3.7.6 \(\int \frac {\sqrt {d+e x} \sqrt {f+g x}}{a+c x^2} \, dx\) [606]

Optimal. Leaf size=342 \[ \frac {2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c}+\frac {\left (c d f-a e g-\sqrt {-a} \sqrt {c} (e f+d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (c d f-a e g+\sqrt {-a} \sqrt {c} (e f+d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g}} \]

[Out]

2*arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))*e^(1/2)*g^(1/2)/c+arctanh((e*x+d)^(1/2)*(-g*(-a)^(1/2)+
f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))^(1/2))*(c*d*f-a*e*g-(d*g+e*f)*(-a)^(1/2)*c^(1/2))/c/(
-a)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))^(1/2)/(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)-arctanh((e*x+d)^(1/2)*(g*(-a)^(1/2)+
f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))^(1/2))*(c*d*f-a*e*g+(d*g+e*f)*(-a)^(1/2)*c^(1/2))/c/(-
a)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))^(1/2)/(g*(-a)^(1/2)+f*c^(1/2))^(1/2)

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Rubi [A]
time = 1.26, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {920, 65, 223, 212, 6857, 95, 214} \begin {gather*} \frac {\left (-\sqrt {-a} \sqrt {c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (\sqrt {-a} \sqrt {c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} c \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}+\frac {2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e]*Sqrt[g]*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/c + ((c*d*f - a*e*g - Sqrt[-a]*Sq
rt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f +
g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - ((c*d*f - a*e*g + Sqrt[-a]*S
qrt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f
+ g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 212

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

Rule 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 920

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[e*(g/c), In
t[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[Simp[c*d*f - a*e*g + (c*e*f + c*d*g)*x, x]*(d +
e*x)^(m - 1)*((f + g*x)^(n - 1)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.91, size = 363, normalized size = 1.06 \begin {gather*} \frac {\frac {\sqrt {c d^2+a e^2} \left (i \sqrt {c} f+\sqrt {a} g\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}+\frac {\sqrt {c d^2+a e^2} \left (-i \sqrt {c} f+\sqrt {a} g\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}+2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[c*d^2 + a*e^2]*(I*Sqrt[c]*f + Sqrt[a]*g)*ArcTan[(Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[-((Sqrt[c]*d
+ I*Sqrt[a]*e)*(Sqrt[c]*f - I*Sqrt[a]*g))]*Sqrt[d + e*x])])/(Sqrt[a]*Sqrt[-((Sqrt[c]*d + I*Sqrt[a]*e)*(Sqrt[c]
*f - I*Sqrt[a]*g))]) + (Sqrt[c*d^2 + a*e^2]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*ArcTan[(Sqrt[c*d^2 + a*e^2]*Sqrt[f +
g*x])/(Sqrt[-((Sqrt[c]*d - I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g))]*Sqrt[d + e*x])])/(Sqrt[a]*Sqrt[-((Sqrt[c]*
d - I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g))]) + 2*Sqrt[e]*Sqrt[g]*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/(Sqrt[g]*Sqr
t[d + e*x])])/c

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1494\) vs. \(2(262)=524\).
time = 0.09, size = 1495, normalized size = 4.37

method	result	size
default	\(-\frac {\sqrt {e x +d}\, \sqrt {g x +f}\, \left (\sqrt {e g}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \ln \left (\frac {c d g x +c e f x -2 \sqrt {-a c}\, e g x +2 c d f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, c -\sqrt {-a c}\, d g -\sqrt {-a c}\, e f}{c x +\sqrt {-a c}}\right ) \sqrt {-a c}\, d g +\sqrt {e g}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \ln \left (\frac {c d g x +c e f x -2 \sqrt {-a c}\, e g x +2 c d f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, c -\sqrt {-a c}\, d g -\sqrt {-a c}\, e f}{c x +\sqrt {-a c}}\right ) \sqrt {-a c}\, e f +\sqrt {e g}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \ln \left (\frac {c d g x +c e f x -2 \sqrt {-a c}\, e g x +2 c d f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, c -\sqrt {-a c}\, d g -\sqrt {-a c}\, e f}{c x +\sqrt {-a c}}\right ) a e g -\sqrt {e g}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \ln \left (\frac {c d g x +c e f x -2 \sqrt {-a c}\, e g x +2 c d f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, c -\sqrt {-a c}\, d g -\sqrt {-a c}\, e f}{c x +\sqrt {-a c}}\right ) c d f +\sqrt {e g}\, \ln \left (\frac {2 \sqrt {-a c}\, e g x +c d g x +c e f x +\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, c +2 c d f}{c x -\sqrt {-a c}}\right ) \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \sqrt {-a c}\, d g +\sqrt {e g}\, \ln \left (\frac {2 \sqrt {-a c}\, e g x +c d g x +c e f x +\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, c +2 c d f}{c x -\sqrt {-a c}}\right ) \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \sqrt {-a c}\, e f -\sqrt {e g}\, \ln \left (\frac {2 \sqrt {-a c}\, e g x +c d g x +c e f x +\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, c +2 c d f}{c x -\sqrt {-a c}}\right ) \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, a e g +\sqrt {e g}\, \ln \left (\frac {2 \sqrt {-a c}\, e g x +c d g x +c e f x +\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, c +2 c d f}{c x -\sqrt {-a c}}\right ) \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, c d f -2 \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) \sqrt {-a c}\, e g \right )}{2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {-a c}\, c \sqrt {e g}\, \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}}\)	\(1495\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(g*x+f)^(1/2)/(c*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d + e x} \sqrt {f + g x}}{a + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.52index.cc index_m i_lex_is_greater Error: Bad Argument Value

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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