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

Optimal. Leaf size=84 \[ \frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{\sqrt{a} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{d^2 x^5}{5 b} \]

[Out]

((b*c - a*d)^2*x)/b^3 + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^5)/(5*b) - (Sqrt[a]*(b*c - a*d)^2*ArcTan[(Sqrt[
b]*x)/Sqrt[a]])/b^(7/2)

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Rubi [A]  time = 0.064104, antiderivative size = 84, 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} \[ \frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{\sqrt{a} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{d^2 x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*x)/b^3 + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^5)/(5*b) - (Sqrt[a]*(b*c - a*d)^2*ArcTan[(Sqrt[
b]*x)/Sqrt[a]])/b^(7/2)

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{x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\int \left (\frac{(b c-a d)^2}{b^3}+\frac{d (2 b c-a d) x^2}{b^2}+\frac{d^2 x^4}{b}+\frac{-a b^2 c^2+2 a^2 b c d-a^3 d^2}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{(b c-a d)^2 x}{b^3}+\frac{d (2 b c-a d) x^3}{3 b^2}+\frac{d^2 x^5}{5 b}-\frac{\left (a (b c-a d)^2\right ) \int \frac{1}{a+b x^2} \, dx}{b^3}\\ &=\frac{(b c-a d)^2 x}{b^3}+\frac{d (2 b c-a d) x^3}{3 b^2}+\frac{d^2 x^5}{5 b}-\frac{\sqrt{a} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0745149, size = 84, normalized size = 1. \[ \frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{\sqrt{a} (a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{d^2 x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*x)/b^3 + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^5)/(5*b) - (Sqrt[a]*(-(b*c) + a*d)^2*ArcTan[(Sq
rt[b]*x)/Sqrt[a]])/b^(7/2)

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Maple [A]  time = 0.003, size = 135, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{5}}{5\,b}}-{\frac{{x}^{3}a{d}^{2}}{3\,{b}^{2}}}+{\frac{2\,c{x}^{3}d}{3\,b}}+{\frac{{a}^{2}{d}^{2}x}{{b}^{3}}}-2\,{\frac{acdx}{{b}^{2}}}+{\frac{{c}^{2}x}{b}}-{\frac{{a}^{3}{d}^{2}}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+2\,{\frac{{a}^{2}cd}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{a{c}^{2}}{b}\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(x^2*(d*x^2+c)^2/(b*x^2+a),x)

[Out]

1/5*d^2*x^5/b-1/3/b^2*x^3*a*d^2+2/3/b*x^3*c*d+1/b^3*a^2*d^2*x-2/b^2*c*a*d*x+1/b*c^2*x-a^3/b^3/(a*b)^(1/2)*arct
an(b*x/(a*b)^(1/2))*d^2+2*a^2/b^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c*d-a/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/
2))*c^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72398, size = 489, normalized size = 5.82 \begin{align*} \left [\frac{6 \, b^{2} d^{2} x^{5} + 10 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{3} + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 30 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{30 \, b^{3}}, \frac{3 \, b^{2} d^{2} x^{5} + 5 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{3} - 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(6*b^2*d^2*x^5 + 10*(2*b^2*c*d - a*b*d^2)*x^3 + 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-a/b)*log((b*x^2
- 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 30*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^3, 1/15*(3*b^2*d^2*x^5 + 5*(2*
b^2*c*d - a*b*d^2)*x^3 - 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 15*(b^2*c^2 -
2*a*b*c*d + a^2*d^2)*x)/b^3]

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Sympy [B]  time = 0.598793, size = 192, normalized size = 2.29 \begin{align*} \frac{\sqrt{- \frac{a}{b^{7}}} \left (a d - b c\right )^{2} \log{\left (- \frac{b^{3} \sqrt{- \frac{a}{b^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{7}}} \left (a d - b c\right )^{2} \log{\left (\frac{b^{3} \sqrt{- \frac{a}{b^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{d^{2} x^{5}}{5 b} - \frac{x^{3} \left (a d^{2} - 2 b c d\right )}{3 b^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a/b**7)*(a*d - b*c)**2*log(-b**3*sqrt(-a/b**7)*(a*d - b*c)**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x)/2
- sqrt(-a/b**7)*(a*d - b*c)**2*log(b**3*sqrt(-a/b**7)*(a*d - b*c)**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x)
/2 + d**2*x**5/(5*b) - x**3*(a*d**2 - 2*b*c*d)/(3*b**2) + x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/b**3

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Giac [A]  time = 1.1495, size = 153, normalized size = 1.82 \begin{align*} -\frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, b^{4} d^{2} x^{5} + 10 \, b^{4} c d x^{3} - 5 \, a b^{3} d^{2} x^{3} + 15 \, b^{4} c^{2} x - 30 \, a b^{3} c d x + 15 \, a^{2} b^{2} d^{2} x}{15 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*b^4*d^2*x^5 + 10*b^4*c*d*
x^3 - 5*a*b^3*d^2*x^3 + 15*b^4*c^2*x - 30*a*b^3*c*d*x + 15*a^2*b^2*d^2*x)/b^5