Optimal. Leaf size=149 \[ \frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}}-\frac{\sqrt{d+e x} (2 c d-b e)}{b^2 (b+c x)}-\frac{d \sqrt{d+e x}}{b x (b+c x)} \]
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Rubi [A] time = 0.611917, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}}-\frac{\sqrt{d+e x} (2 c d-b e)}{b^2 (b+c x)}-\frac{d \sqrt{d+e x}}{b x (b+c x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 63.4677, size = 129, normalized size = 0.87 \[ - \frac{d \sqrt{d + e x}}{b x \left (b + c x\right )} + \frac{\sqrt{d + e x} \left (b e - 2 c d\right )}{b^{2} \left (b + c x\right )} - \frac{\sqrt{d} \left (3 b e - 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} + \frac{\left (b e - 4 c d\right ) \sqrt{b e - c d} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.422972, size = 140, normalized size = 0.94 \[ \frac{-\frac{\left (b^2 e^2-5 b c d e+4 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{c d-b e}}+\frac{b \sqrt{d+e x} (-b d+b e x-2 c d x)}{x (b+c x)}+\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.027, size = 237, normalized size = 1.6 \[{\frac{{e}^{2}}{b \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{ced}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{2}}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-5\,{\frac{ced}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{2}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{d}{{b}^{2}x}\sqrt{ex+d}}-3\,{\frac{e\sqrt{d}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{3/2}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(c*x^2+b*x)^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)/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.262167, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^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)/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.217756, size = 285, normalized size = 1.91 \[ \frac{{\left (4 \, c^{2} d^{2} - 5 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3}} - \frac{{\left (4 \, c d^{2} - 3 \, b d e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c d e - 2 \, \sqrt{x e + d} c d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} + 2 \, \sqrt{x e + d} b d e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]