Optimal. Leaf size=39 \[ -\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0573374, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.5599, size = 37, normalized size = 0.95 \[ - \frac{2 d^{2} \left (b + 2 c x\right )^{3}}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0741743, size = 38, normalized size = 0.97 \[ -\frac{2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^2/(a + b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 38, normalized size = 1. \[{\frac{2\,{d}^{2} \left ( 2\,cx+b \right ) ^{3}}{12\,ac-3\,{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
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((2*c*d*x + b*d)^2/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.303054, size = 197, normalized size = 5.05 \[ -\frac{2 \,{\left (8 \, c^{3} d^{2} x^{3} + 12 \, b c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x + b^{3} d^{2}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.228677, size = 296, normalized size = 7.59 \[ -\frac{2 \,{\left (2 \,{\left (\frac{2 \,{\left (b^{2} c^{3} d^{2} - 4 \, a c^{4} d^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (b^{3} c^{2} d^{2} - 4 \, a b c^{3} d^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (b^{4} c d^{2} - 4 \, a b^{2} c^{2} d^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{5} d^{2} - 4 \, a b^{3} c d^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]