3.951 \(\int \frac{(a+b x)^n}{x \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{b x (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0292743, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b x (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/(x*Sqrt[c*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.3185, size = 39, normalized size = 0.87 \[ \frac{b \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} c x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x/(c*x**2)**(1/2),x)

[Out]

b*sqrt(c*x**2)*(a + b*x)**(n + 1)*hyper((2, n + 1), (n + 2,), 1 + b*x/a)/(a**2*c
*x*(n + 1))

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Mathematica [A]  time = 0.0226481, size = 63, normalized size = 1.4 \[ \frac{c x^2 \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/(x*Sqrt[c*x^2]),x]

[Out]

(c*x^2*(a + b*x)^n*Hypergeometric2F1[1 - n, -n, 2 - n, -(a/(b*x))])/((-1 + n)*(1
+ a/(b*x))^n*(c*x^2)^(3/2))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{x}{\frac{1}{\sqrt{c{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{\sqrt{c x^{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/(sqrt(c*x^2)*x),x, algorithm="maxima")

[Out]

integrate((b*x + a)^n/(sqrt(c*x^2)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{\sqrt{c x^{2}} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/(sqrt(c*x^2)*x),x, algorithm="fricas")

[Out]

integral((b*x + a)^n/(sqrt(c*x^2)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{n}}{x \sqrt{c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/x/(c*x**2)**(1/2),x)

[Out]

Integral((a + b*x)**n/(x*sqrt(c*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{\sqrt{c x^{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/(sqrt(c*x^2)*x),x, algorithm="giac")

[Out]

integrate((b*x + a)^n/(sqrt(c*x^2)*x), x)