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

Optimal. Leaf size=73 \[ \frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{2 a x^2}+\frac{d^3 x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0759506, antiderivative size = 73, 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} \[ \frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{2 a x^2}+\frac{d^3 x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0367246, size = 75, normalized size = 1.03 \[ \frac{-2 b^2 c^2 x^2 \log (x) (b c-3 a d)+a b \left (a d^3 x^4-b c^3\right )+x^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 114, normalized size = 1.6 \begin{align*}{\frac{{d}^{3}{x}^{2}}{2\,b}}-{\frac{{c}^{3}}{2\,a{x}^{2}}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{a}}-{\frac{{c}^{3}\ln \left ( x \right ) b}{{a}^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{2}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ) c{d}^{2}}{2\,b}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ){c}^{2}d}{2\,a}}+{\frac{b\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{a}^{2}}} \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),x)

[Out]

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

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Maxima [A]  time = 0.992942, size = 131, normalized size = 1.79 \begin{align*} \frac{d^{3} x^{2}}{2 \, b} - \frac{c^{3}}{2 \, a x^{2}} - \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.5159, size = 217, normalized size = 2.97 \begin{align*} \frac{a^{2} b d^{3} x^{4} - a b^{2} c^{3} +{\left (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} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x^{2} \log \left (x\right )}{2 \, a^{2} b^{2} x^{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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.38139, size = 63, normalized size = 0.86 \begin{align*} \frac{d^{3} x^{2}}{2 b} - \frac{c^{3}}{2 a x^{2}} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 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),x)

[Out]

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

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Giac [A]  time = 1.14639, size = 162, normalized size = 2.22 \begin{align*} \frac{d^{3} x^{2}}{2 \, b} - \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{2}} + \frac{b c^{3} x^{2} - 3 \, a c^{2} d x^{2} - a c^{3}}{2 \, a^{2} x^{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),x, algorithm="giac")

[Out]

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