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

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

[Out]

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

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Rubi [A]  time = 0.0348365, 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 \[ -\frac{x (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.3229, size = 39, normalized size = 0.81 \[ - \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a c^{2} x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0156203, size = 58, normalized size = 1.21 \[ \frac{x^3 \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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