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

Optimal. Leaf size=98 \[ -\frac{(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}+\frac{(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{2 a^3 b^2}-\frac{c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac{c^3}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.105636, antiderivative size = 98, 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} \[ -\frac{(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}+\frac{(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{2 a^3 b^2}-\frac{c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac{c^3}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0944897, size = 87, normalized size = 0.89 \[ \frac{\frac{a (a d-b c)^3}{b^2 \left (a+b x^2\right )}+\frac{(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{b^2}+2 c^2 \log (x) (3 a d-2 b c)-\frac{a c^3}{x^2}}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 156, normalized size = 1.6 \begin{align*} -{\frac{{c}^{3}}{2\,{a}^{2}{x}^{2}}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{{c}^{3}\ln \left ( x \right ) b}{{a}^{3}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{2}}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ) d{c}^{2}}{2\,{a}^{2}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ){c}^{3}}{{a}^{3}}}+{\frac{a{d}^{3}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,c{d}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{c}^{2}d}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{{c}^{3}b}{2\,{a}^{2} \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^3/(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.01254, size = 190, normalized size = 1.94 \begin{align*} -\frac{a b^{2} c^{3} +{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \,{\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x^{2}\right )}} - \frac{{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac{{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.57421, size = 413, normalized size = 4.21 \begin{align*} -\frac{a^{2} b^{2} c^{3} +{\left (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}\right )} x^{2} -{\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{4} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{4} +{\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.80767, size = 128, normalized size = 1.31 \begin{align*} \frac{- a b^{2} c^{3} + x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{4}} + \frac{c^{2} \left (3 a d - 2 b c\right ) \log{\left (x \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13921, size = 212, normalized size = 2.16 \begin{align*} -\frac{{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac{{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b^{2}} - \frac{a^{2} b d^{3} x^{4} + 4 \, b^{3} c^{3} x^{2} - 6 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{4 \,{\left (b x^{4} + a x^{2}\right )} a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*c^3 - 3*a*c^2*d)*log(x^2)/a^3 + 1/2*(2*b^3*c^3 - 3*a*b^2*c^2*d + a^3*d^3)*log(abs(b*x^2 + a))/(a^3*b
^2) - 1/4*(a^2*b*d^3*x^4 + 4*b^3*c^3*x^2 - 6*a*b^2*c^2*d*x^2 + 6*a^2*b*c*d^2*x^2 - a^3*d^3*x^2 + 2*a*b^2*c^3)/
((b*x^4 + a*x^2)*a^2*b^2)