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

Optimal. Leaf size=65 \[ \frac {2 (b+c x) (d x)^m \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {c x}{b}+1\right )}{c \sqrt {b x+c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {674, 67, 65} \[ \frac {2 (b+c x) (d x)^m \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {c x}{b}+1\right )}{c \sqrt {b x+c x^2}} \]

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

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 674

Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((e*x)^m*(b*x + c*x^2)^p)/(x^(m + p)
*(b + c*x)^p), Int[x^(m + p)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, m}, x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {(d x)^m}{\sqrt {b x+c x^2}} \, dx &=\frac {\left (x^{\frac {1}{2}-m} (d x)^m \sqrt {b+c x}\right ) \int \frac {x^{-\frac {1}{2}+m}}{\sqrt {b+c x}} \, dx}{\sqrt {b x+c x^2}}\\ &=\frac {\left (\left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m \sqrt {b+c x}\right ) \int \frac {\left (-\frac {c x}{b}\right )^{-\frac {1}{2}+m}}{\sqrt {b+c x}} \, dx}{\sqrt {b x+c x^2}}\\ &=\frac {2 \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m (b+c x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};1+\frac {c x}{b}\right )}{c \sqrt {b x+c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 58, normalized size = 0.89 \[ -\frac {2 \sqrt {x (b+c x)} (d x)^m \left (-\frac {c x}{b}\right )^{-m-\frac {1}{2}} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {c x}{b}+1\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{\sqrt {c x^{2} + b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\sqrt {c \,x^{2}+b x}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x\right )}^m}{\sqrt {c\,x^2+b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(b*x + c*x^2)^(1/2),x)

[Out]

int((d*x)^m/(b*x + c*x^2)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{\sqrt {x \left (b + c x\right )}}\, dx \]

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)

________________________________________________________________________________________