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

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

[Out]

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

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Rubi [A]  time = 0.0489539, 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} \[ -\frac{b x^2 (b c-a d)}{2 d^2}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}+\frac{\left (a+b x^2\right )^2}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0227169, size = 49, normalized size = 0.8 \[ \frac{b d x^2 \left (4 a d-2 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (c+d x^2\right )}{4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 85, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{4}}{4\,d}}+{\frac{ab{x}^{2}}{d}}-{\frac{{b}^{2}c{x}^{2}}{2\,{d}^{2}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,d}}-{\frac{\ln \left ( d{x}^{2}+c \right ) cab}{{d}^{2}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}{c}^{2}}{2\,{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992624, size = 88, normalized size = 1.44 \begin{align*} \frac{b^{2} d x^{4} - 2 \,{\left (b^{2} c - 2 \, a b d\right )} x^{2}}{4 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.2664, size = 140, normalized size = 2.3 \begin{align*} \frac{b^{2} d^{2} x^{4} - 2 \,{\left (b^{2} c d - 2 \, 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 (d x^{2} + c\right )}{4 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.481465, size = 51, normalized size = 0.84 \begin{align*} \frac{b^{2} x^{4}}{4 d} + \frac{x^{2} \left (2 a b d - b^{2} c\right )}{2 d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17182, size = 90, normalized size = 1.48 \begin{align*} \frac{b^{2} d x^{4} - 2 \, b^{2} c x^{2} + 4 \, a b d x^{2}}{4 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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