3.965 \(\int \frac{x^4 (a+b x)^n}{(c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=48 \[ -\frac{x (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c^2 (n+1) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0125406, antiderivative size = 48, 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} \[ -\frac{x (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c^2 (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a*c^2*(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{x^4 (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{(a+b x)^n}{x} \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{x (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a c^2 (1+n) \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0084175, size = 47, normalized size = 0.98 \[ -\frac{x^5 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^n/(c*x^2)^(5/2),x)

[Out]

int(x^4*(b*x+a)^n/(c*x^2)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n} x^{4}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^n/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x^4/(c*x^2)^(5/2), 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}}{c^{3} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^n/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

Integral(x**4*(a + b*x)**n/(c*x**2)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n} x^{4}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^n/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^4/(c*x^2)^(5/2), x)