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

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

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Rubi [A]  time = 5.16, antiderivative size = 780, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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 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)]

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*}

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Mathematica [A]  time = 0.41, size = 254, normalized size = 0.33 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [C]  time = 0.44, size = 218, normalized size = 0.28 \begin {gather*} \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^4 c+4 \text {$\#$1}^2 a f-2 \text {$\#$1}^2 c d-4 \text {$\#$1} a e \sqrt {f}+a e^2+c d^2\&,\frac {\text {$\#$1}^2 (-B) \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )+2 \text {$\#$1} A \sqrt {f} \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )-A e \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )+B d \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )}{\text {$\#$1}^3 (-c)-2 \text {$\#$1} a f+\text {$\#$1} c d+a e \sqrt {f}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[c*d^2 + a*e^2 - 4*a*e*Sqrt[f]*#1 - 2*c*d*#1^2 + 4*a*f*#1^2 + c*#1^4 & , (B*d*Log[-(Sqrt[f]*x) + Sqrt[d
+ e*x + f*x^2] - #1] - A*e*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] + 2*A*Sqrt[f]*Log[-(Sqrt[f]*x) + Sq
rt[d + e*x + f*x^2] - #1]*#1 - B*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1^2)/(a*e*Sqrt[f] + c*d*#1 -
2*a*f*#1 - c*#1^3) & ]/2

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fricas [B]  time = 47.87, size = 6861, normalized size = 8.80

result too large to display

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]

-1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f + (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2
*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2
+ a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*
d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*
a^2 - A^4*c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2*(A*B^3*a*c + A^3*B*
c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x + 2*(2*A^2*B*c^3*d^2 + 2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e
+ (B^3*a^2*c - A^2*B*a*c^2)*e^2 - (4*A^2*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f - (B*a*c^4*d^3 - A*a*c^4
*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*c^2*e)*f^2 - (3*B*a^2*c^3*d^2
- 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)
*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4
+ 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^
2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 -
A^2*a*c)*f + (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^
2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2
- A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d
^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*
f^2)) - (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f +
2*(B^2*a^3*c + A^2*a^2*c^2)*d*f^2 + ((B^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 - 2*(B^2*a^2
*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4
*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3
*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 +
a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/x) + 1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d +
(B^2*a^2 - A^2*a*c)*f + (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2
*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d +
(A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(
3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d
*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 - A^4*c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f
- 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2*(A*B^3*a*c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x - 2*(2*A^2*B*
c^3*d^2 + 2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^2 - (4*A^2*B*a*c^2*d
+ (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f - (B*a*c^4*d^3 - A*a*c^4*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c
*f^3 + (3*B*a^3*c^2*d - A*a^3*c^2*e)*f^2 - (3*B*a^2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2
*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4
*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f
^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x^2 + e*x +
d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f + (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*
d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A
^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2
+ a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d
*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)) - (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*
c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f + 2*(B^2*a^3*c + A^2*a^2*c^2)*d*f^2 + ((B^2*a*c^3 +
A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 - 2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2
)*e*f^2)*x)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^
2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3
*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)
*f)))/x) - 1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f - (a*c^3*d^2 + a^2*c^2*e^2 -
2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^
4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c
^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3
+ a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d
*e + (B^4*a^2 - A^4*c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2*(A*B^3*a*
c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x + 2*(2*A^2*B*c^3*d^2 + 2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3
*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^2 - (4*A^2*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f + (B*a*c^4*d^3
- A*a*c^4*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*c^2*e)*f^2 - (3*B*a^
2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c -
A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/
(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f
^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (
B^2*a^2 - A^2*a*c)*f - (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*
B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (
A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3
*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*
f + a^3*c*f^2)) + (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^
3)*d^2*f + 2*(B^2*a^3*c + A^2*a^2*c^2)*d*f^2 + ((B^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 -
2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a
^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3
*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*
c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/x) + 1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2
*c^2)*d + (B^2*a^2 - A^2*a*c)*f - (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d
^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^
2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c
*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 - A^4*c^2)*e^2 - 2*(A*B^3*a^2 + A^3*
B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2*(A*B^3*a*c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x - 2
*(2*A^2*B*c^3*d^2 + 2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^2 - (4*A^2
*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f + (B*a*c^4*d^3 - A*a*c^4*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3
- B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*c^2*e)*f^2 - (3*B*a^2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sq
rt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a
^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x
^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f - (a*c^3*d^2 + a^2*c^2*e^2 -
2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4
*a^2 - 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^
4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 +
a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)) + (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2
*(B^2*a^2*c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f + 2*(B^2*a^3*c + A^2*a^2*c^2)*d*f^2 + ((B
^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 - 2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A
^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2
- 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2
*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*
c^3*d*e^2)*f)))/x)

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

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueConj Error: Bad Argumen
t Type

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maple [A]  time = 0.04, size = 784, normalized size = 1.01 \begin {gather*} -\frac {A \ln \left (\frac {-\frac {2 \left (a f -c d -\sqrt {-a c}\, e \right )}{c}+\frac {\left (c e +2 \sqrt {-a c}\, f \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {a f -c d -\sqrt {-a c}\, e}{c}}\, \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} f +\frac {\left (c e +2 \sqrt {-a c}\, f \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {a f -c d -\sqrt {-a c}\, e}{c}}}{x -\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-a c}\, \sqrt {-\frac {a f -c d -\sqrt {-a c}\, e}{c}}}+\frac {A \ln \left (\frac {-\frac {2 \left (a f -c d +\sqrt {-a c}\, e \right )}{c}+\frac {\left (c e -2 \sqrt {-a c}\, f \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {a f -c d +\sqrt {-a c}\, e}{c}}\, \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} f +\frac {\left (c e -2 \sqrt {-a c}\, f \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {a f -c d +\sqrt {-a c}\, e}{c}}}{x +\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-a c}\, \sqrt {-\frac {a f -c d +\sqrt {-a c}\, e}{c}}}-\frac {B \ln \left (\frac {-\frac {2 \left (a f -c d -\sqrt {-a c}\, e \right )}{c}+\frac {\left (c e +2 \sqrt {-a c}\, f \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {a f -c d -\sqrt {-a c}\, e}{c}}\, \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} f +\frac {\left (c e +2 \sqrt {-a c}\, f \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {a f -c d -\sqrt {-a c}\, e}{c}}}{x -\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-\frac {a f -c d -\sqrt {-a c}\, e}{c}}\, c}-\frac {B \ln \left (\frac {-\frac {2 \left (a f -c d +\sqrt {-a c}\, e \right )}{c}+\frac {\left (c e -2 \sqrt {-a c}\, f \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {a f -c d +\sqrt {-a c}\, e}{c}}\, \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} f +\frac {\left (c e -2 \sqrt {-a c}\, f \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {a f -c d +\sqrt {-a c}\, e}{c}}}{x +\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-\frac {a f -c d +\sqrt {-a c}\, e}{c}}\, c} \end {gather*}

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (c x^{2} + a\right )} \sqrt {f x^{2} + e x + d}}\,{d x} \end {gather*}

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]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{\left (c\,x^2+a\right )\,\sqrt {f\,x^2+e\,x+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (a + c x^{2}\right ) \sqrt {d + e x + f x^{2}}}\, dx \end {gather*}

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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