3.88 \(\int \frac{x^{11} (A+B x^2)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=150 \[ \frac{a^2 x^2 (3 A b-5 a B)}{b^6}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left (a+b x^2\right )}+\frac{a^5 (A b-a B)}{4 b^7 \left (a+b x^2\right )^2}-\frac{5 a^3 (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^7}+\frac{x^6 (A b-3 a B)}{6 b^4}-\frac{3 a x^4 (A b-2 a B)}{4 b^5}+\frac{B x^8}{8 b^3} \]

[Out]

(a^2*(3*A*b - 5*a*B)*x^2)/b^6 - (3*a*(A*b - 2*a*B)*x^4)/(4*b^5) + ((A*b - 3*a*B)*x^6)/(6*b^4) + (B*x^8)/(8*b^3
) + (a^5*(A*b - a*B))/(4*b^7*(a + b*x^2)^2) - (a^4*(5*A*b - 6*a*B))/(2*b^7*(a + b*x^2)) - (5*a^3*(2*A*b - 3*a*
B)*Log[a + b*x^2])/(2*b^7)

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Rubi [A]  time = 0.230432, antiderivative size = 150, 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, 77} \[ \frac{a^2 x^2 (3 A b-5 a B)}{b^6}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left (a+b x^2\right )}+\frac{a^5 (A b-a B)}{4 b^7 \left (a+b x^2\right )^2}-\frac{5 a^3 (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^7}+\frac{x^6 (A b-3 a B)}{6 b^4}-\frac{3 a x^4 (A b-2 a B)}{4 b^5}+\frac{B x^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(a^2*(3*A*b - 5*a*B)*x^2)/b^6 - (3*a*(A*b - 2*a*B)*x^4)/(4*b^5) + ((A*b - 3*a*B)*x^6)/(6*b^4) + (B*x^8)/(8*b^3
) + (a^5*(A*b - a*B))/(4*b^7*(a + b*x^2)^2) - (a^4*(5*A*b - 6*a*B))/(2*b^7*(a + b*x^2)) - (5*a^3*(2*A*b - 3*a*
B)*Log[a + b*x^2])/(2*b^7)

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^{11} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5 (A+B x)}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{2 a^2 (-3 A b+5 a B)}{b^6}+\frac{3 a (-A b+2 a B) x}{b^5}+\frac{(A b-3 a B) x^2}{b^4}+\frac{B x^3}{b^3}+\frac{a^5 (-A b+a B)}{b^6 (a+b x)^3}-\frac{a^4 (-5 A b+6 a B)}{b^6 (a+b x)^2}+\frac{5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 (3 A b-5 a B) x^2}{b^6}-\frac{3 a (A b-2 a B) x^4}{4 b^5}+\frac{(A b-3 a B) x^6}{6 b^4}+\frac{B x^8}{8 b^3}+\frac{a^5 (A b-a B)}{4 b^7 \left (a+b x^2\right )^2}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left (a+b x^2\right )}-\frac{5 a^3 (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^7}\\ \end{align*}

Mathematica [A]  time = 0.087402, size = 136, normalized size = 0.91 \[ \frac{-24 a^2 b x^2 (5 a B-3 A b)+\frac{12 a^4 (6 a B-5 A b)}{a+b x^2}+\frac{6 a^5 (A b-a B)}{\left (a+b x^2\right )^2}+60 a^3 (3 a B-2 A b) \log \left (a+b x^2\right )+4 b^3 x^6 (A b-3 a B)+18 a b^2 x^4 (2 a B-A b)+3 b^4 B x^8}{24 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(-24*a^2*b*(-3*A*b + 5*a*B)*x^2 + 18*a*b^2*(-(A*b) + 2*a*B)*x^4 + 4*b^3*(A*b - 3*a*B)*x^6 + 3*b^4*B*x^8 + (6*a
^5*(A*b - a*B))/(a + b*x^2)^2 + (12*a^4*(-5*A*b + 6*a*B))/(a + b*x^2) + 60*a^3*(-2*A*b + 3*a*B)*Log[a + b*x^2]
)/(24*b^7)

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Maple [A]  time = 0.012, size = 182, normalized size = 1.2 \begin{align*}{\frac{B{x}^{8}}{8\,{b}^{3}}}+{\frac{{x}^{6}A}{6\,{b}^{3}}}-{\frac{{x}^{6}Ba}{2\,{b}^{4}}}-{\frac{3\,{x}^{4}Aa}{4\,{b}^{4}}}+{\frac{3\,{x}^{4}B{a}^{2}}{2\,{b}^{5}}}+3\,{\frac{{a}^{2}A{x}^{2}}{{b}^{5}}}-5\,{\frac{B{x}^{2}{a}^{3}}{{b}^{6}}}+{\frac{{a}^{5}A}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B{a}^{6}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-5\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{b}^{6}}}+{\frac{15\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{7}}}-{\frac{5\,{a}^{4}A}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{{a}^{5}B}{{b}^{7} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

1/8*B*x^8/b^3+1/6/b^3*x^6*A-1/2/b^4*x^6*B*a-3/4/b^4*x^4*A*a+3/2/b^5*x^4*B*a^2+3/b^5*A*x^2*a^2-5/b^6*B*x^2*a^3+
1/4*a^5/b^6/(b*x^2+a)^2*A-1/4*a^6/b^7/(b*x^2+a)^2*B-5*a^3/b^6*ln(b*x^2+a)*A+15/2*a^4/b^7*ln(b*x^2+a)*B-5/2*a^4
/b^6/(b*x^2+a)*A+3*a^5/b^7/(b*x^2+a)*B

