Optimal. Leaf size=158 \[ -\frac{3 e \sqrt{d+e x}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]
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Rubi [A] time = 0.225301, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 e \sqrt{d+e x}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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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((e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
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Mathematica [A] time = 0.185392, size = 110, normalized size = 0.7 \[ \frac{-\frac{3 e^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b d-a e}}-\sqrt{b} \sqrt{d+e x} (3 a e+2 b d+5 b e x)}{4 b^{5/2} (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.021, size = 194, normalized size = 1.2 \[ -{\frac{bx+a}{4\,{b}^{2}} \left ( -3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{2}{e}^{2}-6\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xab{e}^{2}+5\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}b-3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),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((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21846, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} d - a b e}{\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} \sqrt{e x + d} - 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{b^{2} d - a b e}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} \sqrt{e x + d} + 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(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((e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.22631, size = 219, normalized size = 1.39 \[ -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \, \sqrt{-b^{2} d + a b e} b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{5 \,{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} - 3 \, \sqrt{x e + d} b d e^{2} + 3 \, \sqrt{x e + d} a e^{3}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
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