3.135 \(\int \frac{\left (b x+c x^2\right )^p}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right )}{(1-2 p) (d x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0748741, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right )}{(1-2 p) (d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.917, size = 63, normalized size = 1.03 \[ - \frac{2 x^{- p + \frac{3}{2}} x^{p - \frac{1}{2}} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p - \frac{1}{2} \\ p + \frac{1}{2} \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{\left (d x\right )^{\frac{3}{2}} \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**p/(d*x)**(3/2),x)

[Out]

-2*x**(-p + 3/2)*x**(p - 1/2)*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hyper((-p, p -
1/2), (p + 1/2,), -c*x/b)/((d*x)**(3/2)*(-2*p + 1))

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Mathematica [A]  time = 0.0622764, size = 59, normalized size = 0.97 \[ \frac{2 x (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (p-\frac{1}{2},-p;p+\frac{1}{2};-\frac{c x}{b}\right )}{(2 p-1) (d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{ \left ( c{x}^{2}+bx \right ) ^{p} \left ( dx \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x^2+b*x)^p/(d*x)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p/(d*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^p/(sqrt(d*x)*d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**p/(d*x)**(3/2),x)

[Out]

Integral((x*(b + c*x))**p/(d*x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p/(d*x)^(3/2), x)