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

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

[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 = 0.14965, 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 \[ -\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{7/2}}+\frac{x (b c-a d)^2}{d^3}-\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{b^2 x^5}{5 d} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{5}}{5 d} + \frac{b x^{3} \left (2 a d - b c\right )}{3 d^{2}} - \frac{\sqrt{c} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{d^{\frac{7}{2}}} + \left (a d - b c\right )^{2} \int \frac{1}{d^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a)**2/(d*x**2+c),x)

[Out]

b**2*x**5/(5*d) + b*x**3*(2*a*d - b*c)/(3*d**2) - sqrt(c)*(a*d - b*c)**2*atan(sq
rt(d)*x/sqrt(c))/d**(7/2) + (a*d - b*c)**2*Integral(d**(-3), x)

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Mathematica [A]  time = 0.114923, size = 83, normalized size = 1. \[ -\frac{\sqrt{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{7/2}}+\frac{x (a d-b c)^2}{d^3}-\frac{b x^3 (b c-2 a d)}{3 d^2}+\frac{b^2 x^5}{5 d} \]

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

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Maple [A]  time = 0.005, size = 135, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{5}}{5\,d}}+{\frac{2\,ab{x}^{3}}{3\,d}}-{\frac{{x}^{3}{b}^{2}c}{3\,{d}^{2}}}+{\frac{{a}^{2}x}{d}}-2\,{\frac{xabc}{{d}^{2}}}+{\frac{{b}^{2}{c}^{2}x}{{d}^{3}}}-{\frac{{a}^{2}c}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+2\,{\frac{ab{c}^{2}}{{d}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{2}{c}^{3}}{{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

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*a*b*c*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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243301, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, b^{2} d^{2} x^{5} - 10 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) + 30 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{30 \, d^{3}}, \frac{3 \, b^{2} d^{2} x^{5} - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} - 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^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(x/sqrt(c/d)) + 15*(b^2*
c^2 - 2*a*b*c*d + a^2*d^2)*x)/d^3]

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Sympy [A]  time = 2.43984, size = 192, normalized size = 2.31 \[ \frac{b^{2} x^{5}}{5 d} + \frac{\sqrt{- \frac{c}{d^{7}}} \left (a d - b c\right )^{2} \log{\left (- \frac{d^{3} \sqrt{- \frac{c}{d^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{c}{d^{7}}} \left (a d - b c\right )^{2} \log{\left (\frac{d^{3} \sqrt{- \frac{c}{d^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{x^{3} \left (2 a b d - b^{2} c\right )}{3 d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d^{3}} \]

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/XCAS [A]  time = 0.231221, size = 153, normalized size = 1.84 \[ -\frac{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{3}} + \frac{3 \, b^{2} d^{4} x^{5} - 5 \, b^{2} c d^{3} x^{3} + 10 \, a b d^{4} x^{3} + 15 \, b^{2} c^{2} d^{2} x - 30 \, a b c d^{3} x + 15 \, a^{2} d^{4} x}{15 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^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) + 1/1
5*(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