Optimal. Leaf size=81 \[ -\frac{4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac{2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \]
[Out]
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Rubi [A] time = 0.118615, antiderivative size = 81, 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 \[ -\frac{4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac{2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \]
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)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 34.9753, size = 75, normalized size = 0.93 \[ \frac{2 c^{2} d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{3}} + \frac{4 c d \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{3 e^{3}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}}{e^{3}} \]
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)**(5/2),x)
[Out]
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Mathematica [A] time = 0.064554, size = 66, normalized size = 0.81 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^4+10 a c d e^2 (e x-2 d)+c^2 d^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
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)^(5/2),x]
[Out]
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Maple [A] time = 0.009, size = 73, normalized size = 0.9 \[{\frac{6\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+20\,xacd{e}^{3}-8\,x{c}^{2}{d}^{3}e+30\,{a}^{2}{e}^{4}-40\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{15\,{e}^{3}}\sqrt{ex+d}} \]
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)^(5/2),x)
[Out]
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Maxima [A] time = 0.752428, size = 108, normalized size = 1.33 \[ \frac{2 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} d^{2} - 10 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{e x + d}\right )}}{15 \, e^{3}} \]
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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21656, size = 100, normalized size = 1.23 \[ \frac{2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} - 20 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \,{\left (2 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \]
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)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.6311, size = 236, normalized size = 2.91 \[ \begin{cases} - \frac{\frac{2 a^{2} d e^{2}}{\sqrt{d + e x}} + 2 a^{2} e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 4 a c d^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 4 a c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + \frac{2 c^{2} d^{3} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 c^{2} d^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{3}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \]
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)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208737, size = 143, normalized size = 1.77 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d^{2} e^{12} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{3} e^{12} + 15 \, \sqrt{x e + d} c^{2} d^{4} e^{12} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} a c d e^{14} - 30 \, \sqrt{x e + d} a c d^{2} e^{14} + 15 \, \sqrt{x e + d} a^{2} e^{16}\right )} e^{\left (-15\right )} \]
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)^2/(e*x + d)^(5/2),x, algorithm="giac")
[Out]