Optimal. Leaf size=95 \[ \frac{2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt{c+d x}\right )}+\frac{2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{4 a \sqrt{c+d x}}{b^3 d^2}+\frac{x}{b^2 d} \]
[Out]
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Rubi [A] time = 0.199244, antiderivative size = 95, 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 \[ \frac{2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt{c+d x}\right )}+\frac{2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{4 a \sqrt{c+d x}}{b^3 d^2}+\frac{x}{b^2 d} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*Sqrt[c + d*x])^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{4 a \sqrt{c + d x}}{b^{3} d^{2}} + \frac{2 a \left (a^{2} - b^{2} c\right )}{b^{4} d^{2} \left (a + b \sqrt{c + d x}\right )} + \frac{2 \int ^{\sqrt{c + d x}} x\, dx}{b^{2} d^{2}} + \frac{2 \left (3 a^{2} - b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*(d*x+c)**(1/2))**2,x)
[Out]
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Mathematica [A] time = 0.123418, size = 86, normalized size = 0.91 \[ \frac{\frac{2 \left (a^3-a b^2 c\right )}{a+b \sqrt{c+d x}}+2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )-4 a b \sqrt{c+d x}+b^2 (c+d x)}{b^4 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*Sqrt[c + d*x])^2,x]
[Out]
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Maple [A] time = 0.012, size = 125, normalized size = 1.3 \[{\frac{x}{{b}^{2}d}}+{\frac{c}{{b}^{2}{d}^{2}}}-4\,{\frac{a\sqrt{dx+c}}{{b}^{3}{d}^{2}}}-2\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{2}{d}^{2}}}+6\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{2}}{{b}^{4}{d}^{2}}}-2\,{\frac{ac}{{b}^{2}{d}^{2} \left ( a+b\sqrt{dx+c} \right ) }}+2\,{\frac{{a}^{3}}{{b}^{4}{d}^{2} \left ( a+b\sqrt{dx+c} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*(d*x+c)^(1/2))^2,x)
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Maxima [A] time = 0.693321, size = 122, normalized size = 1.28 \[ -\frac{\frac{2 \,{\left (a b^{2} c - a^{3}\right )}}{\sqrt{d x + c} b^{5} + a b^{4}} - \frac{{\left (d x + c\right )} b - 4 \, \sqrt{d x + c} a}{b^{3}} + \frac{2 \,{\left (b^{2} c - 3 \, a^{2}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{4}}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(d*x + c)*b + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266919, size = 163, normalized size = 1.72 \[ -\frac{3 \, a b^{2} d x + 5 \, a b^{2} c - 2 \, a^{3} + 2 \,{\left (a b^{2} c - 3 \, a^{3} +{\left (b^{3} c - 3 \, a^{2} b\right )} \sqrt{d x + c}\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (b^{3} d x + b^{3} c - 4 \, a^{2} b\right )} \sqrt{d x + c}}{\sqrt{d x + c} b^{5} d^{2} + a b^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(d*x + c)*b + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b \sqrt{c + d x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*(d*x+c)**(1/2))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(d*x + c)*b + a)^2,x, algorithm="giac")
[Out]