Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.401916, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]
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)^8,x]
[Out]
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Rubi in Sympy [A] time = 79.5066, size = 218, normalized size = 0.94 \[ \frac{32 c^{3} d^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{1155 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{231 \left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{33 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{11 \left (d + e x\right )^{8} \left (a e^{2} - c d^{2}\right )} \]
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)**8,x)
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Mathematica [A] time = 0.247787, size = 138, normalized size = 0.6 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (-105 a^3 e^6+35 a^2 c d e^4 (11 d+2 e x)-5 a c^2 d^2 e^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )+c^3 d^3 \left (231 d^3+198 d^2 e x+88 d e^2 x^2+16 e^3 x^3\right )\right )}{1155 (d+e x)^8 \left (c d^2-a e^2\right )^4} \]
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)^8,x]
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Maple [A] time = 0.017, size = 217, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+40\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-88\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}-70\,x{a}^{2}cd{e}^{5}+220\,xa{c}^{2}{d}^{3}{e}^{3}-198\,{c}^{3}{d}^{5}ex+105\,{a}^{3}{e}^{6}-385\,{a}^{2}c{d}^{2}{e}^{4}+495\,{c}^{2}{d}^{4}a{e}^{2}-231\,{c}^{3}{d}^{6} \right ) }{1155\, \left ( ex+d \right ) ^{7} \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}}} \]
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)^8,x)
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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)^8,x, algorithm="maxima")
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Fricas [A] time = 13.9808, size = 944, normalized size = 4.09 \[ \frac{2 \,{\left (16 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 495 \, a^{3} c^{2} d^{4} e^{4} + 385 \, a^{4} c d^{2} e^{6} - 105 \, a^{5} e^{8} + 8 \,{\left (11 \, c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4}\right )} x^{4} + 2 \,{\left (99 \, c^{5} d^{7} e - 22 \, a c^{4} d^{5} e^{3} + 3 \, a^{2} c^{3} d^{3} e^{5}\right )} x^{3} +{\left (231 \, c^{5} d^{8} - 99 \, a c^{4} d^{6} e^{2} + 33 \, a^{2} c^{3} d^{4} e^{4} - 5 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (231 \, a c^{4} d^{7} e - 396 \, a^{2} c^{3} d^{5} e^{3} + 275 \, a^{3} c^{2} d^{3} e^{5} - 70 \, a^{4} c d e^{7}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{1155 \,{\left (c^{4} d^{14} - 4 \, a c^{3} d^{12} e^{2} + 6 \, a^{2} c^{2} d^{10} e^{4} - 4 \, a^{3} c d^{8} e^{6} + a^{4} d^{6} e^{8} +{\left (c^{4} d^{8} e^{6} - 4 \, a c^{3} d^{6} e^{8} + 6 \, a^{2} c^{2} d^{4} e^{10} - 4 \, a^{3} c d^{2} e^{12} + a^{4} e^{14}\right )} x^{6} + 6 \,{\left (c^{4} d^{9} e^{5} - 4 \, a c^{3} d^{7} e^{7} + 6 \, a^{2} c^{2} d^{5} e^{9} - 4 \, a^{3} c d^{3} e^{11} + a^{4} d e^{13}\right )} x^{5} + 15 \,{\left (c^{4} d^{10} e^{4} - 4 \, a c^{3} d^{8} e^{6} + 6 \, a^{2} c^{2} d^{6} e^{8} - 4 \, a^{3} c d^{4} e^{10} + a^{4} d^{2} e^{12}\right )} x^{4} + 20 \,{\left (c^{4} d^{11} e^{3} - 4 \, a c^{3} d^{9} e^{5} + 6 \, a^{2} c^{2} d^{7} e^{7} - 4 \, a^{3} c d^{5} e^{9} + a^{4} d^{3} e^{11}\right )} x^{3} + 15 \,{\left (c^{4} d^{12} e^{2} - 4 \, a c^{3} d^{10} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{6} - 4 \, a^{3} c d^{6} e^{8} + a^{4} d^{4} e^{10}\right )} x^{2} + 6 \,{\left (c^{4} d^{13} e - 4 \, a c^{3} d^{11} e^{3} + 6 \, a^{2} c^{2} d^{9} e^{5} - 4 \, a^{3} c d^{7} e^{7} + a^{4} d^{5} e^{9}\right )} x\right )}} \]
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)^8,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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**8,x)
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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)^8,x, algorithm="giac")
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