3.9.51 \(\int \frac {\sqrt {d+e x}}{\sqrt {f+g x} (a+b x+c x^2)} \, dx\) [851]

Optimal. Leaf size=285 \[ -\frac {2 \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}+\frac {2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \]

[Out]

-2*arctanh((e*x+d)^(1/2)*(2*c*f-g*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(g*x+f)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))
^(1/2))*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+2*arc
tanh((e*x+d)^(1/2)*(2*c*f-g*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(g*x+f)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
)*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {923, 95, 214} \begin {gather*} \frac {2 \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {f+g x} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {2 \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {f+g x} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(S
qrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c]
)*g]) + (2*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e
*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*f - (b + Sqrt[b^2 -
4*a*c])*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 923

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Int
[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b
, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m + 1/2, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx\\ &=\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx+\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx\\ &=\left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e-\left (-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )+\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e-\left (-2 c f+\left (b-\sqrt {b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )\\ &=-\frac {2 \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}+\frac {2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(925\) vs. \(2(285)=570\).
time = 10.90, size = 925, normalized size = 3.25 \begin {gather*} \frac {\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{\sqrt {2 c^2 d f+b \left (b-\sqrt {b^2-4 a c}\right ) e g+c \left (\sqrt {b^2-4 a c} e f+\sqrt {b^2-4 a c} d g-2 a e g-b (e f+d g)\right )}}-\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\sqrt {2 c^2 d f+b \left (b+\sqrt {b^2-4 a c}\right ) e g-c \left (b e f+\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g+2 a e g\right )}}-\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \log \left (2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d f+b \left (b-\sqrt {b^2-4 a c}\right ) e g+c \left (\sqrt {b^2-4 a c} e f+\sqrt {b^2-4 a c} d g-2 a e g-b (e f+d g)\right )} \sqrt {d+e x} \sqrt {f+g x}+b^2 (d g+e (f+2 g x))-b \sqrt {b^2-4 a c} (d g+e (f+2 g x))+2 c \left (\sqrt {b^2-4 a c} e f x-2 a e (f+2 g x)+d \left (2 \sqrt {b^2-4 a c} f-2 a g+\sqrt {b^2-4 a c} g x\right )\right )\right )}{\sqrt {2 c^2 d f+b \left (b-\sqrt {b^2-4 a c}\right ) e g+c \left (\sqrt {b^2-4 a c} e f+\sqrt {b^2-4 a c} d g-2 a e g-b (e f+d g)\right )}}+\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \log \left (2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d f+b \left (b+\sqrt {b^2-4 a c}\right ) e g-c \left (b e f+\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g+2 a e g\right )} \sqrt {d+e x} \sqrt {f+g x}-b^2 (d g+e (f+2 g x))-b \sqrt {b^2-4 a c} (d g+e (f+2 g x))+2 c \left (\sqrt {b^2-4 a c} e f x+2 a e (f+2 g x)+d \left (2 \sqrt {b^2-4 a c} f+2 a g+\sqrt {b^2-4 a c} g x\right )\right )\right )}{\sqrt {2 c^2 d f+b \left (b+\sqrt {b^2-4 a c}\right ) e g-c \left (b e f+\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g+2 a e g\right )}}}{\sqrt {2} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/Sqrt[2*c^2*d*f + b*(b - Sqrt[b^2 -
4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f + Sqrt[b^2 - 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))] - ((2*c*d - (b + S
qrt[b^2 - 4*a*c])*e)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x])/Sqrt[2*c^2*d*f + b*(b + Sqrt[b^2 - 4*a*c])*e*g - c*(b
*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)] - ((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*
e)*Log[2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f +
Sqrt[b^2 - 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))]*Sqrt[d + e*x]*Sqrt[f + g*x] + b^2*(d*g + e*(f + 2*g*x)) - b
*Sqrt[b^2 - 4*a*c]*(d*g + e*(f + 2*g*x)) + 2*c*(Sqrt[b^2 - 4*a*c]*e*f*x - 2*a*e*(f + 2*g*x) + d*(2*Sqrt[b^2 -
4*a*c]*f - 2*a*g + Sqrt[b^2 - 4*a*c]*g*x))])/Sqrt[2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*
a*c]*e*f + Sqrt[b^2 - 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))] + ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*Log[2*Sqrt
[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f + b*(b + Sqrt[b^2 - 4*a*c])*e*g - c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*
g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)]*Sqrt[d + e*x]*Sqrt[f + g*x] - b^2*(d*g + e*(f + 2*g*x)) - b*Sqrt[b^2 - 4
*a*c]*(d*g + e*(f + 2*g*x)) + 2*c*(Sqrt[b^2 - 4*a*c]*e*f*x + 2*a*e*(f + 2*g*x) + d*(2*Sqrt[b^2 - 4*a*c]*f + 2*
a*g + Sqrt[b^2 - 4*a*c]*g*x))])/Sqrt[2*c^2*d*f + b*(b + Sqrt[b^2 - 4*a*c])*e*g - c*(b*e*f + Sqrt[b^2 - 4*a*c]*
e*f + b*d*g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)])/(Sqrt[2]*Sqrt[b^2 - 4*a*c])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5481\) vs. \(2(241)=482\).
time = 0.14, size = 5482, normalized size = 19.24

