\(\int \frac {\sqrt {d+e x} (a+b x+c x^2)}{\sqrt {f+g x}} \, dx\) [836]

Optimal result

Integrand size = 29, antiderivative size = 246 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{8 e^2 g^3}-\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}-\frac {(e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{8 e^{5/2} g^{7/2}} \]

[Out]

-1/8*(-d*g+e*f)*(c*(d^2*g^2+2*d*e*f*g+5*e^2*f^2)+2*e*g*(4*a*e*g-b*(d*g+3*e*f)))*arctanh(g^(1/2)*(e*x+d)^(1/2)/
e^(1/2)/(g*x+f)^(1/2))/e^(5/2)/g^(7/2)-1/12*(-6*b*e*g+7*c*d*g+5*c*e*f)*(e*x+d)^(3/2)*(g*x+f)^(1/2)/e^2/g^2+1/3
*c*(e*x+d)^(5/2)*(g*x+f)^(1/2)/e^2/g+1/8*(c*(d^2*g^2+2*d*e*f*g+5*e^2*f^2)+2*e*g*(4*a*e*g-b*(d*g+3*e*f)))*(e*x+
d)^(1/2)*(g*x+f)^(1/2)/e^2/g^3

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {965, 81, 52, 65, 223, 212} \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=-\frac {(e f-d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^{5/2} g^{7/2}}+\frac {\sqrt {d+e x} \sqrt {f+g x} \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^2 g^3}-\frac {(d+e x)^{3/2} \sqrt {f+g x} (-6 b e g+7 c d g+5 c e f)}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g} \]

[In]

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

[Out]

((c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[d + e*x]*Sqrt[f + g*x])/(8*e^2
*g^3) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[f + g*x])/(12*e^2*g^2) + (c*(d + e*x)^(5/2)*Sqrt[f
+ g*x])/(3*e^2*g) - ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Ar
cTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(8*e^(5/2)*g^(7/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 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 965

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], 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] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rubi steps \begin{align*} \text {integral}& = \frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (6 a e^2 g-c d (5 e f+d g)\right )-\frac {1}{2} e (5 c e f+7 c d g-6 b e g) x\right )}{\sqrt {f+g x}} \, dx}{3 e^2 g} \\ & = -\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}+\frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x}} \, dx}{8 e^2 g^2} \\ & = \frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{8 e^2 g^3}-\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}-\frac {\left ((e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{16 e^2 g^3} \\ & = \frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{8 e^2 g^3}-\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}-\frac {\left ((e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{8 e^3 g^3} \\ & = \frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{8 e^2 g^3}-\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}-\frac {\left ((e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{8 e^3 g^3} \\ & = \frac {\left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{8 e^2 g^3}-\frac {(5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {f+g x}}{12 e^2 g^2}+\frac {c (d+e x)^{5/2} \sqrt {f+g x}}{3 e^2 g}-\frac {(e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{8 e^{5/2} g^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.55 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {\sqrt {d+e x} \sqrt {f+g x} \left (6 e g (4 a e g+b (-3 e f+d g+2 e g x))+c \left (-3 d^2 g^2+2 d e g (-2 f+g x)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )}{24 e^2 g^3}+\frac {(e f-d g) \left (c \left (5 e^2 f^2+2 d e f g+d^2 g^2\right )+2 e g (4 a e g-b (3 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}\right )}\right )}{4 e^{5/2} g^{7/2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*Sqrt[f + g*x]*(6*e*g*(4*a*e*g + b*(-3*e*f + d*g + 2*e*g*x)) + c*(-3*d^2*g^2 + 2*d*e*g*(-2*f + g
*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2))))/(24*e^2*g^3) + ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2)
+ 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/(Sqrt[g]*(Sqrt[d - (e*f)/g] - Sqrt[d + e*
x]))])/(4*e^(5/2)*g^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(762\) vs. \(2(214)=428\).

Time = 0.49 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.10

method	result	size
default	\(\frac {\sqrt {e x +d}\, \sqrt {g x +f}\, \left (16 c \,e^{2} g^{2} x^{2} \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+24 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) a d \,e^{2} g^{3}-24 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) a \,e^{3} f \,g^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b \,d^{2} e \,g^{3}-12 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b d \,e^{2} f \,g^{2}+18 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) b \,e^{3} f^{2} g +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} g^{3}+3 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e f \,g^{2}+9 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{2} f^{2} g -15 \ln \left (\frac {2 e g x +2 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,e^{3} f^{3}+24 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, b \,e^{2} g^{2} x +4 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c d e \,g^{2} x -20 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c \,e^{2} f g x +48 \sqrt {e g}\, \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, a \,e^{2} g^{2}+12 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, b d e \,g^{2}-36 \sqrt {e g}\, \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, b \,e^{2} f g -6 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c \,d^{2} g^{2}-8 \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, \sqrt {e g}\, c d e f g +30 \sqrt {e g}\, \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, c \,e^{2} f^{2}\right )}{48 g^{3} \sqrt {\left (g x +f \right ) \left (e x +d \right )}\, e^{2} \sqrt {e g}}\)	\(763\)

