Optimal. Leaf size=47 \[ -\frac{\sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) x} \]
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Rubi [A] time = 0.0317039, antiderivative size = 47, 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 \[ -\frac{\sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) x} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[c*x^2]*(a + b*x)^n)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 14.5106, size = 36, normalized size = 0.77 \[ - \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a x \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(c*x**2)**(1/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0209022, size = 57, normalized size = 1.21 \[ \frac{c x \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[c*x^2]*(a + b*x)^n)/x^2,x]
[Out]
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Maple [F] time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}\sqrt{c{x}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(c*x^2)^(1/2)/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^n/x^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^n/x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n*(c*x**2)**(1/2)/x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^n/x^2,x, algorithm="giac")
[Out]