3.656 \(\int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}+\frac{x \left (3 b^2 c^2-4 a d (3 b c-2 a d)\right )}{3 c^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.210615, antiderivative size = 98, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x \left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right )}{3 c^3 \sqrt{c+d x^2}}-\frac{a^2}{3 c x^3 \sqrt{c+d x^2}}-\frac{2 a (3 b c-2 a d)}{3 c^2 x \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(3*c*x^3*Sqrt[c + d*x^2]) - (2*a*(3*b*c - 2*a*d))/(3*c^2*x*Sqrt[c + d*x^2])
+ ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*x)/(3*c^3*Sqrt[c + d*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.0801, size = 90, normalized size = 0.93 \[ - \frac{a^{2}}{3 c x^{3} \sqrt{c + d x^{2}}} + \frac{2 a \left (2 a d - 3 b c\right )}{3 c^{2} x \sqrt{c + d x^{2}}} + \frac{x \left (4 a d \left (2 a d - 3 b c\right ) + 3 b^{2} c^{2}\right )}{3 c^{3} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.114484, size = 66, normalized size = 0.68 \[ \frac{\sqrt{c+d x^2} \left (-a^2 c+a x^2 (5 a d-6 b c)+\frac{3 x^4 (b c-a d)^2}{c+d x^2}\right )}{3 c^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(-(a^2*c) + a*(-6*b*c + 5*a*d)*x^2 + (3*(b*c - a*d)^2*x^4)/(c +
d*x^2)))/(3*c^3*x^3)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 77, normalized size = 0.8 \[ -{\frac{-8\,{x}^{4}{a}^{2}{d}^{2}+12\,{x}^{4}abcd-3\,{x}^{4}{b}^{2}{c}^{2}-4\,{x}^{2}{a}^{2}cd+6\,a{c}^{2}b{x}^{2}+{a}^{2}{c}^{2}}{3\,{x}^{3}{c}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-8*a^2*d^2*x^4+12*a*b*c*d*x^4-3*b^2*c^2*x^4-4*a^2*c*d*x^2+6*a*b*c^2*x^2+a^
2*c^2)/(d*x^2+c)^(1/2)/x^3/c^3

_______________________________________________________________________________________

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/((d*x^2 + c)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.235962, size = 115, normalized size = 1.19 \[ \frac{{\left ({\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} - a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (c^{3} d x^{5} + c^{4} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*x^4 - a^2*c^2 - 2*(3*a*b*c^2 - 2*a^2*c
*d)*x^2)*sqrt(d*x^2 + c)/(c^3*d*x^5 + c^4*x^3)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{4} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.244736, size = 269, normalized size = 2.77 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt{d x^{2} + c} c^{3}} + \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c \sqrt{d} - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{2} \sqrt{d} + 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c d^{\frac{3}{2}} + 6 \, a b c^{3} \sqrt{d} - 5 \, a^{2} c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(sqrt(d*x^2 + c)*c^3) + 2/3*(6*(sqrt(d)*x - sq
rt(d*x^2 + c))^4*a*b*c*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*d^(3/2) -
12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^2*sqrt(d) + 12*(sqrt(d)*x - sqrt(d*x^2
+ c))^2*a^2*c*d^(3/2) + 6*a*b*c^3*sqrt(d) - 5*a^2*c^2*d^(3/2))/(((sqrt(d)*x - s
qrt(d*x^2 + c))^2 - c)^3*c^2)