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Maxima [A]  time = 1.02869, size = 223, normalized size = 1.49 \begin{align*} \frac{11 \, B a^{6} - 9 \, A a^{5} b + 2 \,{\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x^{2}}{4 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac{3 \, B b^{3} x^{8} - 4 \,{\left (3 \, B a b^{2} - A b^{3}\right )} x^{6} + 18 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{4} - 24 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2}}{24 \, b^{6}} + \frac{5 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/4*(11*B*a^6 - 9*A*a^5*b + 2*(6*B*a^5*b - 5*A*a^4*b^2)*x^2)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7) + 1/24*(3*B*b^3
*x^8 - 4*(3*B*a*b^2 - A*b^3)*x^6 + 18*(2*B*a^2*b - A*a*b^2)*x^4 - 24*(5*B*a^3 - 3*A*a^2*b)*x^2)/b^6 + 5/2*(3*B
*a^4 - 2*A*a^3*b)*log(b*x^2 + a)/b^7

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Fricas [A]  time = 1.48384, size = 489, normalized size = 3.26 \begin{align*} \frac{3 \, B b^{6} x^{12} - 2 \,{\left (3 \, B a b^{5} - 2 \, A b^{6}\right )} x^{10} + 5 \,{\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{8} + 66 \, B a^{6} - 54 \, A a^{5} b - 20 \,{\left (3 \, B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{6} - 6 \,{\left (34 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3}\right )} x^{4} - 12 \,{\left (4 \, B a^{5} b - A a^{4} b^{2}\right )} x^{2} + 60 \,{\left (3 \, B a^{6} - 2 \, A a^{5} b +{\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{4} + 2 \,{\left (3 \, B a^{5} b - 2 \, A a^{4} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/24*(3*B*b^6*x^12 - 2*(3*B*a*b^5 - 2*A*b^6)*x^10 + 5*(3*B*a^2*b^4 - 2*A*a*b^5)*x^8 + 66*B*a^6 - 54*A*a^5*b -
20*(3*B*a^3*b^3 - 2*A*a^2*b^4)*x^6 - 6*(34*B*a^4*b^2 - 21*A*a^3*b^3)*x^4 - 12*(4*B*a^5*b - A*a^4*b^2)*x^2 + 60
*(3*B*a^6 - 2*A*a^5*b + (3*B*a^4*b^2 - 2*A*a^3*b^3)*x^4 + 2*(3*B*a^5*b - 2*A*a^4*b^2)*x^2)*log(b*x^2 + a))/(b^
9*x^4 + 2*a*b^8*x^2 + a^2*b^7)

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Sympy [A]  time = 1.95257, size = 163, normalized size = 1.09 \begin{align*} \frac{B x^{8}}{8 b^{3}} + \frac{5 a^{3} \left (- 2 A b + 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{7}} + \frac{- 9 A a^{5} b + 11 B a^{6} + x^{2} \left (- 10 A a^{4} b^{2} + 12 B a^{5} b\right )}{4 a^{2} b^{7} + 8 a b^{8} x^{2} + 4 b^{9} x^{4}} - \frac{x^{6} \left (- A b + 3 B a\right )}{6 b^{4}} + \frac{x^{4} \left (- 3 A a b + 6 B a^{2}\right )}{4 b^{5}} - \frac{x^{2} \left (- 3 A a^{2} b + 5 B a^{3}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

B*x**8/(8*b**3) + 5*a**3*(-2*A*b + 3*B*a)*log(a + b*x**2)/(2*b**7) + (-9*A*a**5*b + 11*B*a**6 + x**2*(-10*A*a*
*4*b**2 + 12*B*a**5*b))/(4*a**2*b**7 + 8*a*b**8*x**2 + 4*b**9*x**4) - x**6*(-A*b + 3*B*a)/(6*b**4) + x**4*(-3*
A*a*b + 6*B*a**2)/(4*b**5) - x**2*(-3*A*a**2*b + 5*B*a**3)/b**6

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Giac [A]  time = 1.12296, size = 247, normalized size = 1.65 \begin{align*} \frac{5 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{7}} - \frac{45 \, B a^{4} b^{2} x^{4} - 30 \, A a^{3} b^{3} x^{4} + 78 \, B a^{5} b x^{2} - 50 \, A a^{4} b^{2} x^{2} + 34 \, B a^{6} - 21 \, A a^{5} b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{7}} + \frac{3 \, B b^{9} x^{8} - 12 \, B a b^{8} x^{6} + 4 \, A b^{9} x^{6} + 36 \, B a^{2} b^{7} x^{4} - 18 \, A a b^{8} x^{4} - 120 \, B a^{3} b^{6} x^{2} + 72 \, A a^{2} b^{7} x^{2}}{24 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

5/2*(3*B*a^4 - 2*A*a^3*b)*log(abs(b*x^2 + a))/b^7 - 1/4*(45*B*a^4*b^2*x^4 - 30*A*a^3*b^3*x^4 + 78*B*a^5*b*x^2
- 50*A*a^4*b^2*x^2 + 34*B*a^6 - 21*A*a^5*b)/((b*x^2 + a)^2*b^7) + 1/24*(3*B*b^9*x^8 - 12*B*a*b^8*x^6 + 4*A*b^9
*x^6 + 36*B*a^2*b^7*x^4 - 18*A*a*b^8*x^4 - 120*B*a^3*b^6*x^2 + 72*A*a^2*b^7*x^2)/b^12