3.935 \(\int \frac{\left (c x^2\right )^{3/2} (a+b x)^n}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0324117, 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 \[ -\frac{c \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) x} \]

Antiderivative was successfully verified.

[In]  Int[((c*x^2)^(3/2)*(a + b*x)^n)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.6318, size = 37, normalized size = 0.77 \[ - \frac{c \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(3/2)*(b*x+a)**n/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0169201, size = 58, normalized size = 1.21 \[ \frac{\left (c x^2\right )^{3/2} \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x^2)^(3/2)*(a + b*x)^n)/x^4,x]

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)*(b*x+a)^n/x^4,x)

[Out]

int((c*x^2)^(3/2)*(b*x+a)^n/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^n/x^4,x, algorithm="maxima")

[Out]

integrate((c*x^2)^(3/2)*(b*x + a)^n/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^n/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{n}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(3/2)*(b*x+a)**n/x**4,x)

[Out]

Integral((c*x**2)**(3/2)*(a + b*x)**n/x**4, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^n/x^4,x, algorithm="giac")

[Out]

integrate((c*x^2)^(3/2)*(b*x + a)^n/x^4, x)