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

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

[Out]

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

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Rubi [A]  time = 0.0732089, 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 \[ \frac{2 x \left (\frac{c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right )}{(2 p+1) \sqrt{d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0533, size = 61, normalized size = 1. \[ \frac{2 x^{- p + \frac{1}{2}} x^{p + \frac{1}{2}} \left (1 + \frac{c x}{b}\right )^{- p} \left (b x + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + \frac{1}{2} \\ p + \frac{3}{2} \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{\sqrt{d x} \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0691371, size = 59, normalized size = 0.97 \[ \frac{2 \sqrt{d x} (x (b+c x))^p \left (\frac{c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+\frac{1}{2};p+\frac{3}{2};-\frac{c x}{b}\right )}{2 d p+d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{ \left ( c{x}^{2}+bx \right ) ^{p}{\frac{1}{\sqrt{dx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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