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

Optimal. Leaf size=61 \[ \frac{d x^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{\left (c+d x^2\right )^2}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0489202, antiderivative size = 61, 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 = {444, 43} \[ \frac{d x^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{\left (c+d x^2\right )^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x^2)^2)/(a + b*x^2),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0230791, size = 49, normalized size = 0.8 \[ \frac{b d x^2 \left (-2 a d+4 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 85, normalized size = 1.4 \begin{align*}{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{a{d}^{2}{x}^{2}}{2\,{b}^{2}}}+{\frac{d{x}^{2}c}{b}}+{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) cad}{{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988173, size = 89, normalized size = 1.46 \begin{align*} \frac{b d^{2} x^{4} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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*(d*x^2+c)^2/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42728, size = 140, normalized size = 2.3 \begin{align*} \frac{b^{2} d^{2} x^{4} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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*(d*x^2+c)^2/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.503939, size = 51, normalized size = 0.84 \begin{align*} \frac{d^{2} x^{4}}{4 b} - \frac{x^{2} \left (a d^{2} - 2 b c d\right )}{2 b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12555, size = 90, normalized size = 1.48 \begin{align*} \frac{b d^{2} x^{4} + 4 \, b c d x^{2} - 2 \, a d^{2} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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*(d*x^2+c)^2/(b*x^2+a),x, algorithm="giac")

[Out]

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