align*} \frac{1}{20} \, B b^{5} x^{20} + \frac{1}{18} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{18} + \frac{5}{16} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{16} + \frac{5}{7} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{14} + \frac{1}{8} \, A a^{5} x^{8} + \frac{5}{12} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{12} + \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^7*(b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")

[Out]

1/20*B*b^5*x^20 + 1/18*(5*B*a*b^4 + A*b^5)*x^18 + 5/16*(2*B*a^2*b^3 + A*a*b^4)*x^16 + 5/7*(B*a^3*b^2 + A*a^2*b
^3)*x^14 + 1/8*A*a^5*x^8 + 5/12*(B*a^4*b + 2*A*a^3*b^2)*x^12 + 1/10*(B*a^5 + 5*A*a^4*b)*x^10

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Fricas [A]  time = 1.29078, size = 313, normalized size = 2.57 \begin{align*} \frac{1}{20} x^{20} b^{5} B + \frac{5}{18} x^{18} b^{4} a B + \frac{1}{18} x^{18} b^{5} A + \frac{5}{8} x^{16} b^{3} a^{2} B + \frac{5}{16} x^{16} b^{4} a A + \frac{5}{7} x^{14} b^{2} a^{3} B + \frac{5}{7} x^{14} b^{3} a^{2} A + \frac{5}{12} x^{12} b a^{4} B + \frac{5}{6} x^{12} b^{2} a^{3} A + \frac{1}{10} x^{10} a^{5} B + \frac{1}{2} x^{10} 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^2+a)^5*(B*x^2+A),x, algorithm="fricas")

[Out]

1/20*x^20*b^5*B + 5/18*x^18*b^4*a*B + 1/18*x^18*b^5*A + 5/8*x^16*b^3*a^2*B + 5/16*x^16*b^4*a*A + 5/7*x^14*b^2*
a^3*B + 5/7*x^14*b^3*a^2*A + 5/12*x^12*b*a^4*B + 5/6*x^12*b^2*a^3*A + 1/10*x^10*a^5*B + 1/2*x^10*b*a^4*A + 1/8
*x^8*a^5*A

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Sympy [A]  time = 0.083165, size = 136, normalized size = 1.11 \begin{align*} \frac{A a^{5} x^{8}}{8} + \frac{B b^{5} x^{20}}{20} + x^{18} \left (\frac{A b^{5}}{18} + \frac{5 B a b^{4}}{18}\right ) + x^{16} \left (\frac{5 A a b^{4}}{16} + \frac{5 B a^{2} b^{3}}{8}\right ) + x^{14} \left (\frac{5 A a^{2} b^{3}}{7} + \frac{5 B a^{3} b^{2}}{7}\right ) + x^{12} \left (\frac{5 A a^{3} b^{2}}{6} + \frac{5 B a^{4} b}{12}\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**7*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**8/8 + B*b**5*x**20/20 + x**18*(A*b**5/18 + 5*B*a*b**4/18) + x**16*(5*A*a*b**4/16 + 5*B*a**2*b**3/8)
+ x**14*(5*A*a**2*b**3/7 + 5*B*a**3*b**2/7) + x**12*(5*A*a**3*b**2/6 + 5*B*a**4*b/12) + x**10*(A*a**4*b/2 + B*
a**5/10)

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Giac [A]  time = 1.26542, size = 169, normalized size = 1.39 \begin{align*} \frac{1}{20} \, B b^{5} x^{20} + \frac{5}{18} \, B a b^{4} x^{18} + \frac{1}{18} \, A b^{5} x^{18} + \frac{5}{8} \, B a^{2} b^{3} x^{16} + \frac{5}{16} \, A a b^{4} x^{16} + \frac{5}{7} \, B a^{3} b^{2} x^{14} + \frac{5}{7} \, A a^{2} b^{3} x^{14} + \frac{5}{12} \, B a^{4} b x^{12} + \frac{5}{6} \, A a^{3} b^{2} x^{12} + \frac{1}{10} \, B a^{5} x^{10} + \frac{1}{2} \, A a^{4} b x^{10} + \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^2+a)^5*(B*x^2+A),x, algorithm="giac")

[Out]

1/20*B*b^5*x^20 + 5/18*B*a*b^4*x^18 + 1/18*A*b^5*x^18 + 5/8*B*a^2*b^3*x^16 + 5/16*A*a*b^4*x^16 + 5/7*B*a^3*b^2
*x^14 + 5/7*A*a^2*b^3*x^14 + 5/12*B*a^4*b*x^12 + 5/6*A*a^3*b^2*x^12 + 1/10*B*a^5*x^10 + 1/2*A*a^4*b*x^10 + 1/8
*A*a^5*x^8