Optimal. Leaf size=139 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.212427, antiderivative size = 139, 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 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 22.2749, size = 124, normalized size = 0.89 \[ - \frac{3 \sqrt{2} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{8 \sqrt{d} e} + \frac{3 c \sqrt{c d^{2} - c e^{2} x^{2}}}{4 e \left (d + e x\right )^{\frac{3}{2}}} - \frac{\left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{\frac{7}{2}}} \]
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)**(9/2),x)
[Out]
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Mathematica [A] time = 0.246072, size = 109, normalized size = 0.78 \[ \frac{c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{2 (d+5 e x)}{(d+e x)^{5/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} \sqrt{d^2-e^2 x^2}}\right )}{8 e} \]
Antiderivative was successfully verified.
[In] Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(9/2),x]
[Out]
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Maple [A] time = 0.031, size = 190, normalized size = 1.4 \[ -{\frac{c}{8\,e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}c{e}^{2}+6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}-10\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-2\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]
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)^(9/2),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)^(9/2),x, algorithm="maxima")
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Fricas [A] time = 0.22964, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (5 \, c e x + c d\right )} \sqrt{e x + d}}{8 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}, \frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-\frac{c}{d}}}\right ) + \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (5 \, c e x + c d\right )} \sqrt{e x + d}}{4 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}\right ] \]
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)^(9/2),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((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(9/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \]
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)^(9/2),x, algorithm="giac")
[Out]