3.133 \(\int \sqrt{d x} \left (b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=61 \[ \frac{2 x \sqrt{d x} \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right )}{2 p+3} \]

[Out]

(2*x*Sqrt[d*x]*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 3/2 + p, 5/2 + p, -((c*x)/b
)])/((3 + 2*p)*(1 + (c*x)/b)^p)

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Rubi [A]  time = 0.0724093, 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 \[ \frac{2 x \sqrt{d x} \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right )}{2 p+3} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[d*x]*(b*x + c*x^2)^p*Hypergeometric2F1[-p, 3/2 + p, 5/2 + p, -((c*x)/b
)])/((3 + 2*p)*(1 + (c*x)/b)^p)

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Rubi in Sympy [A]  time = 11.8295, size = 63, normalized size = 1.03 \[ \frac{2 x^{- p - \frac{1}{2}} x^{p + \frac{3}{2}} \sqrt{d x} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + \frac{3}{2} \\ p + \frac{5}{2} \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{2 p + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(1/2)*(c*x**2+b*x)**p,x)

[Out]

2*x**(-p - 1/2)*x**(p + 3/2)*sqrt(d*x)*(1 + c*x/b)**(-p)*(b*x + c*x**2)**p*hyper
((-p, p + 3/2), (p + 5/2,), -c*x/b)/(2*p + 3)

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Mathematica [A]  time = 0.0536154, size = 59, normalized size = 0.97 \[ \frac{2 x \sqrt{d x} (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+\frac{3}{2};p+\frac{5}{2};-\frac{c x}{b}\right )}{2 p+3} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[d*x]*(x*(b + c*x))^p*Hypergeometric2F1[-p, 3/2 + p, 5/2 + p, -((c*x)/b
)])/((3 + 2*p)*(1 + (c*x)/b)^p)

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int \sqrt{dx} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(1/2)*(c*x^2+b*x)^p,x)

[Out]

int((d*x)^(1/2)*(c*x^2+b*x)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(1/2)*(c*x**2+b*x)**p,x)

[Out]

Integral(sqrt(d*x)*(x*(b + c*x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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