Optimal. Leaf size=74 \[ \frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.103128, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.0446, size = 61, normalized size = 0.82 \[ \frac{8 d \sqrt{d + e x}}{c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{c e \sqrt{c d^{2} - c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0422803, size = 43, normalized size = 0.58 \[ \frac{2 (3 d-e x) \sqrt{d+e x}}{c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.004, size = 44, normalized size = 0.6 \[ 2\,{\frac{ \left ( -ex+d \right ) \left ( -ex+3\,d \right ) \left ( ex+d \right ) ^{3/2}}{e \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{3/2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.723812, size = 31, normalized size = 0.42 \[ -\frac{2 \,{\left (e x - 3 \, d\right )}}{\sqrt{-e x + d} c^{\frac{3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.225875, size = 68, normalized size = 0.92 \[ -\frac{2 \,{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )}}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{\frac{5}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(5/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.617146, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")
[Out]