Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.279351, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^5 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^6 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^7 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^7,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 53.7516, size = 160, normalized size = 0.94 \[ - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{315 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{8 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{63 \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{9 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**7,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.177947, size = 94, normalized size = 0.55 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 e^4-10 a c d e^2 (9 d+2 e x)+c^2 d^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )}{315 (d+e x)^7 \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^7,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.015, size = 146, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-20\,xacd{e}^{3}+36\,x{c}^{2}{d}^{3}e+35\,{a}^{2}{e}^{4}-90\,ac{d}^{2}{e}^{2}+63\,{c}^{2}{d}^{4} \right ) }{315\, \left ( ex+d \right ) ^{6} \left ({a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{c}^{2}{d}^{4}a{e}^{2}-{c}^{3}{d}^{6} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^7,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*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^7,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 4.60399, size = 637, normalized size = 3.73 \[ \frac{2 \,{\left (8 \, c^{4} d^{4} e^{2} x^{4} + 63 \, a^{2} c^{2} d^{4} e^{2} - 90 \, a^{3} c d^{2} e^{4} + 35 \, a^{4} e^{6} + 4 \,{\left (9 \, c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \,{\left (21 \, c^{4} d^{6} - 6 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (63 \, a c^{3} d^{5} e - 72 \, a^{2} c^{2} d^{3} e^{3} + 25 \, a^{3} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{315 \,{\left (c^{3} d^{11} - 3 \, a c^{2} d^{9} e^{2} + 3 \, a^{2} c d^{7} e^{4} - a^{3} d^{5} e^{6} +{\left (c^{3} d^{6} e^{5} - 3 \, a c^{2} d^{4} e^{7} + 3 \, a^{2} c d^{2} e^{9} - a^{3} e^{11}\right )} x^{5} + 5 \,{\left (c^{3} d^{7} e^{4} - 3 \, a c^{2} d^{5} e^{6} + 3 \, a^{2} c d^{3} e^{8} - a^{3} d e^{10}\right )} x^{4} + 10 \,{\left (c^{3} d^{8} e^{3} - 3 \, a c^{2} d^{6} e^{5} + 3 \, a^{2} c d^{4} e^{7} - a^{3} d^{2} e^{9}\right )} x^{3} + 10 \,{\left (c^{3} d^{9} e^{2} - 3 \, a c^{2} d^{7} e^{4} + 3 \, a^{2} c d^{5} e^{6} - a^{3} d^{3} e^{8}\right )} x^{2} + 5 \,{\left (c^{3} d^{10} e - 3 \, a c^{2} d^{8} e^{3} + 3 \, a^{2} c d^{6} e^{5} - a^{3} d^{4} e^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^7,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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**7,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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^7,x, algorithm="giac")
[Out]