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

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

[Out]

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

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Rubi [A]  time = 0.0347361, antiderivative size = 48, 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 \[ \frac{b x (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 c (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0193465, size = 62, normalized size = 1.29 \[ \frac{x^2 \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{x \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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((b*x + a)^n/(sqrt(c*x^2)*c*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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