Optimal. Leaf size=43 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac{2 c d}{5 e^2 (d+e x)^{5/2}} \]
[Out]
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Rubi [A] time = 0.0645204, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{7 (d+e x)^{7/2}}-\frac{2 c d}{5 e^2 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 15.2769, size = 46, normalized size = 1.07 \[ - \frac{2 c d}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a e^{2} - c d^{2}\right )}{7 e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]
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)/(e*x+d)**(11/2),x)
[Out]
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Mathematica [A] time = 0.0447426, size = 34, normalized size = 0.79 \[ -\frac{2 \left (5 a e^2+c d (2 d+7 e x)\right )}{35 e^2 (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(11/2),x]
[Out]
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Maple [A] time = 0.006, size = 32, normalized size = 0.7 \[ -{\frac{14\,cdex+10\,a{e}^{2}+4\,c{d}^{2}}{35\,{e}^{2}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]
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)/(e*x+d)^(11/2),x)
[Out]
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Maxima [A] time = 0.759117, size = 46, normalized size = 1.07 \[ -\frac{2 \,{\left (7 \,{\left (e x + d\right )} c d - 5 \, c d^{2} + 5 \, a e^{2}\right )}}{35 \,{\left (e x + d\right )}^{\frac{7}{2}} e^{2}} \]
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)/(e*x + d)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222483, size = 85, normalized size = 1.98 \[ -\frac{2 \,{\left (7 \, c d e x + 2 \, c d^{2} + 5 \, a e^{2}\right )}}{35 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )} \sqrt{e x + d}} \]
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)/(e*x + d)^(11/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 39.0533, size = 248, normalized size = 5.77 \[ \begin{cases} - \frac{10 a e^{2}}{35 d^{3} e^{2} \sqrt{d + e x} + 105 d^{2} e^{3} x \sqrt{d + e x} + 105 d e^{4} x^{2} \sqrt{d + e x} + 35 e^{5} x^{3} \sqrt{d + e x}} - \frac{4 c d^{2}}{35 d^{3} e^{2} \sqrt{d + e x} + 105 d^{2} e^{3} x \sqrt{d + e x} + 105 d e^{4} x^{2} \sqrt{d + e x} + 35 e^{5} x^{3} \sqrt{d + e x}} - \frac{14 c d e x}{35 d^{3} e^{2} \sqrt{d + e x} + 105 d^{2} e^{3} x \sqrt{d + e x} + 105 d e^{4} x^{2} \sqrt{d + e x} + 35 e^{5} x^{3} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]
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)/(e*x+d)**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.202316, size = 65, normalized size = 1.51 \[ -\frac{2 \,{\left (7 \,{\left (x e + d\right )}^{2} c d - 5 \,{\left (x e + d\right )} c d^{2} + 5 \,{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{35 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]
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)/(e*x + d)^(11/2),x, algorithm="giac")
[Out]