3.25.9 \(\int \frac {1}{(d+e x) \sqrt {\frac {-c d^2+b d e}{e^2}+b x+c x^2}} \, dx\) [2409]

Optimal. Leaf size=48 \[ \frac {2 e \sqrt {-\frac {d (c d-b e)}{e^2}+b x+c x^2}}{(2 c d-b e) (d+e x)} \]

[Out]

2*e*(-d*(-b*e+c*d)/e^2+b*x+c*x^2)^(1/2)/(-b*e+2*c*d)/(e*x+d)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {664} \begin {gather*} \frac {2 e \sqrt {-\frac {d (c d-b e)}{e^2}+b x+c x^2}}{(d+e x) (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x + c*x^2]),x]

[Out]

(2*e*Sqrt[-((d*(c*d - b*e))/e^2) + b*x + c*x^2])/((2*c*d - b*e)*(d + e*x))

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a +
b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \sqrt {\frac {-c d^2+b d e}{e^2}+b x+c x^2}} \, dx &=\frac {2 e \sqrt {-\frac {d (c d-b e)}{e^2}+b x+c x^2}}{(2 c d-b e) (d+e x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 45, normalized size = 0.94 \begin {gather*} -\frac {2 e \sqrt {\frac {(d+e x) (-c d+b e+c e x)}{e^2}}}{(-2 c d+b e) (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x + c*x^2]),x]

[Out]

(-2*e*Sqrt[((d + e*x)*(-(c*d) + b*e + c*e*x))/e^2])/((-2*c*d + b*e)*(d + e*x))

________________________________________________________________________________________

Maple [A]
time = 0.85, size = 55, normalized size = 1.15

method	result	size
default	\(-\frac {2 \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}\)	\(55\)
trager	\(-\frac {2 e \sqrt {-\frac {-x^{2} c \,e^{2}-b \,e^{2} x -b d e +c \,d^{2}}{e^{2}}}}{\left (b e -2 c d \right ) \left (e x +d \right )}\)	\(55\)
gosper	\(-\frac {2 \left (c e x +b e -c d \right )}{e \left (b e -2 c d \right ) \sqrt {\frac {x^{2} c \,e^{2}+b \,e^{2} x +b d e -c \,d^{2}}{e^{2}}}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/((b*d*e-c*d^2)/e^2+b*x+c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/(b*e-2*c*d)/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e))^(1/2)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*c*d-%e*b>0)', see `assume?`
for more det

________________________________________________________________________________________

Fricas [A]
time = 2.77, size = 63, normalized size = 1.31 \begin {gather*} \frac {2 \, \sqrt {-{\left (c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}\right )} e^{\left (-2\right )}} e}{2 \, c d^{2} - b x e^{2} + {\left (2 \, c d x - b d\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(-(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*e^(-2))*e/(2*c*d^2 - b*x*e^2 + (2*c*d*x - b*d)*e)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (\frac {d}{e} + x\right ) \left (b - \frac {c d}{e} + c x\right )} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/((b*d*e-c*d**2)/e**2+b*x+c*x**2)**(1/2),x)

[Out]

Integral(1/(sqrt((d/e + x)*(b - c*d/e + c*x))*(d + e*x)), x)

________________________________________________________________________________________

Giac [A]
time = 1.18, size = 45, normalized size = 0.94 \begin {gather*} \frac {2}{\sqrt {c} x e + \sqrt {c} d - \sqrt {c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sqrt(c)*x*e + sqrt(c)*d - sqrt(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e))

________________________________________________________________________________________

Mupad [B]
time = 1.18, size = 47, normalized size = 0.98 \begin {gather*} -\frac {2\,e\,\sqrt {b\,x-\frac {c\,d^2-b\,d\,e}{e^2}+c\,x^2}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*e*(b*x - (c*d^2 - b*d*e)/e^2 + c*x^2)^(1/2))/((b*e - 2*c*d)*(d + e*x))

________________________________________________________________________________________