Optimal. Leaf size=69 \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]
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Rubi [A] time = 0.107827, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 30.0643, size = 60, normalized size = 0.87 \[ - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{3}{2}}} - \frac{2}{\sqrt{d + e x} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0968502, size = 69, normalized size = 1. \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Maple [A] time = 0.006, size = 68, normalized size = 1. \[ -2\,{\frac{1}{ \left ( ae-bd \right ) \sqrt{ex+d}}}-2\,{\frac{b}{ \left ( ae-bd \right ) \sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.296398, 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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.293962, size = 101, normalized size = 1.46 \[ \frac{2 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} + \frac{2}{{\left (b d - a e\right )} \sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]