Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0729824, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 18.3452, size = 31, normalized size = 0.91 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{7}{2}}}{7 c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0337355, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4 \left (c (d+e x)^2\right )^{3/2}}{7 e} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.006, size = 95, normalized size = 2.8 \[{\frac{x \left ({e}^{6}{x}^{6}+7\,d{e}^{5}{x}^{5}+21\,{d}^{2}{e}^{4}{x}^{4}+35\,{d}^{3}{e}^{3}{x}^{3}+35\,{d}^{4}{e}^{2}{x}^{2}+21\,{d}^{5}ex+7\,{d}^{6} \right ) }{7\, \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),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((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.232314, size = 139, normalized size = 4.09 \[ \frac{{\left (c e^{6} x^{7} + 7 \, c d e^{5} x^{6} + 21 \, c d^{2} e^{4} x^{5} + 35 \, c d^{3} e^{3} x^{4} + 35 \, c d^{4} e^{2} x^{3} + 21 \, c d^{5} e x^{2} + 7 \, c d^{6} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \,{\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 6.75256, size = 277, normalized size = 8.15 \[ \begin{cases} \frac{c d^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac{6 c d^{5} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{15 c d^{4} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{20 c d^{3} e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{15 c d^{2} e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{6 c d e^{4} x^{5} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac{c e^{5} x^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} & \text{for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.220404, size = 119, normalized size = 3.5 \[ \frac{1}{7} \,{\left (c d^{6} e^{\left (-1\right )} +{\left (6 \, c d^{5} +{\left (15 \, c d^{4} e +{\left (20 \, c d^{3} e^{2} +{\left (15 \, c d^{2} e^{3} +{\left (c x e^{5} + 6 \, c d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*(e*x + d)^3,x, algorithm="giac")
[Out]