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

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

[Out]

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

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Rubi [A]  time = 0.0388987, antiderivative size = 50, 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 \[ \frac{b^3 x (a+b x)^{n+1} \, _2F_1\left (4,n+1;n+2;\frac{b x}{a}+1\right )}{a^4 c (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.1398, size = 42, normalized size = 0.84 \[ \frac{b^{3} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 4, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{4} c^{2} x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*sqrt(c*x**2)*(a + b*x)**(n + 1)*hyper((4, n + 1), (n + 2,), 1 + b*x/a)/(a**
4*c**2*x*(n + 1))

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Mathematica [A]  time = 0.0380463, size = 63, normalized size = 1.26 \[ \frac{c x^2 \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (3-n,-n;4-n;-\frac{a}{b x}\right )}{(n-3) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*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))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{x} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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