3.496 \(\int \frac{\left (a+b \sqrt{c+d x}\right )^p}{x} \, dx\)

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]

-(((a + b*Sqrt[c + d*x])^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[
c + d*x])/(a - b*Sqrt[c])])/((a - b*Sqrt[c])*(1 + p))) - ((a + b*Sqrt[c + d*x])^
(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[c + d*x])/(a + b*Sqrt[c])
])/((a + b*Sqrt[c])*(1 + p))

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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]

-(((a + b*Sqrt[c + d*x])^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[
c + d*x])/(a - b*Sqrt[c])])/((a - b*Sqrt[c])*(1 + p))) - ((a + b*Sqrt[c + d*x])^
(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[c + d*x])/(a + b*Sqrt[c])
])/((a + b*Sqrt[c])*(1 + p))

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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]

-(a + b*sqrt(c + d*x))**(p + 1)*hyper((1, p + 1), (p + 2,), (a + b*sqrt(c + d*x)
)/(a + b*sqrt(c)))/((a + b*sqrt(c))*(p + 1)) - (a + b*sqrt(c + d*x))**(p + 1)*hy
per((1, p + 1), (p + 2,), (a + b*sqrt(c + d*x))/(a - b*sqrt(c)))/((a - b*sqrt(c)
)*(p + 1))

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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]

((a + b*Sqrt[c + d*x])^p*(Hypergeometric2F1[-p, -p, 1 - p, (-a + b*Sqrt[c])/(b*(
Sqrt[c] + Sqrt[c + d*x]))]/((a + b*Sqrt[c + d*x])/(b*Sqrt[c] + b*Sqrt[c + d*x]))
^p + Hypergeometric2F1[-p, -p, 1 - p, (a + b*Sqrt[c])/(b*Sqrt[c] - b*Sqrt[c + d*
x])]/((a + b*Sqrt[c + d*x])/(-(b*Sqrt[c]) + b*Sqrt[c + d*x]))^p))/p

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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]

int((a+b*(d*x+c)^(1/2))^p/x,x)

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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]

integrate((sqrt(d*x + c)*b + a)^p/x, x)

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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]

integral((sqrt(d*x + c)*b + a)^p/x, x)

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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]

Integral((a + b*sqrt(c + d*x))**p/x, x)

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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]

integrate((sqrt(d*x + c)*b + a)^p/x, x)