Optimal. Leaf size=79 \[ -\frac{4 c d \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^3}-\frac{2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt{d+e x}}+\frac{2 c^2 d^2 (d+e x)^{3/2}}{3 e^3} \]
[Out]
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Rubi [A] time = 0.118895, antiderivative size = 79, 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 \[ -\frac{4 c d \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^3}-\frac{2 \left (c d^2-a e^2\right )^2}{e^3 \sqrt{d+e x}}+\frac{2 c^2 d^2 (d+e x)^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 34.9655, size = 73, normalized size = 0.92 \[ \frac{2 c^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{4 c d \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{e^{3}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{2}}{e^{3} \sqrt{d + e x}} \]
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)**2/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.0753528, size = 65, normalized size = 0.82 \[ \frac{2 \left (-3 a^2 e^4+6 a c d e^2 (2 d+e x)+c^2 d^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.009, size = 73, normalized size = 0.9 \[ -{\frac{-2\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-12\,xacd{e}^{3}+8\,x{c}^{2}{d}^{3}e+6\,{a}^{2}{e}^{4}-24\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]
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)^2/(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 0.744881, size = 117, normalized size = 1.48 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c^{2} d^{2} - 6 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]
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)^2/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221812, size = 99, normalized size = 1.25 \[ \frac{2 \,{\left (c^{2} d^{2} e^{2} x^{2} - 8 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} - 2 \,{\left (2 \, c^{2} d^{3} e - 3 \, a c d e^{3}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]
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)^2/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.4127, size = 133, normalized size = 1.68 \[ \begin{cases} - \frac{2 a^{2} e}{\sqrt{d + e x}} + \frac{8 a c d^{2}}{e \sqrt{d + e x}} + \frac{4 a c d x}{\sqrt{d + e x}} - \frac{16 c^{2} d^{4}}{3 e^{3} \sqrt{d + e x}} - \frac{8 c^{2} d^{3} x}{3 e^{2} \sqrt{d + e x}} + \frac{2 c^{2} d^{2} x^{2}}{3 e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{2} \sqrt{d} x^{3}}{3} & \text{otherwise} \end{cases} \]
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)**2/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.20876, size = 155, normalized size = 1.96 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{6} - 6 \, \sqrt{x e + d} c^{2} d^{3} e^{6} + 6 \, \sqrt{x e + d} a c d e^{8}\right )} e^{\left (-9\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{2} c^{2} d^{4} - 2 \,{\left (x e + d\right )}^{2} a c d^{2} e^{2} +{\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{\frac{5}{2}}} \]
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)^2/(e*x + d)^(7/2),x, algorithm="giac")
[Out]