3.90 \(\int \frac{x^7 (A+B x^2)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}+\frac{x^2 (A b-3 a B)}{2 b^4}-\frac{3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac{B x^4}{4 b^3} \]

[Out]

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

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Rubi [A]  time = 0.121235, antiderivative size = 109, 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} \[ -\frac{a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}+\frac{x^2 (A b-3 a B)}{2 b^4}-\frac{3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac{B x^4}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.0652748, size = 94, normalized size = 0.86 \[ \frac{\frac{2 a^2 (4 a B-3 A b)}{a+b x^2}+\frac{a^3 (A b-a B)}{\left (a+b x^2\right )^2}+2 b x^2 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^2\right )+b^2 B x^4}{4 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.011, size = 134, normalized size = 1.2 \begin{align*}{\frac{B{x}^{4}}{4\,{b}^{3}}}-{\frac{3\,B{x}^{2}a}{2\,{b}^{4}}}+{\frac{A{x}^{2}}{2\,{b}^{3}}}+{\frac{{a}^{3}A}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B{a}^{4}}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,a\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{4}}}+3\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) B}{{b}^{5}}}-{\frac{3\,A{a}^{2}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}+2\,{\frac{B{a}^{3}}{{b}^{5} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.99354, size = 157, normalized size = 1.44 \begin{align*} \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}}{4 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac{B b x^{4} - 2 \,{\left (3 \, B a - A b\right )} x^{2}}{4 \, b^{4}} + \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.24866, size = 360, normalized size = 3.3 \begin{align*} \frac{B b^{4} x^{8} - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (2 \, B a^{4} - A a^{3} b +{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{4} + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.65688, size = 116, normalized size = 1.06 \begin{align*} \frac{B x^{4}}{4 b^{3}} + \frac{3 a \left (- A b + 2 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \frac{- 5 A a^{3} b + 7 B a^{4} + x^{2} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{4 a^{2} b^{5} + 8 a b^{6} x^{2} + 4 b^{7} x^{4}} - \frac{x^{2} \left (- A b + 3 B a\right )}{2 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

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

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Giac [A]  time = 1.12029, size = 178, normalized size = 1.63 \begin{align*} \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac{B b^{3} x^{4} - 6 \, B a b^{2} x^{2} + 2 \, A b^{3} x^{2}}{4 \, b^{6}} - \frac{18 \, B a^{2} b^{2} x^{4} - 9 \, A a b^{3} x^{4} + 28 \, B a^{3} b x^{2} - 12 \, A a^{2} b^{2} x^{2} + 11 \, B a^{4} - 4 \, A a^{3} b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

3/2*(2*B*a^2 - A*a*b)*log(abs(b*x^2 + a))/b^5 + 1/4*(B*b^3*x^4 - 6*B*a*b^2*x^2 + 2*A*b^3*x^2)/b^6 - 1/4*(18*B*
a^2*b^2*x^4 - 9*A*a*b^3*x^4 + 28*B*a^3*b*x^2 - 12*A*a^2*b^2*x^2 + 11*B*a^4 - 4*A*a^3*b)/((b*x^2 + a)^2*b^5)