3.227 \(\int \frac{(c+d x^2)^3}{x^4 (a+b x^2)} \, dx\)

Optimal. Leaf size=74 \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0657843, antiderivative size = 74, 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} \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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^4 \left (a+b x^2\right )} \, dx &=\int \left (\frac{d^3}{b}+\frac{c^3}{a x^4}+\frac{c^2 (-b c+3 a d)}{a^2 x^2}-\frac{(-b c+a d)^3}{a^2 b \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c^3}{3 a x^3}+\frac{c^2 (b c-3 a d)}{a^2 x}+\frac{d^3 x}{b}+\frac{(b c-a d)^3 \int \frac{1}{a+b x^2} \, dx}{a^2 b}\\ &=-\frac{c^3}{3 a x^3}+\frac{c^2 (b c-3 a d)}{a^2 x}+\frac{d^3 x}{b}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0390994, size = 74, normalized size = 1. \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 135, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}x}{b}}-{\frac{{c}^{3}}{3\,a{x}^{3}}}-3\,{\frac{{c}^{2}d}{ax}}+{\frac{b{c}^{3}}{{a}^{2}x}}-{\frac{a{d}^{3}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{c{d}^{2}}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{b{c}^{2}d}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{3}}{{a}^{2}}\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^4/(b*x^2+a),x)

[Out]

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

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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^4/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4524, size = 531, normalized size = 7.18 \begin{align*} \left [\frac{6 \, a^{3} b d^{3} x^{4} - 2 \, a^{2} b^{2} 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^{3} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 6 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{6 \, a^{3} b^{2} x^{3}}, \frac{3 \, a^{3} b d^{3} x^{4} - a^{2} b^{2} 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^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{3 \, a^{3} b^{2} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^4/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.23825, size = 221, normalized size = 2.99 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \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^{5} b^{3}}} \left (a d - b c\right )^{3} \log{\left (\frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \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}{b} - \frac{a c^{3} + x^{2} \left (9 a c^{2} d - 3 b c^{3}\right )}{3 a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3/x**4/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.15046, size = 135, normalized size = 1.82 \begin{align*} \frac{d^{3} x}{b} + \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^{2} b} + \frac{3 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^4/(b*x^2+a),x, algorithm="giac")

[Out]

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