Optimal. Leaf size=139 \[ -\frac{\left (a+b \sqrt{c+d x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a-b \sqrt{c}}\right )}{(p+1) \left (a-b \sqrt{c}\right )}-\frac{\left (a+b \sqrt{c+d x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a+b \sqrt{c}}\right )}{(p+1) \left (a+b \sqrt{c}\right )} \]
[Out]
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Rubi [A] time = 0.256577, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{\left (a+b \sqrt{c+d x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a-b \sqrt{c}}\right )}{(p+1) \left (a-b \sqrt{c}\right )}-\frac{\left (a+b \sqrt{c+d x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a+b \sqrt{c}}\right )}{(p+1) \left (a+b \sqrt{c}\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[c + d*x])^p/x,x]
[Out]
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Rubi in Sympy [A] time = 11.6762, size = 105, normalized size = 0.76 \[ - \frac{\left (a + b \sqrt{c + d x}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{a + b \sqrt{c + d x}}{a + b \sqrt{c}}} \right )}}{\left (a + b \sqrt{c}\right ) \left (p + 1\right )} - \frac{\left (a + b \sqrt{c + d x}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{a + b \sqrt{c + d x}}{a - b \sqrt{c}}} \right )}}{\left (a - b \sqrt{c}\right ) \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(d*x+c)**(1/2))**p/x,x)
[Out]
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Mathematica [A] time = 0.474265, size = 189, normalized size = 1.36 \[ \frac{\left (a+b \sqrt{c+d x}\right )^p \left (\left (\frac{a+b \sqrt{c+d x}}{b \sqrt{c+d x}-b \sqrt{c}}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{a+b \sqrt{c}}{b \sqrt{c}-b \sqrt{c+d x}}\right )+\left (\frac{a+b \sqrt{c+d x}}{b \sqrt{c+d x}+b \sqrt{c}}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b \sqrt{c}-a}{b \left (\sqrt{c}+\sqrt{c+d x}\right )}\right )\right )}{p} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[c + d*x])^p/x,x]
[Out]
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Maple [F] time = 0.006, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( a+b\sqrt{dx+c} \right ) ^{p}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(d*x+c)^(1/2))^p/x,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\sqrt{d x + c} b + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p/x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (\sqrt{d x + c} b + a\right )}^{p}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b \sqrt{c + d x}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(d*x+c)**(1/2))**p/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (\sqrt{d x + c} b + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(d*x + c)*b + a)^p/x,x, algorithm="giac")
[Out]