\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx\) [859]

Optimal result

Integrand size = 29, antiderivative size = 490 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx=\frac {\sqrt {a+b x+c x^2}}{(e f-d g) (f+g x)}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} (e f-d g)^2}+\frac {e (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} g (e f-d g)^2}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{g (e f-d g)}+\frac {\sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{(e f-d g)^2}+\frac {(2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 g (e f-d g) \sqrt {c f^2-b f g+a g^2}}-\frac {e \sqrt {c f^2-b f g+a g^2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g (e f-d g)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {974, 748, 857, 635, 212, 738, 746} \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx=\frac {\sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{(e f-d g)^2}-\frac {e \sqrt {a g^2-b f g+c f^2} \text {arctanh}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{g (e f-d g)^2}+\frac {(2 c f-b g) \text {arctanh}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{2 g (e f-d g) \sqrt {a g^2-b f g+c f^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{g (e f-d g)}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} (e f-d g)^2}+\frac {e (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} g (e f-d g)^2}+\frac {\sqrt {a+b x+c x^2}}{(f+g x) (e f-d g)} \]

[In]

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

[Out]

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

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 748

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 857

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

Rule 974

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

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

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx=\frac {\frac {(e f-d g) \sqrt {a+x (b+c x)}}{f+g x}+2 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )-\frac {\sqrt {-c f^2+b f g-a g^2} (2 c d f+2 a e g-b (e f+d g)) \arctan \left (\frac {\sqrt {c} (f+g x)-g \sqrt {a+x (b+c x)}}{\sqrt {-c f^2+g (b f-a g)}}\right )}{c f^2+g (-b f+a g)}}{(e f-d g)^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1358\) vs. \(2(434)=868\).

Time = 0.90 (sec) , antiderivative size = 1359, normalized size of antiderivative = 2.77

method	result	size
default	\(\text {Expression too large to display}\)	\(1359\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (f+g\,x\right )}^2\,\left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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