Optimal. Leaf size=119 \[ -\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e} \]
[Out]
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Rubi [A] time = 0.166112, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 15.3037, size = 102, normalized size = 0.86 \[ - \frac{64 d^{2} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{105 c e \left (d + e x\right )^{\frac{3}{2}}} - \frac{16 d \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{35 c e \sqrt{d + e x}} - \frac{2 \sqrt{d + e x} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{7 c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*(-c*e**2*x**2+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0535847, size = 64, normalized size = 0.54 \[ \frac{2 \left (-71 d^3+17 d^2 e x+39 d e^2 x^2+15 e^3 x^3\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{105 e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 15\,{e}^{2}{x}^{2}+54\,dxe+71\,{d}^{2} \right ) }{105\,e}\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*(-c*e^2*x^2+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.729957, size = 92, normalized size = 0.77 \[ \frac{2 \,{\left (15 \, \sqrt{c} e^{3} x^{3} + 39 \, \sqrt{c} d e^{2} x^{2} + 17 \, \sqrt{c} d^{2} e x - 71 \, \sqrt{c} d^{3}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{105 \,{\left (e^{2} x + d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221074, size = 117, normalized size = 0.98 \[ -\frac{2 \,{\left (15 \, c e^{5} x^{5} + 39 \, c d e^{4} x^{4} + 2 \, c d^{2} e^{3} x^{3} - 110 \, c d^{3} e^{2} x^{2} - 17 \, c d^{4} e x + 71 \, c d^{5}\right )}}{105 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(3/2)*(-c*e**2*x**2+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]