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

Optimal. Leaf size=50 \[ \frac{b^3 x (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 c (n+1) \sqrt{c x^2}} \]

[Out]

(b^3*x*(a + b*x)^(1 + n)*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (b*x)/a])/(a^4*c*(1 + n)*Sqrt[c*x^2])

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Rubi [A]  time = 0.0141199, antiderivative size = 50, 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} \[ \frac{b^3 x (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 c (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x*(c*x^2)^(3/2)),x]

[Out]

(b^3*x*(a + b*x)^(1 + n)*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (b*x)/a])/(a^4*c*(1 + n)*Sqrt[c*x^2])

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{(a+b x)^n}{x \left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^n}{x^4} \, dx}{c \sqrt{c x^2}}\\ &=\frac{b^3 x (a+b x)^{1+n} \, _2F_1\left (4,1+n;2+n;1+\frac{b x}{a}\right )}{a^4 c (1+n) \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0134882, size = 50, normalized size = 1. \[ \frac{b^3 c x^5 (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 (n+1) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*c*x^5*(a + b*x)^(1 + n)*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (b*x)/a])/(a^4*(1 + n)*(c*x^2)^(5/2))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{x} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n/((c*x^2)^(3/2)*x), 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}}{c^{2} x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2)*(b*x + a)^n/(c^2*x^5), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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