Optimal. Leaf size=136 \[ -\frac{4 \sqrt{2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}+\frac{4 c d \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.212471, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{4 \sqrt{2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}+\frac{4 c d \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 23.1196, size = 124, normalized size = 0.91 \[ - \frac{4 \sqrt{2} c^{\frac{3}{2}} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{e} + \frac{4 c d \sqrt{c d^{2} - c e^{2} x^{2}}}{e \sqrt{d + e x}} + \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 0.316952, size = 110, normalized size = 0.81 \[ \frac{2 c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{7 d-e x}{\sqrt{d+e x}}-\frac{6 \sqrt{2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}\right )}{3 e} \]
Antiderivative was successfully verified.
[In] Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.022, size = 122, normalized size = 0.9 \[ -{\frac{2\,c}{3\,e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}+xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-7\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(5/2),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((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228795, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (c^{2} e^{3} x^{3} - 7 \, c^{2} d e^{2} x^{2} - c^{2} d^{2} e x + 7 \, c^{2} d^{3} + 3 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} c d \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}, \frac{2 \,{\left (c^{2} e^{3} x^{3} - 7 \, c^{2} d e^{2} x^{2} - c^{2} d^{2} e x + 7 \, c^{2} d^{3} + 6 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} c d \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-c d}}\right )\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]