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

Optimal. Leaf size=131 \[ -\frac{3 (b c-a d)^2 (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}-\frac{c^2 (3 b c-a d)}{2 a^2 b x}-\frac{d^2 x (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.133165, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {468, 570, 205} \[ -\frac{3 (b c-a d)^2 (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}-\frac{c^2 (3 b c-a d)}{2 a^2 b x}-\frac{d^2 x (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

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

Mathematica [A]  time = 0.0595166, size = 94, normalized size = 0.72 \[ \frac{x (a d-b c)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac{3 (a d-b c)^2 (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}-\frac{c^3}{a^2 x}+\frac{d^3 x}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 189, normalized size = 1.4 \begin{align*}{\frac{{d}^{3}x}{{b}^{2}}}-{\frac{{c}^{3}}{{a}^{2}x}}+{\frac{ax{d}^{3}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,cx{d}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{3\,x{c}^{2}d}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bx{c}^{3}}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a{d}^{3}}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c{d}^{2}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{2}d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,b{c}^{3}}{2\,{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^2/(b*x^2+a)^2,x)

[Out]

d^3/b^2*x-c^3/a^2/x+1/2*a/b^2*x/(b*x^2+a)*d^3-3/2/b*x/(b*x^2+a)*c*d^2+3/2/a*x/(b*x^2+a)*c^2*d-1/2/a^2*b*x/(b*x
^2+a)*c^3-3/2*a/b^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^3+3/2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c*d^2+3/
2/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^2*d-3/2/a^2*b/(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^2/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67409, size = 798, normalized size = 6.09 \begin{align*} \left [\frac{4 \, a^{3} b^{2} d^{3} x^{4} - 4 \, a^{2} b^{3} c^{3} - 6 \,{\left (a b^{4} c^{3} - a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} - 3 \,{\left ({\left (b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{3} +{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + a^{4} d^{3}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{4 \,{\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}}, \frac{2 \, a^{3} b^{2} d^{3} x^{4} - 2 \, a^{2} b^{3} c^{3} - 3 \,{\left (a b^{4} c^{3} - a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} - 3 \,{\left ({\left (b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{3} +{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + a^{4} d^{3}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}}\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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.84766, size = 309, normalized size = 2.36 \begin{align*} \frac{3 \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \left (a d + b c\right ) \log{\left (- \frac{3 a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \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 + 3 b^{3} c^{3}} + x \right )}}{4} - \frac{3 \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \left (a d + b c\right ) \log{\left (\frac{3 a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \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 + 3 b^{3} c^{3}} + x \right )}}{4} + \frac{- 2 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 - 3 b^{3} c^{3}\right )}{2 a^{3} b^{2} x + 2 a^{2} b^{3} x^{3}} + \frac{d^{3} x}{b^{2}} \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)**2,x)

[Out]

3*sqrt(-1/(a**5*b**5))*(a*d - b*c)**2*(a*d + b*c)*log(-3*a**3*b**2*sqrt(-1/(a**5*b**5))*(a*d - b*c)**2*(a*d +
b*c)/(3*a**3*d**3 - 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + 3*b**3*c**3) + x)/4 - 3*sqrt(-1/(a**5*b**5))*(a*d - b*
c)**2*(a*d + b*c)*log(3*a**3*b**2*sqrt(-1/(a**5*b**5))*(a*d - b*c)**2*(a*d + b*c)/(3*a**3*d**3 - 3*a**2*b*c*d*
*2 - 3*a*b**2*c**2*d + 3*b**3*c**3) + x)/4 + (-2*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 - 3*b**3*c**3))/(2*a**3*b**2*x + 2*a**2*b**3*x**3) + d**3*x/b**2

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Giac [A]  time = 1.12446, size = 193, normalized size = 1.47 \begin{align*} \frac{d^{3} x}{b^{2}} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} b^{2}} - \frac{3 \, b^{3} c^{3} x^{2} - 3 \, a b^{2} c^{2} d x^{2} + 3 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{2 \,{\left (b x^{3} + a x\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^2/(b*x^2+a)^2,x, algorithm="giac")

[Out]

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