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

Optimal. Leaf size=51 \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 c^2 (n+1) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0389531, antiderivative size = 51, 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 \[ -\frac{b^2 x (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 c^2 (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8144, size = 44, normalized size = 0.86 \[ - \frac{b^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{3} c^{3} x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2*sqrt(c*x**2)*(a + b*x)**(n + 1)*hyper((3, n + 1), (n + 2,), 1 + b*x/a)/(a*
*3*c**3*x*(n + 1))

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Mathematica [A]  time = 0.0281441, size = 62, normalized size = 1.22 \[ \frac{x^3 \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (2-n,-n;3-n;-\frac{a}{b x}\right )}{(n-2) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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