Optimal. Leaf size=119 \[ -\frac{6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \]
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Rubi [A] time = 0.239967, antiderivative size = 119, 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 \[ -\frac{6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
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Rubi in Sympy [A] time = 47.4749, size = 110, normalized size = 0.92 \[ \frac{2 c^{3} d^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{4}} + \frac{6 c^{2} d^{2} \left (d + e x\right )^{\frac{13}{2}} \left (a e^{2} - c d^{2}\right )}{13 e^{4}} + \frac{6 c d \left (d + e x\right )^{\frac{11}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{11 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} - c d^{2}\right )^{3}}{9 e^{4}} \]
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*(e*x+d)**(1/2),x)
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Mathematica [A] time = 0.217912, size = 111, normalized size = 0.93 \[ \frac{2 (d+e x)^{9/2} \left (715 a^3 e^6-195 a^2 c d e^4 (2 d-9 e x)+15 a c^2 d^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
[Out]
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Maple [A] time = 0.011, size = 131, normalized size = 1.1 \[{\frac{858\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+2970\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-396\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+3510\,x{a}^{2}cd{e}^{5}-1080\,xa{c}^{2}{d}^{3}{e}^{3}+144\,{c}^{3}{d}^{5}ex+1430\,{a}^{3}{e}^{6}-780\,{a}^{2}c{d}^{2}{e}^{4}+240\,{c}^{2}{d}^{4}a{e}^{2}-32\,{c}^{3}{d}^{6}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]
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*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.753883, size = 185, normalized size = 1.55 \[ \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} d^{3} - 1485 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1755 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \]
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*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220659, size = 452, normalized size = 3.8 \[ \frac{2 \,{\left (429 \, c^{3} d^{3} e^{7} x^{7} - 16 \, c^{3} d^{10} + 120 \, a c^{2} d^{8} e^{2} - 390 \, a^{2} c d^{6} e^{4} + 715 \, a^{3} d^{4} e^{6} + 33 \,{\left (46 \, c^{3} d^{4} e^{6} + 45 \, a c^{2} d^{2} e^{8}\right )} x^{6} + 9 \,{\left (206 \, c^{3} d^{5} e^{5} + 600 \, a c^{2} d^{3} e^{7} + 195 \, a^{2} c d e^{9}\right )} x^{5} + 5 \,{\left (160 \, c^{3} d^{6} e^{4} + 1374 \, a c^{2} d^{4} e^{6} + 1326 \, a^{2} c d^{2} e^{8} + 143 \, a^{3} e^{10}\right )} x^{4} + 5 \,{\left (c^{3} d^{7} e^{3} + 636 \, a c^{2} d^{5} e^{5} + 1794 \, a^{2} c d^{3} e^{7} + 572 \, a^{3} d e^{9}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{8} e^{2} - 15 \, a c^{2} d^{6} e^{4} - 1560 \, a^{2} c d^{4} e^{6} - 1430 \, a^{3} d^{2} e^{8}\right )} x^{2} +{\left (8 \, c^{3} d^{9} e - 60 \, a c^{2} d^{7} e^{3} + 195 \, a^{2} c d^{5} e^{5} + 2860 \, a^{3} d^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \]
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*sqrt(e*x + d),x, algorithm="fricas")
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Sympy [A] time = 4.54536, size = 165, normalized size = 1.39 \[ \frac{2 \left (\frac{c^{3} d^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{3}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{13 e^{3}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{11 e^{3}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right )}{9 e^{3}}\right )}{e} \]
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*(e*x+d)**(1/2),x)
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GIAC/XCAS [A] time = 0.225466, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
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*sqrt(e*x + d),x, algorithm="giac")
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