Optimal. Leaf size=81 \[ \frac{4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac{2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 c^2 d^2}{e^3 \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 0.117844, 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 \[ \frac{4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac{2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 c^2 d^2}{e^3 \sqrt{d+e x}} \]
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)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 35.0259, size = 76, normalized size = 0.94 \[ - \frac{2 c^{2} d^{2}}{e^{3} \sqrt{d + e x}} - \frac{4 c d \left (a e^{2} - c d^{2}\right )}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{2}}{5 e^{3} \left (d + e x\right )^{\frac{5}{2}}} \]
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)**2/(e*x+d)**(11/2),x)
[Out]
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Mathematica [A] time = 0.0773904, size = 67, normalized size = 0.83 \[ -\frac{2 \left (3 a^2 e^4+2 a c d e^2 (2 d+5 e x)+c^2 d^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
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)^(11/2),x]
[Out]
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Maple [A] time = 0.009, size = 73, normalized size = 0.9 \[ -{\frac{30\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+20\,xacd{e}^{3}+40\,x{c}^{2}{d}^{3}e+6\,{a}^{2}{e}^{4}+8\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
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)^2/(e*x+d)^(11/2),x)
[Out]
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Maxima [A] time = 0.74176, size = 104, normalized size = 1.28 \[ -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c^{2} d^{2} + 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 10 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \]
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)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249757, size = 127, normalized size = 1.57 \[ -\frac{2 \,{\left (15 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]
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)^(11/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 48.7122, size = 388, normalized size = 4.79 \[ \begin{cases} - \frac{6 a^{2} e^{4}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{8 a c d^{2} e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{20 a c d e^{3} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c^{2} d^{4}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c^{2} d^{3} e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{2} x^{3}}{3 d^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
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)**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.209527, size = 146, normalized size = 1.8 \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{4} c^{2} d^{2} - 10 \,{\left (x e + d\right )}^{3} c^{2} d^{3} + 3 \,{\left (x e + d\right )}^{2} c^{2} d^{4} + 10 \,{\left (x e + d\right )}^{3} a c d e^{2} - 6 \,{\left (x e + d\right )}^{2} a c d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]
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)^(11/2),x, algorithm="giac")
[Out]