Optimal. Leaf size=132 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
[Out]
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Rubi [A] time = 0.151591, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 23.9581, size = 114, normalized size = 0.86 \[ - \frac{35 d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{35 d x \sqrt{d^{2} - e^{2} x^{2}}}{2} - \frac{35 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e} - \frac{14 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{2}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}}{e \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.168079, size = 85, normalized size = 0.64 \[ \frac{1}{6} \sqrt{d^2-e^2 x^2} \left (-\frac{96 d^3}{e (d+e x)}-\frac{70 d^2}{e}+15 d x-2 e x^2\right )-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.018, size = 364, normalized size = 2.8 \[ -{\frac{1}{{e}^{6}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-5}}-4\,{\frac{1}{{e}^{5}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{20}{3\,{e}^{4}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-8\,{\frac{1}{{e}^{3}{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}-8\,{\frac{1}{e{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2}}-{\frac{28\,x}{3\,{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{35\,x}{3\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,dx}{2}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{35\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^5,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((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240807, size = 471, normalized size = 3.57 \[ \frac{2 \, e^{7} x^{7} - 7 \, d e^{6} x^{6} + 261 \, d^{3} e^{4} x^{4} + 624 \, d^{4} e^{3} x^{3} - 324 \, d^{5} e^{2} x^{2} - 888 \, d^{6} e x + 210 \,{\left (d^{3} e^{4} x^{4} - 3 \, d^{4} e^{3} x^{3} - 8 \, d^{5} e^{2} x^{2} + 4 \, d^{6} e x + 8 \, d^{7} +{\left (d^{3} e^{3} x^{3} + 4 \, d^{4} e^{2} x^{2} - 4 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (2 \, e^{6} x^{6} - 21 \, d e^{5} x^{5} + 99 \, d^{2} e^{4} x^{4} + 180 \, d^{3} e^{3} x^{3} - 324 \, d^{4} e^{2} x^{2} - 888 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{5} x^{4} - 3 \, d e^{4} x^{3} - 8 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + 8 \, d^{4} e +{\left (e^{4} x^{3} + 4 \, d e^{3} x^{2} - 4 \, d^{2} e^{2} x - 8 \, d^{3} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^5,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**2*x**2+d**2)**(7/2)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [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^2*x^2 + d^2)^(7/2)/(e*x + d)^5,x, algorithm="giac")
[Out]