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

Optimal. Leaf size=66 \[ -\frac{A b-2 a B}{2 b^3 \left (a+b x^2\right )}+\frac{a (A b-a B)}{4 b^3 \left (a+b x^2\right )^2}+\frac{B \log \left (a+b x^2\right )}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.06588, antiderivative size = 66, 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} \[ -\frac{A b-2 a B}{2 b^3 \left (a+b x^2\right )}+\frac{a (A b-a B)}{4 b^3 \left (a+b x^2\right )^2}+\frac{B \log \left (a+b x^2\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0240274, size = 64, normalized size = 0.97 \[ \frac{3 a^2 B-a b \left (A-4 B x^2\right )+2 B \left (a+b x^2\right )^2 \log \left (a+b x^2\right )-2 A b^2 x^2}{4 b^3 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 80, normalized size = 1.2 \begin{align*}{\frac{Aa}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{{a}^{2}B}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}}-{\frac{A}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Ba}{{b}^{3} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.996254, size = 97, normalized size = 1.47 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x^{2}}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac{B \log \left (b x^{2} + a\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.29813, size = 184, normalized size = 2.79 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x^{2} + 2 \,{\left (B b^{2} x^{4} + 2 \, B a b x^{2} + B a^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.09992, size = 70, normalized size = 1.06 \begin{align*} \frac{B \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{- A a b + 3 B a^{2} + x^{2} \left (- 2 A b^{2} + 4 B a b\right )}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13556, size = 82, normalized size = 1.24 \begin{align*} \frac{B \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x^{2} + \frac{3 \, B a^{2} - A a b}{b}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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