Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac{8 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.285568, antiderivative size = 171, 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 \[ \frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac{8 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 55.212, size = 158, normalized size = 0.92 \[ - \frac{16 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{8 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
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Mathematica [A] time = 0.148265, size = 94, normalized size = 0.55 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (3 a^2 e^4-2 a c d e^2 (5 d+2 e x)+c^2 d^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
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Maple [A] time = 0.013, size = 146, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-4\,xacd{e}^{3}+20\,x{c}^{2}{d}^{3}e+3\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}+15\,{c}^{2}{d}^{4} \right ) }{15\, \left ({a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{c}^{2}{d}^{4}a{e}^{2}-{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.475873, size = 377, normalized size = 2.2 \[ \frac{2 \,{\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{15 \,{\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} +{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \,{\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.280133, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3),x, algorithm="giac")
[Out]