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

Optimal. Leaf size=48 \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.034002, antiderivative size = 48, 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 \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3552, size = 42, normalized size = 0.88 \[ - \frac{b^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{3} c x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2*sqrt(c*x**2)*(a + b*x)**(n + 1)*hyper((3, n + 1), (n + 2,), 1 + b*x/a)/(a*
*3*c*x*(n + 1))

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Mathematica [A]  time = 0.0297098, size = 61, normalized size = 1.27 \[ \frac{c x \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (2-n,-n;3-n;-\frac{a}{b x}\right )}{(n-2) \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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