3.53 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{21}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{5 a^2 b^2 (a B+A b)}{7 x^{14}}-\frac{a^4 (a B+5 A b)}{18 x^{18}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{a^5 A}{20 x^{20}}-\frac{5 a b^3 (2 a B+A b)}{12 x^{12}}-\frac{b^4 (5 a B+A b)}{10 x^{10}}-\frac{b^5 B}{8 x^8} \]

[Out]

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

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Rubi [A]  time = 0.0825582, antiderivative size = 117, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{7 x^{14}}-\frac{a^4 (a B+5 A b)}{18 x^{18}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{a^5 A}{20 x^{20}}-\frac{5 a b^3 (2 a B+A b)}{12 x^{12}}-\frac{b^4 (5 a B+A b)}{10 x^{10}}-\frac{b^5 B}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^21,x]

[Out]

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

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 \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{21}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^{11}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^5 A}{x^{11}}+\frac{a^4 (5 A b+a B)}{x^{10}}+\frac{5 a^3 b (2 A b+a B)}{x^9}+\frac{10 a^2 b^2 (A b+a B)}{x^8}+\frac{5 a b^3 (A b+2 a B)}{x^7}+\frac{b^4 (A b+5 a B)}{x^6}+\frac{b^5 B}{x^5}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 A}{20 x^{20}}-\frac{a^4 (5 A b+a B)}{18 x^{18}}-\frac{5 a^3 b (2 A b+a B)}{16 x^{16}}-\frac{5 a^2 b^2 (A b+a B)}{7 x^{14}}-\frac{5 a b^3 (A b+2 a B)}{12 x^{12}}-\frac{b^4 (A b+5 a B)}{10 x^{10}}-\frac{b^5 B}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0312151, size = 121, normalized size = 1.03 \[ -\frac{600 a^2 b^3 x^6 \left (6 A+7 B x^2\right )+450 a^3 b^2 x^4 \left (7 A+8 B x^2\right )+175 a^4 b x^2 \left (8 A+9 B x^2\right )+28 a^5 \left (9 A+10 B x^2\right )+420 a b^4 x^8 \left (5 A+6 B x^2\right )+126 b^5 x^{10} \left (4 A+5 B x^2\right )}{5040 x^{20}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^21,x]

[Out]

-(126*b^5*x^10*(4*A + 5*B*x^2) + 420*a*b^4*x^8*(5*A + 6*B*x^2) + 600*a^2*b^3*x^6*(6*A + 7*B*x^2) + 450*a^3*b^2
*x^4*(7*A + 8*B*x^2) + 175*a^4*b*x^2*(8*A + 9*B*x^2) + 28*a^5*(9*A + 10*B*x^2))/(5040*x^20)

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Maple [A]  time = 0.007, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{20\,{x}^{20}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{18\,{x}^{18}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{16\,{x}^{16}}}-{\frac{5\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{7\,{x}^{14}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{12\,{x}^{12}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{10\,{x}^{10}}}-{\frac{B{b}^{5}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^21,x)

[Out]

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

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Maxima [A]  time = 1.02775, size = 163, normalized size = 1.39 \begin{align*} -\frac{630 \, B b^{5} x^{12} + 504 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 2100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 3600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 252 \, A a^{5} + 1575 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 280 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{5040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^21,x, algorithm="maxima")

[Out]

-1/5040*(630*B*b^5*x^12 + 504*(5*B*a*b^4 + A*b^5)*x^10 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 3600*(B*a^3*b^2 +
A*a^2*b^3)*x^6 + 252*A*a^5 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 280*(B*a^5 + 5*A*a^4*b)*x^2)/x^20

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Fricas [A]  time = 1.39844, size = 282, normalized size = 2.41 \begin{align*} -\frac{630 \, B b^{5} x^{12} + 504 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 2100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 3600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 252 \, A a^{5} + 1575 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 280 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{5040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^21,x, algorithm="fricas")

[Out]

-1/5040*(630*B*b^5*x^12 + 504*(5*B*a*b^4 + A*b^5)*x^10 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 3600*(B*a^3*b^2 +
A*a^2*b^3)*x^6 + 252*A*a^5 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 280*(B*a^5 + 5*A*a^4*b)*x^2)/x^20

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**21,x)

[Out]

Timed out

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Giac [A]  time = 1.11512, size = 171, normalized size = 1.46 \begin{align*} -\frac{630 \, B b^{5} x^{12} + 2520 \, B a b^{4} x^{10} + 504 \, A b^{5} x^{10} + 4200 \, B a^{2} b^{3} x^{8} + 2100 \, A a b^{4} x^{8} + 3600 \, B a^{3} b^{2} x^{6} + 3600 \, A a^{2} b^{3} x^{6} + 1575 \, B a^{4} b x^{4} + 3150 \, A a^{3} b^{2} x^{4} + 280 \, B a^{5} x^{2} + 1400 \, A a^{4} b x^{2} + 252 \, A a^{5}}{5040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^21,x, algorithm="giac")

[Out]

-1/5040*(630*B*b^5*x^12 + 2520*B*a*b^4*x^10 + 504*A*b^5*x^10 + 4200*B*a^2*b^3*x^8 + 2100*A*a*b^4*x^8 + 3600*B*
a^3*b^2*x^6 + 3600*A*a^2*b^3*x^6 + 1575*B*a^4*b*x^4 + 3150*A*a^3*b^2*x^4 + 280*B*a^5*x^2 + 1400*A*a^4*b*x^2 +
252*A*a^5)/x^20