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

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

[Out]

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

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Rubi [A]  time = 0.0286116, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{x (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/Sqrt[c*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.0349, size = 37, normalized size = 0.82 \[ - \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a c x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0147851, size = 56, normalized size = 1.24 \[ \frac{x \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{n}{\frac{1}{\sqrt{c{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/(c*x^2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/sqrt(c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n/sqrt(c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/sqrt(c*x^2), x)