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

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

[Out]

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

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Rubi [A]  time = 0.0110567, 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, Rules used = {15, 65} \[ -\frac{\sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) x} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^n)/x^2,x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)^n}{x^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{(a+b x)^n}{x} \, dx}{x}\\ &=-\frac{\sqrt{c x^2} (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a (1+n) x}\\ \end{align*}

Mathematica [A]  time = 0.0104535, size = 46, normalized size = 0.98 \[ -\frac{c x (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*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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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}\sqrt{c{x}^{2}}}\, dx \end{align*}

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(c*x^2)^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(c*x^2)^(1/2)/x^2,x, algorithm="fricas")

[Out]

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

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

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n*(c*x^2)^(1/2)/x^2,x, algorithm="giac")

[Out]

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