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

Optimal. Leaf size=780 \[ \frac{\sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac{\sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}} \]

[Out]

(Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt
[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 +
a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*S
qrt[a]*Sqrt[c]*Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*
d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt
[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]) - (Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*
c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*
(a*(A*c*e - B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f + Sqrt[c^2*
d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2
+ a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqr
t[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])

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Rubi [A]  time = 5.16229, antiderivative size = 780, 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 = {1036, 1030, 208} \[ \frac{\sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac{\sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt
[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 +
a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*S
qrt[a]*Sqrt[c]*Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*
d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt
[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]) - (Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*
c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*
(a*(A*c*e - B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f + Sqrt[c^2*
d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^2*d^2
+ a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqr
t[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])

Rule 1036

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

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

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

Mathematica [A]  time = 0.43835, size = 254, normalized size = 0.33 \[ \frac{\frac{\left (A \sqrt{c}-\sqrt{-a} B\right ) \tanh ^{-1}\left (\frac{\sqrt{c} (2 d+e x)-\sqrt{-a} (e+2 f x)}{2 \sqrt{d+x (e+f x)} \sqrt{-\sqrt{-a} \sqrt{c} e-a f+c d}}\right )}{\sqrt{-\sqrt{-a} \sqrt{c} e-a f+c d}}-\frac{\left (\sqrt{-a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{-a} (e+2 f x)+\sqrt{c} (2 d+e x)}{2 \sqrt{d+x (e+f x)} \sqrt{\sqrt{-a} \sqrt{c} e-a f+c d}}\right )}{\sqrt{\sqrt{-a} \sqrt{c} e-a f+c d}}}{2 \sqrt{-a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.355, size = 784, normalized size = 1. \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+a)/(f*x^2+e*x+d)^(1/2),x)

[Out]

-1/2/(-a*c)^(1/2)/(-(-(-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2)*ln((-2*(-(-a*c)^(1/2)*e+a*f-c*d)/c+(2*f*(-a*c)^(1/2)+c*
e)/c*(x-1/c*(-a*c)^(1/2))+2*(-(-(-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2)*((x-1/c*(-a*c)^(1/2))^2*f+(2*f*(-a*c)^(1/2)+c
*e)/c*(x-1/c*(-a*c)^(1/2))-(-(-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2))/(x-1/c*(-a*c)^(1/2)))*A-1/2/c/(-(-(-a*c)^(1/2)*
e+a*f-c*d)/c)^(1/2)*ln((-2*(-(-a*c)^(1/2)*e+a*f-c*d)/c+(2*f*(-a*c)^(1/2)+c*e)/c*(x-1/c*(-a*c)^(1/2))+2*(-(-(-a
*c)^(1/2)*e+a*f-c*d)/c)^(1/2)*((x-1/c*(-a*c)^(1/2))^2*f+(2*f*(-a*c)^(1/2)+c*e)/c*(x-1/c*(-a*c)^(1/2))-(-(-a*c)
^(1/2)*e+a*f-c*d)/c)^(1/2))/(x-1/c*(-a*c)^(1/2)))*B+1/2/(-a*c)^(1/2)/(-((-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2)*ln((-
2*((-a*c)^(1/2)*e+a*f-c*d)/c+1/c*(-2*f*(-a*c)^(1/2)+c*e)*(x+1/c*(-a*c)^(1/2))+2*(-((-a*c)^(1/2)*e+a*f-c*d)/c)^
(1/2)*((x+1/c*(-a*c)^(1/2))^2*f+1/c*(-2*f*(-a*c)^(1/2)+c*e)*(x+1/c*(-a*c)^(1/2))-((-a*c)^(1/2)*e+a*f-c*d)/c)^(
1/2))/(x+1/c*(-a*c)^(1/2)))*A-1/2/c/(-((-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2)*ln((-2*((-a*c)^(1/2)*e+a*f-c*d)/c+1/c*
(-2*f*(-a*c)^(1/2)+c*e)*(x+1/c*(-a*c)^(1/2))+2*(-((-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2)*((x+1/c*(-a*c)^(1/2))^2*f+1
/c*(-2*f*(-a*c)^(1/2)+c*e)*(x+1/c*(-a*c)^(1/2))-((-a*c)^(1/2)*e+a*f-c*d)/c)^(1/2))/(x+1/c*(-a*c)^(1/2)))*B

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

[Out]

Exception raised: ValueError

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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+a)/(f*x^2+e*x+d)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+a)/(f*x^2+e*x+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError