Optimal. Leaf size=172 \[ -\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac{3 e \sqrt{d+e x}}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.301718, antiderivative size = 172, 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 \[ -\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac{3 e \sqrt{d+e x}}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[d + e*x]*(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(1/(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)
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Mathematica [A] time = 0.180275, size = 121, normalized size = 0.7 \[ \frac{\sqrt{b} \sqrt{d+e x} \sqrt{b d-a e} (5 a e-2 b d+3 b e x)-3 e^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (a+b x) \sqrt{(a+b x)^2} (b d-a e)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
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Maple [A] time = 0.016, size = 203, normalized size = 1.2 \[{\frac{bx+a}{4\, \left ( ae-bd \right ) ^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{2}{e}^{2}+6\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xab{e}^{2}+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xbe+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}+5\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae-2\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(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/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.225951, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} d - a b e}{\left (3 \, b e x - 2 \, b d + 5 \, a e\right )} \sqrt{e x + d} + 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e}}, \frac{\sqrt{-b^{2} d + a b e}{\left (3 \, b e x - 2 \, b d + 5 \, a e\right )} \sqrt{e x + d} - 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.226034, size = 392, normalized size = 2.28 \[ -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{2} d^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 2 \, a b d e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} - 5 \, \sqrt{x e + d} b d e^{2} + 5 \, \sqrt{x e + d} a e^{3}}{4 \,{\left (b^{2} d^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 2 \, a b d e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d)),x, algorithm="giac")
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