Optimal. Leaf size=295 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d} \]
[Out]
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Rubi [A] time = 0.706701, 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 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15015 c^5 d^5 (d+e x)^{5/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3003 c^4 d^4 (d+e x)^{3/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{429 c^3 d^3 \sqrt{d+e x}}+\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{143 c^2 d^2}+\frac{2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 105.818, size = 279, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{13 c d} - \frac{16 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{143 c^{2} d^{2}} + \frac{32 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{429 c^{3} d^{3} \sqrt{d + e x}} - \frac{128 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{3003 c^{4} d^{4} \left (d + e x\right )^{\frac{3}{2}}} + \frac{256 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{15015 c^{5} d^{5} \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.289004, size = 187, normalized size = 0.63 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (128 a^4 e^8-64 a^3 c d e^6 (13 d+5 e x)+16 a^2 c^2 d^2 e^4 \left (143 d^2+130 d e x+35 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (429 d^3+715 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right )+c^4 d^4 \left (3003 d^4+8580 d^3 e x+10010 d^2 e^2 x^2+5460 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 243, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1155\,{x}^{4}{c}^{4}{d}^{4}{e}^{4}-840\,{x}^{3}a{c}^{3}{d}^{3}{e}^{5}+5460\,{x}^{3}{c}^{4}{d}^{5}{e}^{3}+560\,{x}^{2}{a}^{2}{c}^{2}{d}^{2}{e}^{6}-3640\,{x}^{2}a{c}^{3}{d}^{4}{e}^{4}+10010\,{x}^{2}{c}^{4}{d}^{6}{e}^{2}-320\,x{a}^{3}cd{e}^{7}+2080\,x{a}^{2}{c}^{2}{d}^{3}{e}^{5}-5720\,xa{c}^{3}{d}^{5}{e}^{3}+8580\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-832\,{a}^{3}c{d}^{2}{e}^{6}+2288\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-3432\,a{c}^{3}{d}^{6}{e}^{2}+3003\,{c}^{4}{d}^{8} \right ) }{15015\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.764775, size = 504, normalized size = 1.71 \[ \frac{2 \,{\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \,{\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \,{\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \,{\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \,{\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \,{\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{15015 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223881, size = 753, normalized size = 2.55 \[ \frac{2 \,{\left (1155 \, c^{7} d^{7} e^{5} x^{8} + 3003 \, a^{3} c^{4} d^{9} e^{3} - 3432 \, a^{4} c^{3} d^{7} e^{5} + 2288 \, a^{5} c^{2} d^{5} e^{7} - 832 \, a^{6} c d^{3} e^{9} + 128 \, a^{7} d e^{11} + 105 \,{\left (63 \, c^{7} d^{8} e^{4} + 25 \, a c^{6} d^{6} e^{6}\right )} x^{7} + 35 \,{\left (442 \, c^{7} d^{9} e^{3} + 439 \, a c^{6} d^{7} e^{5} + 43 \, a^{2} c^{5} d^{5} e^{7}\right )} x^{6} + 5 \,{\left (3718 \, c^{7} d^{10} e^{2} + 7410 \, a c^{6} d^{8} e^{4} + 1809 \, a^{2} c^{5} d^{6} e^{6} - a^{3} c^{4} d^{4} e^{8}\right )} x^{5} +{\left (11583 \, c^{7} d^{11} e + 46618 \, a c^{6} d^{9} e^{3} + 22698 \, a^{2} c^{5} d^{7} e^{5} - 57 \, a^{3} c^{4} d^{5} e^{7} + 8 \, a^{4} c^{3} d^{3} e^{9}\right )} x^{4} +{\left (3003 \, c^{7} d^{12} + 31317 \, a c^{6} d^{10} e^{2} + 30602 \, a^{2} c^{5} d^{8} e^{4} - 338 \, a^{3} c^{4} d^{6} e^{6} + 112 \, a^{4} c^{3} d^{4} e^{8} - 16 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} +{\left (9009 \, a c^{6} d^{11} e + 24453 \, a^{2} c^{5} d^{9} e^{3} - 2002 \, a^{3} c^{4} d^{7} e^{5} + 1248 \, a^{4} c^{3} d^{5} e^{7} - 432 \, a^{5} c^{2} d^{3} e^{9} + 64 \, a^{6} c d e^{11}\right )} x^{2} +{\left (9009 \, a^{2} c^{5} d^{10} e^{2} + 1287 \, a^{3} c^{4} d^{8} e^{4} - 2288 \, a^{4} c^{3} d^{6} e^{6} + 1872 \, a^{5} c^{2} d^{4} e^{8} - 768 \, a^{6} c d^{2} e^{10} + 128 \, a^{7} e^{12}\right )} x\right )}}{15015 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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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)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
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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)^(5/2),x, algorithm="giac")
[Out]