3.7.12 \(\int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} (a+c x^2)} \, dx\) [612]

Optimal. Leaf size=230 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g}} \]

[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))/(-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))/(-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 = 0.14, antiderivative size = 230, 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 = {926, 95, 214} \begin {gather*} \frac {\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 {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\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 {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*(a + c*x^2)),x]

[Out]

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]*S
qrt[Sqrt[c]*d - Sqrt[-a]*e]*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[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[
-a]*g])

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 926

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \left (a+c x^2\right )} \, dx &=\int \left (\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a}}\\ &=-\frac {\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}}-\frac {\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}}\\ &=\frac {\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 {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\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 {\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 = 0.86, size = 286, normalized size = 1.24 \begin {gather*} \frac {\sqrt [4]{-1} \left (-\frac {\sqrt {-i \sqrt {c} d+\sqrt {a} e} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-i \sqrt {c} d+\sqrt {a} e} \sqrt {\sqrt {c} f-i \sqrt {a} g} \sqrt {d+e x}}\right )}{\sqrt {\sqrt {c} f-i \sqrt {a} g}}+\frac {\sqrt {i \sqrt {c} d+\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {i \sqrt {c} d+\sqrt {a} e} \sqrt {\sqrt {c} f+i \sqrt {a} g} \sqrt {d+e x}}\right )}{\sqrt {\sqrt {c} f+i \sqrt {a} g}}\right )}{\sqrt {a} \sqrt {c d^2+a e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*(a + c*x^2)),x]

[Out]

((-1)^(1/4)*(-((Sqrt[(-I)*Sqrt[c]*d + Sqrt[a]*e]*ArcTan[((-1)^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[(
-I)*Sqrt[c]*d + Sqrt[a]*e]*Sqrt[Sqrt[c]*f - I*Sqrt[a]*g]*Sqrt[d + e*x])])/Sqrt[Sqrt[c]*f - I*Sqrt[a]*g]) + (Sq
rt[I*Sqrt[c]*d + Sqrt[a]*e]*ArcTanh[((-1)^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[I*Sqrt[c]*d + Sqrt[a]
*e]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*Sqrt[d + e*x])])/Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]))/(Sqrt[a]*Sqrt[c*d^2 + a*e^2
])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1414\) vs. \(2(170)=340\).
time = 0.10, size = 1415, normalized size = 6.15

