Optimal. Leaf size=171 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 c^3 d^3 (d+e x)^{7/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.337385, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 c^3 d^3 (d+e x)^{7/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 54.1242, size = 160, normalized size = 0.94 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{11 c d \left (d + e x\right )^{\frac{3}{2}}} - \frac{8 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{99 c^{2} d^{2} \left (d + e x\right )^{\frac{5}{2}}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{693 c^{3} d^{3} \left (d + e x\right )^{\frac{7}{2}}} \]
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)**(5/2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.17146, size = 98, normalized size = 0.57 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 e^4-4 a c d e^2 (11 d+7 e x)+c^2 d^2 \left (99 d^2+154 d e x+63 e^2 x^2\right )\right )}{693 c^3 d^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.011, size = 110, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 63\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-28\,xacd{e}^{3}+154\,x{c}^{2}{d}^{3}e+8\,{a}^{2}{e}^{4}-44\,ac{d}^{2}{e}^{2}+99\,{c}^{2}{d}^{4} \right ) }{693\,{c}^{3}{d}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
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)^(5/2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.792962, size = 293, normalized size = 1.71 \[ \frac{2 \,{\left (63 \, c^{5} d^{5} e^{2} x^{5} + 99 \, a^{3} c^{2} d^{4} e^{3} - 44 \, a^{4} c d^{2} e^{5} + 8 \, a^{5} e^{7} + 7 \,{\left (22 \, c^{5} d^{6} e + 23 \, a c^{4} d^{4} e^{3}\right )} x^{4} +{\left (99 \, c^{5} d^{7} + 418 \, a c^{4} d^{5} e^{2} + 113 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} + 3 \,{\left (99 \, a c^{4} d^{6} e + 110 \, a^{2} c^{3} d^{4} e^{3} + a^{3} c^{2} d^{2} e^{5}\right )} x^{2} +{\left (297 \, a^{2} c^{3} d^{5} e^{2} + 22 \, a^{3} c^{2} d^{3} e^{4} - 4 \, a^{4} c d e^{6}\right )} x\right )} \sqrt{c d x + a e}}{693 \, c^{3} d^{3}} \]
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)^(5/2)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213768, size = 512, normalized size = 2.99 \[ \frac{2 \,{\left (63 \, c^{6} d^{6} e^{3} x^{7} + 99 \, a^{4} c^{2} d^{5} e^{4} - 44 \, a^{5} c d^{3} e^{6} + 8 \, a^{6} d e^{8} + 7 \,{\left (31 \, c^{6} d^{7} e^{2} + 32 \, a c^{5} d^{5} e^{4}\right )} x^{6} +{\left (253 \, c^{6} d^{8} e + 796 \, a c^{5} d^{6} e^{3} + 274 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{5} +{\left (99 \, c^{6} d^{9} + 968 \, a c^{5} d^{7} e^{2} + 1022 \, a^{2} c^{4} d^{5} e^{4} + 116 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{4} +{\left (396 \, a c^{5} d^{8} e + 1342 \, a^{2} c^{4} d^{6} e^{3} + 468 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x^{3} +{\left (594 \, a^{2} c^{4} d^{7} e^{2} + 748 \, a^{3} c^{3} d^{5} e^{4} - 23 \, a^{4} c^{2} d^{3} e^{6} + 4 \, a^{5} c d e^{8}\right )} x^{2} +{\left (396 \, a^{3} c^{3} d^{6} e^{3} + 77 \, a^{4} c^{2} d^{4} e^{5} - 40 \, a^{5} c d^{2} e^{7} + 8 \, a^{6} e^{9}\right )} x\right )}}{693 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]
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)^(5/2)/sqrt(e*x + d),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)**(5/2)/(e*x+d)**(1/2),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)^(5/2)/sqrt(e*x + d),x, algorithm="giac")
[Out]