3.120 \(\int (d x)^m \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0864399, antiderivative size = 67, 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 \[ \frac{2 (b+c x) \sqrt{b x+c x^2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c x}{b}+1\right )}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7773, size = 71, normalized size = 1.06 \[ \frac{2 x^{- m - \frac{1}{2}} x^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{c x}{b}\right )^{- m - \frac{1}{2}} \left (b + c x\right ) \sqrt{b x + c x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{1}{2}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{c x}{b}} \right )}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0421613, size = 59, normalized size = 0.88 \[ \frac{2 x \sqrt{x (b+c x)} (d x)^m \, _2F_1\left (-\frac{1}{2},m+\frac{3}{2};m+\frac{5}{2};-\frac{c x}{b}\right )}{(2 m+3) \sqrt{\frac{c x}{b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{c{x}^{2}+bx}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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