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

Optimal. Leaf size=88 \[ -\frac{(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{2 a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0844912, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 88} \[ -\frac{(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{2 a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*x^2)/(2*b^2) + (b*c - a*d)^3/(2*a*b^3*(a + b*x^2)) + (c^3*Log[x])/a^2 - ((b*c - a*d)^2*(b*c + 2*a*d)*Log[
a + b*x^2])/(2*a^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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0990077, size = 111, normalized size = 1.26 \[ \frac{\frac{\frac{a \left (a^2 b d^2 \left (3 c+d x^2\right )-a^3 d^3+a b^2 \left (d^3 x^4-3 c^2 d\right )+b^3 c^3\right )}{a+b x^2}-(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{b^3}+2 c^3 \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.017, size = 146, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{x}^{2}}{2\,{b}^{2}}}+{\frac{{c}^{3}\ln \left ( x \right ) }{{a}^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){d}^{3}}{{b}^{3}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){d}^{2}c}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{a}^{2}}}-{\frac{{a}^{2}{d}^{3}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,a{d}^{2}c}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,d{c}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{{c}^{3}}{2\,a \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.993607, size = 165, normalized size = 1.88 \begin{align*} \frac{d^{3} x^{2}}{2 \, b^{2}} + \frac{c^{3} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} - \frac{{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.57065, size = 352, normalized size = 4. \begin{align*} \frac{a^{2} b^{2} d^{3} x^{4} + a^{3} b d^{3} x^{2} + a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} -{\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{4} c^{3} x^{2} + a b^{3} c^{3}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.8078, size = 110, normalized size = 1.25 \begin{align*} - \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} + \frac{d^{3} x^{2}}{2 b^{2}} + \frac{c^{3} \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{2} \left (2 a d + b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13824, size = 203, normalized size = 2.31 \begin{align*} \frac{d^{3} x^{2}}{2 \, b^{2}} + \frac{c^{3} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{3}} + \frac{b^{4} c^{3} x^{2} - 3 \, a^{2} b^{2} c d^{2} x^{2} + 2 \, a^{3} b d^{3} x^{2} + 2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}}{2 \,{\left (b x^{2} + a\right )} a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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