Optimal. Leaf size=130 \[ -\frac{a-b \sqrt{c+d x}}{x \left (a^2-b^2 c\right )}+\frac{a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac{2 a b^2 d \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac{b d \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^2} \]
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Rubi [A] time = 0.363361, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{a-b \sqrt{c+d x}}{x \left (a^2-b^2 c\right )}+\frac{a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac{2 a b^2 d \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac{b d \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*Sqrt[c + d*x])),x]
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Rubi in Sympy [A] time = 26.3431, size = 119, normalized size = 0.92 \[ \frac{a b^{2} d \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{2}} - \frac{2 a b^{2} d \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{2}} + \frac{b d \left (a^{2} + b^{2} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c} \left (a^{2} - b^{2} c\right )^{2}} - \frac{a - b \sqrt{c + d x}}{x \left (a^{2} - b^{2} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*(d*x+c)**(1/2)),x)
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Mathematica [A] time = 0.325795, size = 144, normalized size = 1.11 \[ \frac{\sqrt{c} \left (-\left (a^2-b^2 c\right ) \left (a-b \sqrt{c+d x}\right )-a b^2 d x \log \left (a^2-b^2 (c+d x)\right )+a b^2 d x \log (x)\right )+b d x \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )-2 a b^2 \sqrt{c} d x \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )}{\sqrt{c} x \left (a^2-b^2 c\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*Sqrt[c + d*x])),x]
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Maple [A] time = 0.027, size = 216, normalized size = 1.7 \[ -{\frac{{b}^{3}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}\sqrt{dx+c}}+{\frac{{a}^{2}b}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}\sqrt{dx+c}}+{\frac{a{b}^{2}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}}-{\frac{{a}^{3}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}}+{\frac{a{b}^{2}d\ln \left ( dx \right ) }{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}}+{\frac{{b}^{3}d}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }+{\frac{{a}^{2}bd}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-2\,{\frac{a{b}^{2}d\ln \left ( a+b\sqrt{dx+c} \right ) }{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*(d*x+c)^(1/2)),x)
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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(d*x + c)*b + a)*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.352942, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, a b^{2} \sqrt{c} d x \log \left (\sqrt{d x + c} b + a\right ) -{\left (b^{3} c + a^{2} b\right )} d x \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x + c} c}{x}\right ) + 2 \,{\left (b^{3} c - a^{2} b\right )} \sqrt{d x + c} \sqrt{c} - 2 \,{\left (a b^{2} d x \log \left (x\right ) + a b^{2} c - a^{3}\right )} \sqrt{c}}{2 \,{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{c} x}, -\frac{2 \, a b^{2} \sqrt{-c} d x \log \left (\sqrt{d x + c} b + a\right ) +{\left (b^{3} c + a^{2} b\right )} d x \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) +{\left (b^{3} c - a^{2} b\right )} \sqrt{d x + c} \sqrt{-c} -{\left (a b^{2} d x \log \left (x\right ) + a b^{2} c - a^{3}\right )} \sqrt{-c}}{{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{-c} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((sqrt(d*x + c)*b + a)*x^2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + b \sqrt{c + d x}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a+b*(d*x+c)**(1/2)),x)
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GIAC/XCAS [A] time = 0.286877, size = 342, normalized size = 2.63 \[ -\frac{2 \, a b^{3} d{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b} + \frac{a b^{2} d{\rm ln}\left (-d x\right )}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}} - \frac{{\left (b^{3} c d + a^{2} b d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{-c}} - \frac{a b^{2} c d{\rm ln}\left (c\right ) - 2 \, a b^{2} c d{\rm ln}\left ({\left | a \right |}\right ) - a b^{2} c d + a^{3} d}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c} + \frac{a b^{2} c d - a^{3} d -{\left (b^{3} c d - a^{2} b d\right )} \sqrt{d x + c}}{{\left (b^{2} c - a^{2}\right )}^{2} d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((sqrt(d*x + c)*b + a)*x^2),x, algorithm="giac")
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