3.51 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{19}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{a^5 A}{18 x^{18}}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^4 (5 a B+A b)}{8 x^8}-\frac{b^5 B}{6 x^6} \]

[Out]

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

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Rubi [A]  time = 0.0838325, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{a^5 A}{18 x^{18}}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^4 (5 a B+A b)}{8 x^8}-\frac{b^5 B}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^19,x]

[Out]

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

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^{19}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^5 A}{x^{10}}+\frac{a^4 (5 A b+a B)}{x^9}+\frac{5 a^3 b (2 A b+a B)}{x^8}+\frac{10 a^2 b^2 (A b+a B)}{x^7}+\frac{5 a b^3 (A b+2 a B)}{x^6}+\frac{b^4 (A b+5 a B)}{x^5}+\frac{b^5 B}{x^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 A}{18 x^{18}}-\frac{a^4 (5 A b+a B)}{16 x^{16}}-\frac{5 a^3 b (2 A b+a B)}{14 x^{14}}-\frac{5 a^2 b^2 (A b+a B)}{6 x^{12}}-\frac{a b^3 (A b+2 a B)}{2 x^{10}}-\frac{b^4 (A b+5 a B)}{8 x^8}-\frac{b^5 B}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0293235, size = 121, normalized size = 1.03 \[ -\frac{168 a^2 b^3 x^6 \left (5 A+6 B x^2\right )+120 a^3 b^2 x^4 \left (6 A+7 B x^2\right )+45 a^4 b x^2 \left (7 A+8 B x^2\right )+7 a^5 \left (8 A+9 B x^2\right )+126 a b^4 x^8 \left (4 A+5 B x^2\right )+42 b^5 x^{10} \left (3 A+4 B x^2\right )}{1008 x^{18}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^19,x]

[Out]

-(42*b^5*x^10*(3*A + 4*B*x^2) + 126*a*b^4*x^8*(4*A + 5*B*x^2) + 168*a^2*b^3*x^6*(5*A + 6*B*x^2) + 120*a^3*b^2*
x^4*(6*A + 7*B*x^2) + 45*a^4*b*x^2*(7*A + 8*B*x^2) + 7*a^5*(8*A + 9*B*x^2))/(1008*x^18)

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Maple [A]  time = 0.006, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{18\,{x}^{18}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{16\,{x}^{16}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{5\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{6\,{x}^{12}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{2\,{x}^{10}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{8\,{x}^{8}}}-{\frac{B{b}^{5}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^19,x)

[Out]

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

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Maxima [A]  time = 0.994185, size = 163, normalized size = 1.39 \begin{align*} -\frac{168 \, B b^{5} x^{12} + 126 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^19,x, algorithm="maxima")

[Out]

-1/1008*(168*B*b^5*x^12 + 126*(5*B*a*b^4 + A*b^5)*x^10 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 840*(B*a^3*b^2 + A*
a^2*b^3)*x^6 + 56*A*a^5 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^18

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Fricas [A]  time = 1.45371, size = 275, normalized size = 2.35 \begin{align*} -\frac{168 \, B b^{5} x^{12} + 126 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^19,x, algorithm="fricas")

[Out]

-1/1008*(168*B*b^5*x^12 + 126*(5*B*a*b^4 + A*b^5)*x^10 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 840*(B*a^3*b^2 + A*
a^2*b^3)*x^6 + 56*A*a^5 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^18

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Sympy [A]  time = 111.735, size = 128, normalized size = 1.09 \begin{align*} - \frac{56 A a^{5} + 168 B b^{5} x^{12} + x^{10} \left (126 A b^{5} + 630 B a b^{4}\right ) + x^{8} \left (504 A a b^{4} + 1008 B a^{2} b^{3}\right ) + x^{6} \left (840 A a^{2} b^{3} + 840 B a^{3} b^{2}\right ) + x^{4} \left (720 A a^{3} b^{2} + 360 B a^{4} b\right ) + x^{2} \left (315 A a^{4} b + 63 B a^{5}\right )}{1008 x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**19,x)

[Out]

-(56*A*a**5 + 168*B*b**5*x**12 + x**10*(126*A*b**5 + 630*B*a*b**4) + x**8*(504*A*a*b**4 + 1008*B*a**2*b**3) +
x**6*(840*A*a**2*b**3 + 840*B*a**3*b**2) + x**4*(720*A*a**3*b**2 + 360*B*a**4*b) + x**2*(315*A*a**4*b + 63*B*a
**5))/(1008*x**18)

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Giac [A]  time = 1.12449, size = 171, normalized size = 1.46 \begin{align*} -\frac{168 \, B b^{5} x^{12} + 630 \, B a b^{4} x^{10} + 126 \, A b^{5} x^{10} + 1008 \, B a^{2} b^{3} x^{8} + 504 \, A a b^{4} x^{8} + 840 \, B a^{3} b^{2} x^{6} + 840 \, A a^{2} b^{3} x^{6} + 360 \, B a^{4} b x^{4} + 720 \, A a^{3} b^{2} x^{4} + 63 \, B a^{5} x^{2} + 315 \, A a^{4} b x^{2} + 56 \, A a^{5}}{1008 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^19,x, algorithm="giac")

[Out]

-1/1008*(168*B*b^5*x^12 + 630*B*a*b^4*x^10 + 126*A*b^5*x^10 + 1008*B*a^2*b^3*x^8 + 504*A*a*b^4*x^8 + 840*B*a^3
*b^2*x^6 + 840*A*a^2*b^3*x^6 + 360*B*a^4*b*x^4 + 720*A*a^3*b^2*x^4 + 63*B*a^5*x^2 + 315*A*a^4*b*x^2 + 56*A*a^5
)/x^18