method	result	size
default	\(\text {Expression too large to display}\)	\(5482\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4495 vs. \(2 (247) = 494\).
time = 100.48, size = 4495, normalized size = 15.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*sqrt((2*c*d*f - b*d*g - (b*f - 2*a*g)*e + ((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 -
4*a^2*c)*g^2)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (
b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*
c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2))*log((b*d^2*g^2*x + 2*b*d^2*f*g - 2*a*d^2*g^2 + sqrt(2
)*((b^2 - 4*a*c)*d*f*g - (b^2 - 4*a*c)*f^2*e - ((b^3*c - 4*a*b*c^2)*f^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g
+ 3*(a*b^3 - 4*a^2*b*c)*f*g^2 - 2*(a^2*b^2 - 4*a^3*c)*g^3)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*
a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3
+ (a^2*b^2 - 4*a^3*c)*g^4)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt((2*c*d*f - b*d*g - (b*f - 2*a*g)*e + ((b^2*c -
4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4
*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^
3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)) + (2*a*f
^2 - (b*f^2 - 4*a*f*g)*x)*e^2 - 2*(2*a*d*g^2*x + b*d*f^2)*e - (2*(b^2*c - 4*a*c^2)*d*f^3 - 2*(b^3 - 4*a*b*c)*d
*f^2*g + 2*(a*b^2 - 4*a^2*c)*d*f*g^2 + ((b^2*c - 4*a*c^2)*f^3 - (b^3 - 4*a*b*c)*f^2*g + (a*b^2 - 4*a^2*c)*f*g^
2)*x*e + ((b^2*c - 4*a*c^2)*d*f^2*g - (b^3 - 4*a*b*c)*d*f*g^2 + (a*b^2 - 4*a^2*c)*d*g^3)*x)*sqrt((d^2*g^2 - 2*
d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*
g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/x) + 1/4*sqrt(2)*sqrt((2*c*d*f - b*d*g - (b*f -
2*a*g)*e + ((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((d^2*g^2 - 2*d*f*g*e +
f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a
*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4
*a^2*c)*g^2))*log((b*d^2*g^2*x + 2*b*d^2*f*g - 2*a*d^2*g^2 - sqrt(2)*((b^2 - 4*a*c)*d*f*g - (b^2 - 4*a*c)*f^2*
e - ((b^3*c - 4*a*b*c^2)*f^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g + 3*(a*b^3 - 4*a^2*b*c)*f*g^2 - 2*(a^2*b^2
- 4*a^3*c)*g^3)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g +
(b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))*sqrt(g*x + f)
*sqrt(x*e + d)*sqrt((2*c*d*f - b*d*g - (b*f - 2*a*g)*e + ((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2
- 4*a^2*c)*g^2)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g +
(b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*
a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)) + (2*a*f^2 - (b*f^2 - 4*a*f*g)*x)*e^2 - 2*(2*a*d*g^
2*x + b*d*f^2)*e - (2*(b^2*c - 4*a*c^2)*d*f^3 - 2*(b^3 - 4*a*b*c)*d*f^2*g + 2*(a*b^2 - 4*a^2*c)*d*f*g^2 + ((b^
2*c - 4*a*c^2)*f^3 - (b^3 - 4*a*b*c)*f^2*g + (a*b^2 - 4*a^2*c)*f*g^2)*x*e + ((b^2*c - 4*a*c^2)*d*f^2*g - (b^3
- 4*a*b*c)*d*f*g^2 + (a*b^2 - 4*a^2*c)*d*g^3)*x)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4
- 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^
2 - 4*a^3*c)*g^4)))/x) - 1/4*sqrt(2)*sqrt((2*c*d*f - b*d*g - (b*f - 2*a*g)*e - ((b^2*c - 4*a*c^2)*f^2 - (b^3 -
4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*
c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*
c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2))*log((b*d^2*g^2*x + 2*b*d^2*f*
g - 2*a*d^2*g^2 + sqrt(2)*((b^2 - 4*a*c)*d*f*g - (b^2 - 4*a*c)*f^2*e + ((b^3*c - 4*a*b*c^2)*f^3 - (b^4 - 2*a*b
^2*c - 8*a^2*c^2)*f^2*g + 3*(a*b^3 - 4*a^2*b*c)*f*g^2 - 2*(a^2*b^2 - 4*a^3*c)*g^3)*sqrt((d^2*g^2 - 2*d*f*g*e +
f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(
a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))*sqrt(g*x + f)*sqrt(x*e + d)*sqrt((2*c*d*f - b*d*g - (b*f
- 2*a*g)*e - ((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((d^2*g^2 - 2*d*f*g*e
+ f^2*e^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*
(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 -
4*a^2*c)*g^2)) + (2*a*f^2 - (b*f^2 - 4*a*f*g)*x)*e^2 - 2*(2*a*d*g^2*x + b*d*f^2)*e + (2*(b^2*c - 4*a*c^2)*d*f
^3 - 2*(b^3 - 4*a*b*c)*d*f^2*g + 2*(a*b^2 - 4*a^2*c)*d*f*g^2 + ((b^2*c - 4*a*c^2)*f^3 - (b^3 - 4*a*b*c)*f^2*g
+ (a*b^2 - 4*a^2*c)*f*g^2)*x*e + ((b^2*c - 4*a*c^2)*d*f^2*g - (b^3 - 4*a*b*c)*d*f*g^2 + (a*b^2 - 4*a^2*c)*d*g^
3)*x)*sqrt((d^2*g^2 - 2*d*f*g*e + f^2*e^2)/((b^...

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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} \left (a + b x + c x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

\text{Hanged}

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