3.7.7 \(\int \frac {\sqrt {f+g x}}{\sqrt {d+e x} (a+c x^2)} \, dx\) [607]

Optimal. Leaf size=240 \[ \frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {-a} e}} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {924, 95, 214} \begin {gather*} \frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \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} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\sqrt {\sqrt {-a} g+\sqrt {c} f} \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} \sqrt {c} \sqrt {\sqrt {-a} e+\sqrt {c} d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 924

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegr
and[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &
& NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx &=\int \left (\frac {\sqrt {-a} f-\frac {a g}{\sqrt {c}}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {\sqrt {-a} f+\frac {a g}{\sqrt {c}}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx\\ &=\frac {1}{2} \left (\frac {a f}{(-a)^{3/2}}-\frac {g}{\sqrt {c}}\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx+\frac {1}{2} \left (\frac {a f}{(-a)^{3/2}}+\frac {g}{\sqrt {c}}\right ) \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx\\ &=\left (\frac {a f}{(-a)^{3/2}}-\frac {g}{\sqrt {c}}\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 )+\left (\frac {a f}{(-a)^{3/2}}+\frac {g}{\sqrt {c}}\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 )\\ &=\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \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} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \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} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {-a} e}}\\ \end {align*}

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Mathematica [A]
time = 10.33, size = 229, normalized size = 0.95 \begin {gather*} \frac {\frac {\sqrt {-\sqrt {c} f+\sqrt {-a} g} \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 {-\sqrt {c} d+\sqrt {-a} e}}-\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \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 {\sqrt {c} d+\sqrt {-a} e}}}{\sqrt {-a} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1386\) vs. \(2(176)=352\).
time = 0.11, size = 1387, normalized size = 5.78

method	result	size
default	\(\frac {\sqrt {g x +f}\, \sqrt {e x +d}\, \left (\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 c \,e^{2} f -\sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \sqrt {-a 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^{2} 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^{2} d^{2} f -\sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \sqrt {-a 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^{2} g -\sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \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 ) a c \,e^{2} f -\sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \sqrt {-a c}\, \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 ) a \,e^{2} g -\sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \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 ) c^{2} d^{2} f -\sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \sqrt {-a c}\, \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 ) c \,d^{2} g \right )}{2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \left (\sqrt {-a c}\, e +c d \right ) \left (c d -\sqrt {-a c}\, e \right ) \sqrt {-a c}\, \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}}}\)	\(1387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1865 vs. \(2 (182) = 364\).
time = 19.26, size = 1865, normalized size = 7.77 \begin {gather*} -\frac {1}{4} \, \sqrt {-\frac {c d f + a g e + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} \log \left (-\frac {2 \, d g^{2} x e + d^{2} g^{2} + 2 \, {\left (c d^{2} g - c d f e + {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {-\frac {c d f + a g e + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} - {\left (2 \, f g x + f^{2}\right )} e^{2} - {\left (c^{2} d^{3} g x + c^{2} d^{2} f x e + 2 \, c^{2} d^{3} f + a c f x e^{3} + {\left (a c d g x + 2 \, a c d f\right )} e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{x}\right ) + \frac {1}{4} \, \sqrt {-\frac {c d f + a g e + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} \log \left (-\frac {2 \, d g^{2} x e + d^{2} g^{2} - 2 \, {\left (c d^{2} g - c d f e + {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {-\frac {c d f + a g e + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} - {\left (2 \, f g x + f^{2}\right )} e^{2} - {\left (c^{2} d^{3} g x + c^{2} d^{2} f x e + 2 \, c^{2} d^{3} f + a c f x e^{3} + {\left (a c d g x + 2 \, a c d f\right )} e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{x}\right ) - \frac {1}{4} \, \sqrt {-\frac {c d f + a g e - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} \log \left (-\frac {2 \, d g^{2} x e + d^{2} g^{2} + 2 \, {\left (c d^{2} g - c d f e - {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {-\frac {c d f + a g e - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} - {\left (2 \, f g x + f^{2}\right )} e^{2} + {\left (c^{2} d^{3} g x + c^{2} d^{2} f x e + 2 \, c^{2} d^{3} f + a c f x e^{3} + {\left (a c d g x + 2 \, a c d f\right )} e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{x}\right ) + \frac {1}{4} \, \sqrt {-\frac {c d f + a g e - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} \log \left (-\frac {2 \, d g^{2} x e + d^{2} g^{2} - 2 \, {\left (c d^{2} g - c d f e - {\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {-\frac {c d f + a g e - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{a c^{2} d^{2} + a^{2} c e^{2}}} - {\left (2 \, f g x + f^{2}\right )} e^{2} + {\left (c^{2} d^{3} g x + c^{2} d^{2} f x e + 2 \, c^{2} d^{3} f + a c f x e^{3} + {\left (a c d g x + 2 \, a c d f\right )} e^{2}\right )} \sqrt {-\frac {d^{2} g^{2} - 2 \, d f g e + f^{2} e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-(c*d*f + a*g*e + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c
^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(2*d*g^2*x*e + d^2*g^2 + 2*(c*d^2*g - c*d*f*e + (a*c^2
*d^2*e + a^2*c*e^3)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(g
*x + f)*sqrt(x*e + d)*sqrt(-(c*d*f + a*g*e + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*
c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2)) - (2*f*g*x + f^2)*e^2 - (c^2*d^3*g*x + c^2
*d^2*f*x*e + 2*c^2*d^3*f + a*c*f*x*e^3 + (a*c*d*g*x + 2*a*c*d*f)*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a
*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) + 1/4*sqrt(-(c*d*f + a*g*e + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(d^2
*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(2*d*g
^2*x*e + d^2*g^2 - 2*(c*d^2*g - c*d*f*e + (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c
^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt(-(c*d*f + a*g*e + (a*c^2*d^2 + a^2*
c*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*
e^2)) - (2*f*g*x + f^2)*e^2 - (c^2*d^3*g*x + c^2*d^2*f*x*e + 2*c^2*d^3*f + a*c*f*x*e^3 + (a*c*d*g*x + 2*a*c*d*
f)*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) - 1/4*sqrt(-(c*
d*f + a*g*e - (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a
^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(2*d*g^2*x*e + d^2*g^2 + 2*(c*d^2*g - c*d*f*e - (a*c^2*d^2*e + a^2*c
*e^3)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(g*x + f)*sqrt(x
*e + d)*sqrt(-(c*d*f + a*g*e - (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^
2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2)) - (2*f*g*x + f^2)*e^2 + (c^2*d^3*g*x + c^2*d^2*f*x*e + 2
*c^2*d^3*f + a*c*f*x*e^3 + (a*c*d*g*x + 2*a*c*d*f)*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a
^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) + 1/4*sqrt(-(c*d*f + a*g*e - (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(d^2*g^2 - 2*d*f*g
*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(2*d*g^2*x*e + d^2*g
^2 - 2*(c*d^2*g - c*d*f*e - (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2
*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt(-(c*d*f + a*g*e - (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(
d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2)) - (2*f*g
*x + f^2)*e^2 + (c^2*d^3*g*x + c^2*d^2*f*x*e + 2*c^2*d^3*f + a*c*f*x*e^3 + (a*c*d*g*x + 2*a*c*d*f)*e^2)*sqrt(-
(d^2*g^2 - 2*d*f*g*e + f^2*e^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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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)/((a + c*x^2)*(d + e*x)^(1/2)),x)

[Out]

\text{Hanged}

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