3.663 \(\int \frac{x \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{2 b (b c-a d)}{d^3 \sqrt{c+d x^2}}-\frac{(b c-a d)^2}{3 d^3 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.158596, antiderivative size = 72, 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 \[ \frac{2 b (b c-a d)}{d^3 \sqrt{c+d x^2}}-\frac{(b c-a d)^2}{3 d^3 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \sqrt{c+d x^2}}{d^3} \]

Antiderivative was successfully verified.

[In]  Int[(x*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.8012, size = 63, normalized size = 0.88 \[ \frac{b^{2} \sqrt{c + d x^{2}}}{d^{3}} - \frac{2 b \left (a d - b c\right )}{d^{3} \sqrt{c + d x^{2}}} - \frac{\left (a d - b c\right )^{2}}{3 d^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0763108, size = 67, normalized size = 0.93 \[ \frac{-a^2 d^2-2 a b d \left (2 c+3 d x^2\right )+b^2 \left (8 c^2+12 c d x^2+3 d^2 x^4\right )}{3 d^3 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 68, normalized size = 0.9 \[ -{\frac{-3\,{b}^{2}{d}^{2}{x}^{4}+6\,ab{d}^{2}{x}^{2}-12\,{b}^{2}cd{x}^{2}+{a}^{2}{d}^{2}+4\,cabd-8\,{b}^{2}{c}^{2}}{3\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

-1/3*(-3*b^2*d^2*x^4+6*a*b*d^2*x^2-12*b^2*c*d*x^2+a^2*d^2+4*a*b*c*d-8*b^2*c^2)/(
d*x^2+c)^(3/2)/d^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*x/(d*x^2 + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.216283, size = 123, normalized size = 1.71 \[ \frac{{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 4 \, a b c d - a^{2} d^{2} + 6 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x/(d*x^2 + c)^(5/2),x, algorithm="fricas")

[Out]

1/3*(3*b^2*d^2*x^4 + 8*b^2*c^2 - 4*a*b*c*d - a^2*d^2 + 6*(2*b^2*c*d - a*b*d^2)*x
^2)*sqrt(d*x^2 + c)/(d^5*x^4 + 2*c*d^4*x^2 + c^2*d^3)

_______________________________________________________________________________________

Sympy [A]  time = 4.25115, size = 303, normalized size = 4.21 \[ \begin{cases} - \frac{a^{2} d^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} - \frac{4 a b c d}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} - \frac{6 a b d^{2} x^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} + \frac{8 b^{2} c^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} + \frac{12 b^{2} c d x^{2}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} + \frac{3 b^{2} d^{2} x^{4}}{3 c d^{3} \sqrt{c + d x^{2}} + 3 d^{4} x^{2} \sqrt{c + d x^{2}}} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Piecewise((-a**2*d**2/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2))
- 4*a*b*c*d/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) - 6*a*b*
d**2*x**2/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) + 8*b**2*c*
*2/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) + 12*b**2*c*d*x**2
/(3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)) + 3*b**2*d**2*x**4/(
3*c*d**3*sqrt(c + d*x**2) + 3*d**4*x**2*sqrt(c + d*x**2)), Ne(d, 0)), ((a**2*x**
2/2 + a*b*x**4/2 + b**2*x**6/6)/c**(5/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.233521, size = 105, normalized size = 1.46 \[ \frac{3 \, \sqrt{d x^{2} + c} b^{2} + \frac{6 \,{\left (d x^{2} + c\right )} b^{2} c - b^{2} c^{2} - 6 \,{\left (d x^{2} + c\right )} a b d + 2 \, a b c d - a^{2} d^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x/(d*x^2 + c)^(5/2),x, algorithm="giac")

[Out]

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