Optimal. Leaf size=80 \[ \frac{a b d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2}}-\frac{\left (a+b \sqrt{c+d x}\right )^2}{2 x^2}-\frac{b d \left (a \sqrt{c+d x}+b c\right )}{2 c x} \]
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Rubi [A] time = 0.179639, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{a b d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2}}-\frac{\left (a+b \sqrt{c+d x}\right )^2}{2 x^2}-\frac{b d \left (a \sqrt{c+d x}+b c\right )}{2 c x} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[c + d*x])^2/x^3,x]
[Out]
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Rubi in Sympy [A] time = 10.6154, size = 68, normalized size = 0.85 \[ \frac{a b d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}}} - \frac{b d \left (a \sqrt{c + d x} + b c\right )}{2 c x} - \frac{\left (a + b \sqrt{c + d x}\right )^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(d*x+c)**(1/2))**2/x**3,x)
[Out]
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Mathematica [A] time = 0.129455, size = 77, normalized size = 0.96 \[ \frac{a b d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2}}-\frac{a^2 c+a b \sqrt{c+d x} (2 c+d x)+b^2 c (c+2 d x)}{2 c x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[c + d*x])^2/x^3,x]
[Out]
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Maple [A] time = 0.017, size = 81, normalized size = 1. \[{b}^{2} \left ( -{\frac{d}{x}}-{\frac{c}{2\,{x}^{2}}} \right ) +4\,ab{d}^{2} \left ({\frac{1}{{d}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( dx+c \right ) ^{3/2}}{c}}-1/8\,\sqrt{dx+c} \right ) }+1/8\,{\frac{1}{{c}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) } \right ) -{\frac{{a}^{2}}{2\,{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(d*x+c)^(1/2))^2/x^3,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((sqrt(d*x + c)*b + a)^2/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278916, size = 1, normalized size = 0.01 \[ \left [\frac{a b d^{2} x^{2} \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x + c} c}{x}\right ) - 2 \,{\left (a b d x + 2 \, a b c\right )} \sqrt{d x + c} \sqrt{c} - 2 \,{\left (2 \, b^{2} c d x + b^{2} c^{2} + a^{2} c\right )} \sqrt{c}}{4 \, c^{\frac{3}{2}} x^{2}}, -\frac{a b d^{2} x^{2} \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) +{\left (a b d x + 2 \, a b c\right )} \sqrt{d x + c} \sqrt{-c} +{\left (2 \, b^{2} c d x + b^{2} c^{2} + a^{2} c\right )} \sqrt{-c}}{2 \, \sqrt{-c} c x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^2/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 38.4755, size = 292, normalized size = 3.65 \[ - \frac{a^{2}}{2 x^{2}} - \frac{20 a b c^{2} d^{2} \sqrt{c + d x}}{- 8 c^{4} - 16 c^{3} d x + 8 c^{2} \left (c + d x\right )^{2}} + \frac{12 a b c d^{2} \left (c + d x\right )^{\frac{3}{2}}}{- 8 c^{4} - 16 c^{3} d x + 8 c^{2} \left (c + d x\right )^{2}} + \frac{3 a b c d^{2} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{c + d x} \right )}}{4} - \frac{3 a b c d^{2} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{c + d x} \right )}}{4} - a b d^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )} + a b d^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )} - \frac{2 a b d \sqrt{c + d x}}{c x} - \frac{b^{2} c}{2 x^{2}} - \frac{b^{2} d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(d*x+c)**(1/2))**2/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.2944, size = 170, normalized size = 2.12 \[ -\frac{\frac{a b d^{3} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{b^{2} c d^{3} - a^{2} d^{3}}{c^{2}} + \frac{2 \,{\left (d x + c\right )} b^{2} c d^{3} - b^{2} c^{2} d^{3} +{\left (d x + c\right )}^{\frac{3}{2}} a b d^{3} + \sqrt{d x + c} a b c d^{3} + a^{2} c d^{3}}{c d^{2} x^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^2/x^3,x, algorithm="giac")
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