Optimal. Leaf size=90 \[ -\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]
[Out]
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Rubi [A] time = 0.178688, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2 a \int ^{\sqrt{c + d x}} x\, dx}{b^{2} d^{2}} - \frac{2 a \left (a^{2} - b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} + \frac{2 \left (a^{2} - b^{2} c\right ) \int ^{\sqrt{c + d x}} \frac{1}{b^{3}}\, dx}{d^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*(d*x+c)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.083584, size = 85, normalized size = 0.94 \[ \frac{b \left (6 a^2 \sqrt{c+d x}-3 a b (c+d x)+2 b^2 (d x-2 c) \sqrt{c+d x}\right )-6 \left (a^3-a b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{3 b^4 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0.006, size = 116, normalized size = 1.3 \[{\frac{2}{3\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{{b}^{2}d}}-{\frac{ac}{{b}^{2}{d}^{2}}}-2\,{\frac{c\sqrt{dx+c}}{b{d}^{2}}}+2\,{\frac{\sqrt{dx+c}{a}^{2}}{{b}^{3}{d}^{2}}}+2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{2}{d}^{2}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ) }{{b}^{4}{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*(d*x+c)^(1/2)),x)
[Out]
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Maxima [A] time = 0.699251, size = 109, normalized size = 1.21 \[ \frac{\frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} - 3 \,{\left (d x + c\right )} a b - 6 \,{\left (b^{2} c - a^{2}\right )} \sqrt{d x + c}}{b^{3}} + \frac{6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{4}}}{3 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(d*x + c)*b + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28963, size = 96, normalized size = 1.07 \[ -\frac{3 \, a b^{2} d x - 6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 2 \,{\left (b^{3} d x - 2 \, b^{3} c + 3 \, a^{2} b\right )} \sqrt{d x + c}}{3 \, b^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(d*x + c)*b + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{a + b \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*(d*x+c)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.276789, size = 177, normalized size = 1.97 \[ \frac{\frac{6 \,{\left (a b^{2} c - a^{3}\right )}{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{4} d} - \frac{6 \,{\left (a b^{2} c{\rm ln}\left ({\left | a \right |}\right ) - a^{3}{\rm ln}\left ({\left | a \right |}\right )\right )}}{b^{4} d} + \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{2} - 6 \, \sqrt{d x + c} b^{2} c d^{2} - 3 \,{\left (d x + c\right )} a b d^{2} + 6 \, \sqrt{d x + c} a^{2} d^{2}}{b^{3} d^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(d*x + c)*b + a),x, algorithm="giac")
[Out]