3.968 \(\int \frac{x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0376066, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{b^3 x (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 c^2 (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0185133, size = 62, normalized size = 1.24 \[ \frac{x^2 \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (3-n,-n;4-n;-\frac{a}{b x}\right )}{(n-3) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{x \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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