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

Optimal. Leaf size=73 \[ \frac{d^2 x^2 (3 b c-a d)}{2 b^2}-\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 a b^3}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0772613, 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, 72} \[ \frac{d^2 x^2 (3 b c-a d)}{2 b^2}-\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 a b^3}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(x*(a + b*x^2)),x]

[Out]

(d^2*(3*b*c - a*d)*x^2)/(2*b^2) + (d^3*x^4)/(4*b) + (c^3*Log[x])/a - ((b*c - a*d)^3*Log[a + b*x^2])/(2*a*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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0302298, size = 65, normalized size = 0.89 \[ \frac{a b d^2 x^2 \left (-2 a d+6 b c+b d x^2\right )-2 (b c-a d)^3 \log \left (a+b x^2\right )+4 b^3 c^3 \log (x)}{4 a b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*d^2*x^2*(6*b*c - 2*a*d + b*d*x^2) + 4*b^3*c^3*Log[x] - 2*(b*c - a*d)^3*Log[a + b*x^2])/(4*a*b^3)

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Maple [A]  time = 0.004, size = 116, normalized size = 1.6 \begin{align*}{\frac{{d}^{3}{x}^{4}}{4\,b}}-{\frac{{d}^{3}a{x}^{2}}{2\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,b}}+{\frac{{c}^{3}\ln \left ( x \right ) }{a}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{3}}}-{\frac{3\,a\ln \left ( b{x}^{2}+a \right ) c{d}^{2}}{2\,{b}^{2}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){c}^{2}d}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01955, size = 132, normalized size = 1.81 \begin{align*} \frac{c^{3} \log \left (x^{2}\right )}{2 \, a} + \frac{b d^{3} x^{4} + 2 \,{\left (3 \, b c d^{2} - a d^{3}\right )} x^{2}}{4 \, b^{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 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.52583, size = 209, normalized size = 2.86 \begin{align*} \frac{a b^{2} d^{3} x^{4} + 4 \, b^{3} c^{3} \log \left (x\right ) + 2 \,{\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{2} - 2 \,{\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 )}{4 \, a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.91193, size = 63, normalized size = 0.86 \begin{align*} \frac{d^{3} x^{4}}{4 b} - \frac{x^{2} \left (a d^{3} - 3 b c d^{2}\right )}{2 b^{2}} + \frac{c^{3} \log{\left (x \right )}}{a} + \frac{\left (a d - b c\right )^{3} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a 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),x)

[Out]

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

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Giac [A]  time = 1.15656, size = 134, normalized size = 1.84 \begin{align*} \frac{c^{3} \log \left (x^{2}\right )}{2 \, a} + \frac{b d^{3} x^{4} + 6 \, b c d^{2} x^{2} - 2 \, a d^{3} x^{2}}{4 \, b^{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 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),x, algorithm="giac")

[Out]

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