3.23 \(\int \frac{A+B x}{(a+b x+c x^2) \sqrt{d+f x^2}} \, dx\)

Optimal. Leaf size=302 \[ \frac{\left (-B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 c d-f x \left (b-\sqrt{b^2-4 a c}\right )}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left (b-\sqrt{b^2-4 a c}\right )-2 a c f+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left (b-\sqrt{b^2-4 a c}\right )-2 a c f+2 c^2 d}}+\frac{\left (2 A c-B \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{2 c d-f x \left (\sqrt{b^2-4 a c}+b\right )}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left (\sqrt{b^2-4 a c}+b\right )-2 a c f+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left (\sqrt{b^2-4 a c}+b\right )-2 a c f+2 c^2 d}} \]

[Out]

((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d - (b - Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a
*c*f + b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - 2*a*c*f + b*(
b - Sqrt[b^2 - 4*a*c])*f]) + ((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(2*c*d - (b + Sqrt[b^2 - 4*a*c])*f*x
)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]
*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f])

________________________________________________________________________________________

Rubi [A]  time = 0.842636, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1034, 725, 206} \[ \frac{\left (-B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 c d-f x \left (b-\sqrt{b^2-4 a c}\right )}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left (b-\sqrt{b^2-4 a c}\right )-2 a c f+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left (b-\sqrt{b^2-4 a c}\right )-2 a c f+2 c^2 d}}+\frac{\left (2 A c-B \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{2 c d-f x \left (\sqrt{b^2-4 a c}+b\right )}{\sqrt{2} \sqrt{d+f x^2} \sqrt{b f \left (\sqrt{b^2-4 a c}+b\right )-2 a c f+2 c^2 d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b f \left (\sqrt{b^2-4 a c}+b\right )-2 a c f+2 c^2 d}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d - (b - Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a
*c*f + b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - 2*a*c*f + b*(
b - Sqrt[b^2 - 4*a*c])*f]) + ((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(2*c*d - (b + Sqrt[b^2 - 4*a*c])*f*x
)/(Sqrt[2]*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]
*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f])

Rule 1034

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.49046, size = 283, normalized size = 0.94 \[ \frac{\sqrt{2} \left (-\frac{\left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \tanh ^{-1}\left (\frac{f x \left (\sqrt{b^2-4 a c}-b\right )+2 c d}{\sqrt{d+f x^2} \sqrt{2 b f \left (b-\sqrt{b^2-4 a c}\right )-4 a c f+4 c^2 d}}\right )}{2 \sqrt{b f \left (b-\sqrt{b^2-4 a c}\right )-2 a c f+2 c^2 d}}-\frac{\left (B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac{2 c d-f x \left (\sqrt{b^2-4 a c}+b\right )}{\sqrt{d+f x^2} \sqrt{2 b f \left (\sqrt{b^2-4 a c}+b\right )-4 a c f+4 c^2 d}}\right )}{2 \sqrt{b f \left (\sqrt{b^2-4 a c}+b\right )-2 a c f+2 c^2 d}}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

(Sqrt[2]*(-((-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[4*c^2*
d - 4*a*c*f + 2*b*(b - Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(2*Sqrt[2*c^2*d - 2*a*c*f + b*(b - Sqrt[b^2 -
4*a*c])*f]) - ((b*B - 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*d - (b + Sqrt[b^2 - 4*a*c])*f*x)/(Sqrt[4*c^2*d
- 4*a*c*f + 2*b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[d + f*x^2])])/(2*Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4
*a*c])*f])))/Sqrt[b^2 - 4*a*c]

________________________________________________________________________________________

Maple [B]  time = 0.344, size = 1771, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x)

[Out]

-2/(-4*a*c+b^2)^(1/2)/(-2*(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(b*f*(-4*a*c+b^2)^(1/
2)+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))+1/2*(-2*(b*f*(-4*a*
c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*f-4*f*(b-(-4*a*c+b^2)^(1
/2))/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))-2*(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x-1/2/c
*(-b+(-4*a*c+b^2)^(1/2))))*A-1/c/(-2*(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(b*f*(-4*a
*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))+1/2*(-2*
(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^2*f-4*f*(b-(-4*
a*c+b^2)^(1/2))/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))-2*(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2
))/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))*B+1/(-4*a*c+b^2)^(1/2)/c/(-2*(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2
*d)/c^2)^(1/2)*ln((-(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2/c*(-b
+(-4*a*c+b^2)^(1/2)))+1/2*(-2*(b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2/c*(-b+(-4*a*
c+b^2)^(1/2)))^2*f-4*f*(b-(-4*a*c+b^2)^(1/2))/c*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))-2*(b*f*(-4*a*c+b^2)^(1/2)+2*
a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))*B*b+2/(-4*a*c+b^2)^(1/2)/(-2*(-b*f*(-4*a*c
+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b+(-
4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c
^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c
)-2*(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*A-1/c/(-2*(-
b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/
c^2-f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f
-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^
2)^(1/2))/c)-2*(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*B
-1/(-4*a*c+b^2)^(1/2)/c/(-2*(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*ln((-(-b*f*(-4*a*c+b^2)
^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2-f*(b+(-4*a*c+b^2)^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-b*f*(-
4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*f*(b+(-4*a*c+b^2)
^(1/2))/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-b*f*(-4*a*c+b^2)^(1/2)+2*a*c*f-b^2*f-2*c^2*d)/c^2)^(1/2))/(x+1/
2*(b+(-4*a*c+b^2)^(1/2))/c))*B*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((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\sqrt{d + f x^{2}} \left (a + b x + c x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+b*x+a)/(f*x**2+d)**(1/2),x)

[Out]

Integral((A + B*x)/(sqrt(d + f*x**2)*(a + b*x + c*x**2)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x+a)/(f*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError