3.46 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^{14}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{a^5 A}{13 x^{13}}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{3 x^3}-\frac{b^5 B}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.062222, antiderivative size = 113, 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} \[ -\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{9 x^9}-\frac{a^5 A}{13 x^{13}}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{3 x^3}-\frac{b^5 B}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^14,x]

[Out]

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

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

Mathematica [A]  time = 0.0325218, size = 119, normalized size = 1.05 \[ -\frac{2574 a^2 b^3 x^6 \left (5 A+7 B x^2\right )+1430 a^3 b^2 x^4 \left (7 A+9 B x^2\right )+455 a^4 b x^2 \left (9 A+11 B x^2\right )+63 a^5 \left (11 A+13 B x^2\right )+3003 a b^4 x^8 \left (3 A+5 B x^2\right )+3003 b^5 x^{10} \left (A+3 B x^2\right )}{9009 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^14,x]

[Out]

-(3003*b^5*x^10*(A + 3*B*x^2) + 3003*a*b^4*x^8*(3*A + 5*B*x^2) + 2574*a^2*b^3*x^6*(5*A + 7*B*x^2) + 1430*a^3*b
^2*x^4*(7*A + 9*B*x^2) + 455*a^4*b*x^2*(9*A + 11*B*x^2) + 63*a^5*(11*A + 13*B*x^2))/(9009*x^13)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{13\,{x}^{13}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{9\,{x}^{9}}}-{\frac{10\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{{x}^{5}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{3\,{x}^{3}}}-{\frac{B{b}^{5}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^14,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00434, size = 163, normalized size = 1.44 \begin{align*} -\frac{9009 \, B b^{5} x^{12} + 3003 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9009 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 12870 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 693 \, A a^{5} + 5005 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 819 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{9009 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^14,x, algorithm="maxima")

[Out]

-1/9009*(9009*B*b^5*x^12 + 3003*(5*B*a*b^4 + A*b^5)*x^10 + 9009*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 12870*(B*a^3*b^2
+ A*a^2*b^3)*x^6 + 693*A*a^5 + 5005*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 819*(B*a^5 + 5*A*a^4*b)*x^2)/x^13

________________________________________________________________________________________

Fricas [A]  time = 1.43882, size = 286, normalized size = 2.53 \begin{align*} -\frac{9009 \, B b^{5} x^{12} + 3003 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 9009 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 12870 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 693 \, A a^{5} + 5005 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 819 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{9009 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^14,x, algorithm="fricas")

[Out]

-1/9009*(9009*B*b^5*x^12 + 3003*(5*B*a*b^4 + A*b^5)*x^10 + 9009*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 12870*(B*a^3*b^2
+ A*a^2*b^3)*x^6 + 693*A*a^5 + 5005*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 819*(B*a^5 + 5*A*a^4*b)*x^2)/x^13

________________________________________________________________________________________

Sympy [A]  time = 16.9007, size = 128, normalized size = 1.13 \begin{align*} - \frac{693 A a^{5} + 9009 B b^{5} x^{12} + x^{10} \left (3003 A b^{5} + 15015 B a b^{4}\right ) + x^{8} \left (9009 A a b^{4} + 18018 B a^{2} b^{3}\right ) + x^{6} \left (12870 A a^{2} b^{3} + 12870 B a^{3} b^{2}\right ) + x^{4} \left (10010 A a^{3} b^{2} + 5005 B a^{4} b\right ) + x^{2} \left (4095 A a^{4} b + 819 B a^{5}\right )}{9009 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**14,x)

[Out]

-(693*A*a**5 + 9009*B*b**5*x**12 + x**10*(3003*A*b**5 + 15015*B*a*b**4) + x**8*(9009*A*a*b**4 + 18018*B*a**2*b
**3) + x**6*(12870*A*a**2*b**3 + 12870*B*a**3*b**2) + x**4*(10010*A*a**3*b**2 + 5005*B*a**4*b) + x**2*(4095*A*
a**4*b + 819*B*a**5))/(9009*x**13)

________________________________________________________________________________________

Giac [A]  time = 1.16323, size = 171, normalized size = 1.51 \begin{align*} -\frac{9009 \, B b^{5} x^{12} + 15015 \, B a b^{4} x^{10} + 3003 \, A b^{5} x^{10} + 18018 \, B a^{2} b^{3} x^{8} + 9009 \, A a b^{4} x^{8} + 12870 \, B a^{3} b^{2} x^{6} + 12870 \, A a^{2} b^{3} x^{6} + 5005 \, B a^{4} b x^{4} + 10010 \, A a^{3} b^{2} x^{4} + 819 \, B a^{5} x^{2} + 4095 \, A a^{4} b x^{2} + 693 \, A a^{5}}{9009 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^14,x, algorithm="giac")

[Out]

-1/9009*(9009*B*b^5*x^12 + 15015*B*a*b^4*x^10 + 3003*A*b^5*x^10 + 18018*B*a^2*b^3*x^8 + 9009*A*a*b^4*x^8 + 128
70*B*a^3*b^2*x^6 + 12870*A*a^2*b^3*x^6 + 5005*B*a^4*b*x^4 + 10010*A*a^3*b^2*x^4 + 819*B*a^5*x^2 + 4095*A*a^4*b
*x^2 + 693*A*a^5)/x^13