Optimal. Leaf size=133 \[ \frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.209897, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 22.2709, size = 119, normalized size = 0.89 \[ \frac{3 \sqrt{2} c^{\frac{3}{2}} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{e} - \frac{3 c \sqrt{c d^{2} - c e^{2} x^{2}}}{e \sqrt{d + e x}} - \frac{\left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )^{\frac{5}{2}}} \]
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)**(7/2),x)
[Out]
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Mathematica [A] time = 0.299854, size = 107, normalized size = 0.8 \[ \frac{c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{3 \sqrt{2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}-\frac{2 (2 d+e x)}{(d+e x)^{3/2}}\right )}{e} \]
Antiderivative was successfully verified.
[In] Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.03, size = 154, normalized size = 1.2 \[{\frac{c}{e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}-2\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-4\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]
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)^(7/2),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)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230334, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, c^{2} e^{2} x^{2} + 4 \, c^{2} d e x - 8 \, c^{2} d^{2} + 3 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} c \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}, \frac{2 \, c^{2} e^{2} x^{2} + 2 \, c^{2} d e x - 4 \, c^{2} d^{2} - 3 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} c \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-c d}}\right )}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}\right ] \]
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)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]
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)**(7/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]
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)^(7/2),x, algorithm="giac")
[Out]