Optimal. Leaf size=171 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.331083, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 (d+e x)^{3/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 52.8861, size = 160, normalized size = 0.94 \[ \frac{2 \sqrt{d + e x} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{7 c d} - \frac{8 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{35 c^{2} d^{2} \sqrt{d + e x}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{105 c^{3} d^{3} \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0996149, size = 118, normalized size = 0.69 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^3 e^5-4 a^2 c d e^3 (7 d+e x)+a c^2 d^2 e \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 x \left (35 d^2+42 d e x+15 e^2 x^2\right )\right )}{105 c^3 d^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 110, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 15\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-12\,xacd{e}^{3}+42\,x{c}^{2}{d}^{3}e+8\,{a}^{2}{e}^{4}-28\,ac{d}^{2}{e}^{2}+35\,{c}^{2}{d}^{4} \right ) }{105\,{c}^{3}{d}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.756308, size = 189, normalized size = 1.11 \[ \frac{2 \,{\left (15 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 28 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + 3 \,{\left (14 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (35 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - 4 \, a^{2} c d e^{4}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{105 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.251755, size = 347, normalized size = 2.03 \[ \frac{2 \,{\left (15 \, c^{4} d^{4} e^{3} x^{5} + 35 \, a^{2} c^{2} d^{5} e^{2} - 28 \, a^{3} c d^{3} e^{4} + 8 \, a^{4} d e^{6} + 3 \,{\left (19 \, c^{4} d^{5} e^{2} + 6 \, a c^{3} d^{3} e^{4}\right )} x^{4} +{\left (77 \, c^{4} d^{6} e + 74 \, a c^{3} d^{4} e^{3} - a^{2} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (35 \, c^{4} d^{7} + 126 \, a c^{3} d^{5} e^{2} - 15 \, a^{2} c^{2} d^{3} e^{4} + 4 \, a^{3} c d e^{6}\right )} x^{2} +{\left (70 \, a c^{3} d^{6} e + 21 \, a^{2} c^{2} d^{4} e^{3} - 24 \, a^{3} c d^{2} e^{5} + 8 \, a^{4} e^{7}\right )} x\right )}}{105 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2),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((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]