### 3.364 $$\int \frac{\sqrt{d+e x}}{b x+c x^2} \, dx$$

Optimal. Leaf size=77 $\frac{2 \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0723879, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {699, 1130, 208} $\frac{2 \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]/(b*x + c*x^2),x]

[Out]

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

Rule 699

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2
- b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{b x+c x^2} \, dx &=(2 e) \operatorname{Subst}\left (\int \frac{x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=\frac{(2 c d) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}+\left (e \left (1+\frac{-2 c d+b e}{b e}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )\\ &=-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 \sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.036094, size = 75, normalized size = 0.97 $\frac{2 \left (\frac{\sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c}}-\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.217, size = 100, normalized size = 1.3 \begin{align*} 2\,{\frac{e}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{cd}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{\sqrt{d}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.38493, size = 788, normalized size = 10.23 \begin{align*} \left [\frac{\sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b}, \frac{2 \, \sqrt{-\frac{c d - b e}{c}} \arctan \left (-\frac{\sqrt{e x + d} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b}, \frac{2 \, \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) + \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b}, \frac{2 \,{\left (\sqrt{-\frac{c d - b e}{c}} \arctan \left (-\frac{\sqrt{e x + d} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right ) + \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right )\right )}}{b}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + sqrt(d)*l
og((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/b, (2*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b
*e)/c)/(c*d - b*e)) + sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/b, (2*sqrt(-d)*arctan(sqrt(e*x + d
)*sqrt(-d)/d) + sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b
)))/b, 2*(sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + sqrt(-d)*arctan(sqr
t(e*x + d)*sqrt(-d)/d))/b]

________________________________________________________________________________________

Sympy [A]  time = 4.90654, size = 78, normalized size = 1.01 \begin{align*} \frac{2 \left (\frac{d e \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b \sqrt{- d}} + \frac{e \left (b e - c d\right ) \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{b c \sqrt{\frac{b e - c d}{c}}}\right )}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17986, size = 108, normalized size = 1.4 \begin{align*} -\frac{2 \,{\left (c d - b e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b} + \frac{2 \, d \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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