3.200 \(\int \frac{x^3 (c+d x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{x^2 (b c-a d)}{2 b^2}-\frac{a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^4}{4 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0559814, antiderivative size = 54, 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} \[ \frac{x^2 (b c-a d)}{2 b^2}-\frac{a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x^2))/(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0184562, size = 47, normalized size = 0.87 \[ \frac{b x^2 \left (-2 a d+2 b c+b d x^2\right )+2 a (a d-b c) \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(c + d*x^2))/(a + b*x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 62, normalized size = 1.2 \begin{align*}{\frac{d{x}^{4}}{4\,b}}-{\frac{ad{x}^{2}}{2\,{b}^{2}}}+{\frac{c{x}^{2}}{2\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\,{b}^{3}}}-{\frac{ac\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d*x^2+c)/(b*x^2+a),x)

[Out]

1/4*d*x^4/b-1/2/b^2*a*d*x^2+1/2*c*x^2/b+1/2*a^2/b^3*ln(b*x^2+a)*d-1/2*a*c*ln(b*x^2+a)/b^2

________________________________________________________________________________________

Maxima [A]  time = 0.990317, size = 68, normalized size = 1.26 \begin{align*} \frac{b d x^{4} + 2 \,{\left (b c - a d\right )} x^{2}}{4 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \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*(d*x^2+c)/(b*x^2+a),x, algorithm="maxima")

[Out]

1/4*(b*d*x^4 + 2*(b*c - a*d)*x^2)/b^2 - 1/2*(a*b*c - a^2*d)*log(b*x^2 + a)/b^3

________________________________________________________________________________________

Fricas [A]  time = 1.44177, size = 108, normalized size = 2. \begin{align*} \frac{b^{2} d x^{4} + 2 \,{\left (b^{2} c - a b d\right )} x^{2} - 2 \,{\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^2*d*x^4 + 2*(b^2*c - a*b*d)*x^2 - 2*(a*b*c - a^2*d)*log(b*x^2 + a))/b^3

________________________________________________________________________________________

Sympy [A]  time = 0.412109, size = 44, normalized size = 0.81 \begin{align*} \frac{a \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{d x^{4}}{4 b} - \frac{x^{2} \left (a d - b c\right )}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(d*x**2+c)/(b*x**2+a),x)

[Out]

a*(a*d - b*c)*log(a + b*x**2)/(2*b**3) + d*x**4/(4*b) - x**2*(a*d - b*c)/(2*b**2)

________________________________________________________________________________________

Giac [A]  time = 1.16664, size = 70, normalized size = 1.3 \begin{align*} \frac{b d x^{4} + 2 \, b c x^{2} - 2 \, a d x^{2}}{4 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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