3.928 \(\int \frac{\sqrt{c x^2} (a+b x)^n}{x^3} \, dx\)

Optimal. Leaf size=47 \[ \frac{b \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0312537, 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 \[ \frac{b \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1) x} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c*x^2]*(a + b*x)^n)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.7019, size = 37, normalized size = 0.79 \[ \frac{b \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n*(c*x**2)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0252166, size = 62, normalized size = 1.32 \[ \frac{\sqrt{c x^2} \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c*x^2]*(a + b*x)^n)/x^3,x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{3}}\sqrt{c{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^n*(c*x^2)^(1/2)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^n/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2)*(b*x + a)^n/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^n/x^3,x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2)*(b*x + a)^n/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(c*x**2)**(1/2)/x**3,x)

[Out]

Integral(sqrt(c*x**2)*(a + b*x)**n/x**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^n/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2)*(b*x + a)^n/x^3, x)