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

Optimal. Leaf size=77 \[ -\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]

[Out]

-(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) - ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3
/2)*b^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.064137, antiderivative size = 77, 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 = {461, 205} \[ -\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) - ((b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3
/2)*b^(5/2))

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx &=\int \left (\frac{d^2 (3 b c-a d)}{b^2}+\frac{c^3}{a x^2}+\frac{d^3 x^2}{b}+\frac{(-b c+a d)^3}{a b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c^3}{a x}+\frac{d^2 (3 b c-a d) x}{b^2}+\frac{d^3 x^3}{3 b}-\frac{(b c-a d)^3 \int \frac{1}{a+b x^2} \, dx}{a b^2}\\ &=-\frac{c^3}{a x}+\frac{d^2 (3 b c-a d) x}{b^2}+\frac{d^3 x^3}{3 b}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0322889, size = 76, normalized size = 0.99 \[ \frac{(a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) + ((-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a
^(3/2)*b^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 135, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,b}}-{\frac{{d}^{3}ax}{{b}^{2}}}+3\,{\frac{{d}^{2}xc}{b}}-{\frac{{c}^{3}}{ax}}+{\frac{{a}^{2}{d}^{3}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{ac{d}^{2}}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+3\,{\frac{{c}^{2}d}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{b{c}^{3}}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.51257, size = 520, normalized size = 6.75 \begin{align*} \left [\frac{2 \, a^{2} b^{2} d^{3} x^{4} - 6 \, a b^{3} c^{3} + 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 )} \sqrt{-a b} x \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 6 \,{\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{6 \, a^{2} b^{3} x}, \frac{a^{2} b^{2} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 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 )} \sqrt{a b} x \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \,{\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{3 \, a^{2} b^{3} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(2*a^2*b^2*d^3*x^4 - 6*a*b^3*c^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a*b)*x*log
((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 6*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^2)/(a^2*b^3*x), 1/3*(a^2*b^2*d^
3*x^4 - 3*a*b^3*c^3 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b)*x*arctan(sqrt(a*b)*x/a)
+ 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^2)/(a^2*b^3*x)]

________________________________________________________________________________________

Sympy [B]  time = 0.931007, size = 221, normalized size = 2.87 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log{\left (\frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{3}}{3 b} - \frac{x \left (a d^{3} - 3 b c d^{2}\right )}{b^{2}} - \frac{c^{3}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(a**3*b**5))*(a*d - b*c)**3*log(-a**2*b**2*sqrt(-1/(a**3*b**5))*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*
c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 + sqrt(-1/(a**3*b**5))*(a*d - b*c)**3*log(a**2*b**2*sqrt(-1/(a**3
*b**5))*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 + d**3*x**3/(3*b) -
x*(a*d**3 - 3*b*c*d**2)/b**2 - c**3/(a*x)

________________________________________________________________________________________

Giac [A]  time = 1.13286, size = 140, normalized size = 1.82 \begin{align*} -\frac{c^{3}}{a x} - \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 )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a b^{2}} + \frac{b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x - 3 \, a b d^{3} x}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3/(a*x) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^2) + 1/3
*(b^2*d^3*x^3 + 9*b^2*c*d^2*x - 3*a*b*d^3*x)/b^3