### 3.365 $$\int \frac{1}{\sqrt{d+e x} (b x+c x^2)} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0576517, 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 = {707, 1093, 208} $\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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 1093

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

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

Mathematica [A]  time = 0.0841091, size = 75, normalized size = 0.97 $\frac{\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.25, size = 62, normalized size = 0.8 \begin{align*} -2\,{\frac{c}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{1}{b\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.33209, size = 880, normalized size = 11.43 \begin{align*} \left [\frac{d \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b d}, \frac{2 \, d \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{-\frac{c}{c d - b e}}}{c e x + c d}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b d}, \frac{d \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + 2 \, \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right )}{b d}, \frac{2 \,{\left (d \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{-\frac{c}{c d - b e}}}{c e x + c d}\right ) + \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right )\right )}}{b d}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 17.2026, size = 80, normalized size = 1.04 \begin{align*} \frac{2 c \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{b \sqrt{\frac{c}{b e - c d}} \left (b e - c d\right )} + \frac{2 \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{b d \sqrt{- \frac{1}{d}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*atan(1/(sqrt(c/(b*e - c*d))*sqrt(d + e*x)))/(b*sqrt(c/(b*e - c*d))*(b*e - c*d)) + 2*atan(1/(sqrt(-1/d)*sqr
t(d + e*x)))/(b*d*sqrt(-1/d))

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Giac [A]  time = 1.17955, size = 96, normalized size = 1.25 \begin{align*} -\frac{2 \, c \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 \, \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(1/(e*x+d)^(1/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

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