Optimal. Leaf size=160 \[ -\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}} \]
[Out]
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Rubi [A] time = 0.231663, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 21.5651, size = 136, normalized size = 0.85 \[ \frac{256 d^{3} \sqrt{d + e x}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{64 d^{2} \left (d + e x\right )^{\frac{3}{2}}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{8 d \left (d + e x\right )^{\frac{5}{2}}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{7}{2}}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(9/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0675603, size = 66, normalized size = 0.41 \[ -\frac{2 \sqrt{d+e x} \left (-91 d^3+43 d^2 e x+7 d e^2 x^2+e^3 x^3\right )}{5 c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 66, normalized size = 0.4 \[{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{3}{x}^{3}-7\,d{e}^{2}{x}^{2}-43\,{d}^{2}xe+91\,{d}^{3} \right ) }{5\,e} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.730802, size = 61, normalized size = 0.38 \[ -\frac{2 \,{\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )}}{5 \, \sqrt{-e x + d} c^{\frac{3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(9/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2111, size = 97, normalized size = 0.61 \[ -\frac{2 \,{\left (e^{4} x^{4} + 8 \, d e^{3} x^{3} + 50 \, d^{2} e^{2} x^{2} - 48 \, d^{3} e x - 91 \, d^{4}\right )}}{5 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(9/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(9/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.612206, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(9/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")
[Out]