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

Optimal. Leaf size=83 \[ -\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{x (b c-a d)^2}{d^3}-\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{7/2}}+\frac{b^2 x^5}{5 d} \]

[Out]

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

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Rubi [A]  time = 0.0644815, antiderivative size = 83, 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} \[ -\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{x (b c-a d)^2}{d^3}-\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{7/2}}+\frac{b^2 x^5}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0693865, size = 83, normalized size = 1. \[ -\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{x (a d-b c)^2}{d^3}-\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{7/2}}+\frac{b^2 x^5}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5/d+2/3/d*x^3*a*b-1/3/d^2*x^3*b^2*c+1/d*a^2*x-2/d^2*c*a*b*x+1/d^3*b^2*c^2*x-c/d/(c*d)^(1/2)*arctan(x
*d/(c*d)^(1/2))*a^2+2*c^2/d^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b-c^3/d^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/
2))*b^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*(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28082, size = 489, normalized size = 5.89 \begin{align*} \left [\frac{6 \, b^{2} d^{2} x^{5} - 10 \,{\left (b^{2} c d - 2 \, 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{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) + 30 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{30 \, d^{3}}, \frac{3 \, b^{2} d^{2} x^{5} - 5 \,{\left (b^{2} c d - 2 \, 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{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, d^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.614831, size = 192, normalized size = 2.31 \begin{align*} \frac{b^{2} x^{5}}{5 d} + \frac{\sqrt{- \frac{c}{d^{7}}} \left (a d - b c\right )^{2} \log{\left (- \frac{d^{3} \sqrt{- \frac{c}{d^{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{c}{d^{7}}} \left (a d - b c\right )^{2} \log{\left (\frac{d^{3} \sqrt{- \frac{c}{d^{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{x^{3} \left (2 a b d - b^{2} c\right )}{3 d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15543, size = 153, normalized size = 1.84 \begin{align*} -\frac{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{3}} + \frac{3 \, b^{2} d^{4} x^{5} - 5 \, b^{2} c d^{3} x^{3} + 10 \, a b d^{4} x^{3} + 15 \, b^{2} c^{2} d^{2} x - 30 \, a b c d^{3} x + 15 \, a^{2} d^{4} x}{15 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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