Optimal. Leaf size=119 \[ \frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
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Rubi [A] time = 0.191186, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0864799, size = 96, normalized size = 0.81 \[ \frac{2 (a+b x) \left (\sqrt{b d-a e}-\sqrt{b} \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{\sqrt{(a+b x)^2} \sqrt{d+e x} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
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Maple [A] time = 0.014, size = 90, normalized size = 0.8 \[ -2\,{\frac{bx+a}{\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ae-bd \right ) \sqrt{ex+d}\sqrt{b \left ( ae-bd \right ) }} \left ( b\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}+\sqrt{b \left ( ae-bd \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2),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(1/(sqrt((b*x + a)^2)*(e*x + d)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.218439, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) - 2}{{\left (b d - a e\right )} \sqrt{e x + d}}, -\frac{2 \,{\left (\sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) - 1\right )}}{{\left (b d - a e\right )} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(3/2)),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(1/(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214503, size = 109, normalized size = 0.92 \[ 2 \,{\left (\frac{b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} + \frac{1}{{\left (b d - a e\right )} \sqrt{x e + d}}\right )}{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]