method	result	size
default	\(\frac {c^{2} \left (\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^{2} e^{2} g^{2} \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 \,d^{2} g^{2} \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^{2} \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^{2} \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^{2} e^{2} g^{2} \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 \,d^{2} g^{2} \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^{2} \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^{2} \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\right ) \sqrt {g x +f}\, \sqrt {e x +d}}{2 \sqrt {-\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f +a e g -c d f}{c}}\, \left (c f -g \sqrt {-a c}\right ) \sqrt {\frac {\sqrt {-a c}\, d g +\sqrt {-a c}\, e f -a e g +c d f}{c}}\, \left (g \sqrt {-a c}+c f \right ) \sqrt {-a c}\, \left (c d -\sqrt {-a c}\, e \right ) \left (\sqrt {-a c}\, e +c d \right ) \sqrt {\left (e x +d \right ) \left (g x +f \right )}}\)	\(1415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4509 vs. \(2 (176) = 352\).
time = 61.51, size = 4509, normalized size = 19.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-(c*d*f - a*g*e + (a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2)*sqrt(-(c*d^2*g^2 + 2*c
*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^
2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^4)*e^2)))/(a*c^2*d^2*f^2 + a^2*c*d
^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2))*log((d^2*g^2 + 2*(c*d^2*f*g - a*f*g*e^2 + (c*d*f^2 - a*d*g^2)*e - (a*c^2*
d^3*f^2*g + a^2*c*d^3*g^3 + (a^2*c*f^3 + a^3*f*g^2)*e^3 + (a^2*c*d*f^2*g + a^3*d*g^3)*e^2 + (a*c^2*d^2*f^3 + a
^2*c*d^2*f*g^2)*e)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^
2*d^4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c
*d^2*g^4)*e^2)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt(-(c*d*f - a*g*e + (a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^
2 + a^3*g^2)*e^2)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2
*d^4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*
d^2*g^4)*e^2)))/(a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2)) + (2*f*g*x + f^2)*e^2 + 2*(d*g^2*
x + d*f*g)*e + (2*c^2*d^3*f^3 + 2*a*c*d^3*f*g^2 + (a*c*f^3 + a^2*f*g^2)*x*e^3 + (c^2*d^2*f^3 + a*c*d^2*f*g^2)*
x*e + (c^2*d^3*f^2*g + a*c*d^3*g^3)*x + (2*a*c*d*f^3 + 2*a^2*d*f*g^2 + (a*c*d*f^2*g + a^2*d*g^3)*x)*e^2)*sqrt(
-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g^4 + (a^3*c^2*f^4
+ 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^4)*e^2)))/x) + 1/
4*sqrt(-(c*d*f - a*g*e + (a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2)*sqrt(-(c*d^2*g^2 + 2*c*d*
f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 +
a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^4)*e^2)))/(a*c^2*d^2*f^2 + a^2*c*d^2*
g^2 + (a^2*c*f^2 + a^3*g^2)*e^2))*log((d^2*g^2 - 2*(c*d^2*f*g - a*f*g*e^2 + (c*d*f^2 - a*d*g^2)*e - (a*c^2*d^3
*f^2*g + a^2*c*d^3*g^3 + (a^2*c*f^3 + a^3*f*g^2)*e^3 + (a^2*c*d*f^2*g + a^3*d*g^3)*e^2 + (a*c^2*d^2*f^3 + a^2*
c*d^2*f*g^2)*e)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d
^4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^
2*g^4)*e^2)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt(-(c*d*f - a*g*e + (a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 +
a^3*g^2)*e^2)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^
4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2
*g^4)*e^2)))/(a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2)) + (2*f*g*x + f^2)*e^2 + 2*(d*g^2*x +
d*f*g)*e + (2*c^2*d^3*f^3 + 2*a*c*d^3*f*g^2 + (a*c*f^3 + a^2*f*g^2)*x*e^3 + (c^2*d^2*f^3 + a*c*d^2*f*g^2)*x*e
+ (c^2*d^3*f^2*g + a*c*d^3*g^3)*x + (2*a*c*d*f^3 + 2*a^2*d*f*g^2 + (a*c*d*f^2*g + a^2*d*g^3)*x)*e^2)*sqrt(-(c
*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g^4 + (a^3*c^2*f^4 +
2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^4)*e^2)))/x) - 1/4*s
qrt(-(c*d*f - a*g*e - (a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g
*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^
5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^4)*e^2)))/(a*c^2*d^2*f^2 + a^2*c*d^2*g^2
+ (a^2*c*f^2 + a^3*g^2)*e^2))*log((d^2*g^2 + 2*(c*d^2*f*g - a*f*g*e^2 + (c*d*f^2 - a*d*g^2)*e + (a*c^2*d^3*f^
2*g + a^2*c*d^3*g^3 + (a^2*c*f^3 + a^3*f*g^2)*e^3 + (a^2*c*d*f^2*g + a^3*d*g^3)*e^2 + (a*c^2*d^2*f^3 + a^2*c*d
^2*f*g^2)*e)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*
g^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g
^4)*e^2)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt(-(c*d*f - a*g*e - (a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^
3*g^2)*e^2)*sqrt(-(c*d^2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g
^4 + (a^3*c^2*f^4 + 2*a^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^
4)*e^2)))/(a*c^2*d^2*f^2 + a^2*c*d^2*g^2 + (a^2*c*f^2 + a^3*g^2)*e^2)) + (2*f*g*x + f^2)*e^2 + 2*(d*g^2*x + d*
f*g)*e - (2*c^2*d^3*f^3 + 2*a*c*d^3*f*g^2 + (a*c*f^3 + a^2*f*g^2)*x*e^3 + (c^2*d^2*f^3 + a*c*d^2*f*g^2)*x*e +
(c^2*d^3*f^2*g + a*c*d^3*g^3)*x + (2*a*c*d*f^3 + 2*a^2*d*f*g^2 + (a*c*d*f^2*g + a^2*d*g^3)*x)*e^2)*sqrt(-(c*d^
2*g^2 + 2*c*d*f*g*e + c*f^2*e^2)/(a*c^4*d^4*f^4 + 2*a^2*c^3*d^4*f^2*g^2 + a^3*c^2*d^4*g^4 + (a^3*c^2*f^4 + 2*a
^4*c*f^2*g^2 + a^5*g^4)*e^4 + 2*(a^2*c^3*d^2*f^4 + 2*a^3*c^2*d^2*f^2*g^2 + a^4*c*d^2*g^4)*e^2)))/x) + 1/4*sqrt
(-(c*d*f - a*g*e - (a*c^2*d^2*f^2 + a^2*c*d^2*g...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right ) \sqrt {d + e x} \sqrt {f + g x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

\text{Hanged}

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