Optimal. Leaf size=295 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.71276, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 102.566, size = 279, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{11 c d} - \frac{16 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{99 c^{2} d^{2}} + \frac{32 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{231 c^{3} d^{3}} - \frac{128 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{1155 c^{4} d^{4} \sqrt{d + e x}} + \frac{256 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3465 c^{5} d^{5} \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(7/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.198712, size = 239, normalized size = 0.81 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (128 a^5 e^9-64 a^4 c d e^7 (11 d+e x)+16 a^3 c^2 d^2 e^5 \left (99 d^2+22 d e x+3 e^2 x^2\right )-8 a^2 c^3 d^3 e^3 \left (231 d^3+99 d^2 e x+33 d e^2 x^2+5 e^3 x^3\right )+a c^4 d^4 e \left (1155 d^4+924 d^3 e x+594 d^2 e^2 x^2+220 d e^3 x^3+35 e^4 x^4\right )+c^5 d^5 x \left (1155 d^4+2772 d^3 e x+2970 d^2 e^2 x^2+1540 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 c^5 d^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 243, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 315\,{x}^{4}{c}^{4}{d}^{4}{e}^{4}-280\,{x}^{3}a{c}^{3}{d}^{3}{e}^{5}+1540\,{x}^{3}{c}^{4}{d}^{5}{e}^{3}+240\,{x}^{2}{a}^{2}{c}^{2}{d}^{2}{e}^{6}-1320\,{x}^{2}a{c}^{3}{d}^{4}{e}^{4}+2970\,{x}^{2}{c}^{4}{d}^{6}{e}^{2}-192\,x{a}^{3}cd{e}^{7}+1056\,x{a}^{2}{c}^{2}{d}^{3}{e}^{5}-2376\,xa{c}^{3}{d}^{5}{e}^{3}+2772\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-704\,{a}^{3}c{d}^{2}{e}^{6}+1584\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-1848\,a{c}^{3}{d}^{6}{e}^{2}+1155\,{c}^{4}{d}^{8} \right ) }{3465\,{c}^{5}{d}^{5}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.737935, size = 400, normalized size = 1.36 \[ \frac{2 \,{\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \,{\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \,{\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \,{\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{3465 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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)^(7/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.241849, size = 632, normalized size = 2.14 \[ \frac{2 \,{\left (315 \, c^{6} d^{6} e^{5} x^{7} + 1155 \, a^{2} c^{4} d^{9} e^{2} - 1848 \, a^{3} c^{3} d^{7} e^{4} + 1584 \, a^{4} c^{2} d^{5} e^{6} - 704 \, a^{5} c d^{3} e^{8} + 128 \, a^{6} d e^{10} + 35 \,{\left (53 \, c^{6} d^{7} e^{4} + 10 \, a c^{5} d^{5} e^{6}\right )} x^{6} + 5 \,{\left (902 \, c^{6} d^{8} e^{3} + 422 \, a c^{5} d^{6} e^{5} - a^{2} c^{4} d^{4} e^{7}\right )} x^{5} +{\left (5742 \, c^{6} d^{9} e^{2} + 5324 \, a c^{5} d^{7} e^{4} - 49 \, a^{2} c^{4} d^{5} e^{6} + 8 \, a^{3} c^{3} d^{3} e^{8}\right )} x^{4} +{\left (3927 \, c^{6} d^{10} e + 7260 \, a c^{5} d^{8} e^{3} - 242 \, a^{2} c^{4} d^{6} e^{5} + 96 \, a^{3} c^{3} d^{4} e^{7} - 16 \, a^{4} c^{2} d^{2} e^{9}\right )} x^{3} +{\left (1155 \, c^{6} d^{11} + 6006 \, a c^{5} d^{9} e^{2} - 1122 \, a^{2} c^{4} d^{7} e^{4} + 880 \, a^{3} c^{3} d^{5} e^{6} - 368 \, a^{4} c^{2} d^{3} e^{8} + 64 \, a^{5} c d e^{10}\right )} x^{2} +{\left (2310 \, a c^{5} d^{10} e + 231 \, a^{2} c^{4} d^{8} e^{3} - 1056 \, a^{3} c^{3} d^{6} e^{5} + 1232 \, a^{4} c^{2} d^{4} e^{7} - 640 \, a^{5} c d^{2} e^{9} + 128 \, a^{6} e^{11}\right )} x\right )}}{3465 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{5} d^{5}} \]
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)^(7/2),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)**(7/2)*(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)^(7/2),x, algorithm="giac")
[Out]