[In]

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

[Out]

1/48*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(16*c*e^2*g^2*x^2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+24*ln(1/2*(2*e*g*x+2*((
g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*d*e^2*g^3-24*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/
2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*e^3*f*g^2-6*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e
*f)/(e*g)^(1/2))*b*d^2*e*g^3-12*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*
d*e^2*f*g^2+18*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*e^3*f^2*g+3*ln(1/
2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*g^3+3*ln(1/2*(2*e*g*x+2*((g*x+f)*
(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e*f*g^2+9*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*
g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^2*f^2*g-15*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)
/(e*g)^(1/2))*c*e^3*f^3+24*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*b*e^2*g^2*x+4*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/
2)*c*d*e*g^2*x-20*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*e^2*f*g*x+48*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*a*e^2
*g^2+12*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*b*d*e*g^2-36*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*b*e^2*f*g-6*((g*x
+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d^2*g^2-8*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d*e*f*g+30*(e*g)^(1/2)*((g*x+
f)*(e*x+d))^(1/2)*c*e^2*f^2)/g^3/((g*x+f)*(e*x+d))^(1/2)/e^2/(e*g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\left [-\frac {3 \, {\left (5 \, c e^{3} f^{3} - 3 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} - {\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) - 4 \, {\left (8 \, c e^{3} g^{3} x^{2} + 15 \, c e^{3} f^{2} g - 2 \, {\left (2 \, c d e^{2} + 9 \, b e^{3}\right )} f g^{2} - 3 \, {\left (c d^{2} e - 2 \, b d e^{2} - 8 \, a e^{3}\right )} g^{3} - 2 \, {\left (5 \, c e^{3} f g^{2} - {\left (c d e^{2} + 6 \, b e^{3}\right )} g^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{96 \, e^{3} g^{4}}, \frac {3 \, {\left (5 \, c e^{3} f^{3} - 3 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} - {\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, c e^{3} g^{3} x^{2} + 15 \, c e^{3} f^{2} g - 2 \, {\left (2 \, c d e^{2} + 9 \, b e^{3}\right )} f g^{2} - 3 \, {\left (c d^{2} e - 2 \, b d e^{2} - 8 \, a e^{3}\right )} g^{3} - 2 \, {\left (5 \, c e^{3} f g^{2} - {\left (c d e^{2} + 6 \, b e^{3}\right )} g^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{48 \, e^{3} g^{4}}\right ] \]

[In]

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

[Out]

[-1/96*(3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^
2*e + 8*a*d*e^2)*g^3)*sqrt(e*g)*log(8*e^2*g^2*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 4*(2*e*g*x + e*f + d*g)*sq
rt(e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(e^2*f*g + d*e*g^2)*x) - 4*(8*c*e^3*g^3*x^2 + 15*c*e^3*f^2*g - 2*(2*c*
d*e^2 + 9*b*e^3)*f*g^2 - 3*(c*d^2*e - 2*b*d*e^2 - 8*a*e^3)*g^3 - 2*(5*c*e^3*f*g^2 - (c*d*e^2 + 6*b*e^3)*g^3)*x
)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^4), 1/48*(3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*d^2*e - 4*b*
d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*g^3)*sqrt(-e*g)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt
(-e*g)*sqrt(e*x + d)*sqrt(g*x + f)/(e^2*g^2*x^2 + d*e*f*g + (e^2*f*g + d*e*g^2)*x)) + 2*(8*c*e^3*g^3*x^2 + 15*
c*e^3*f^2*g - 2*(2*c*d*e^2 + 9*b*e^3)*f*g^2 - 3*(c*d^2*e - 2*b*d*e^2 - 8*a*e^3)*g^3 - 2*(5*c*e^3*f*g^2 - (c*d*
e^2 + 6*b*e^3)*g^3)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^4)]

Sympy [F]

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

[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)*(a + b*x + c*x**2)/sqrt(f + g*x), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {{\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {e x + d} {\left (2 \, {\left (e x + d\right )} {\left (\frac {4 \, {\left (e x + d\right )} c}{e^{3} g} - \frac {5 \, c e^{7} f g^{3} + 7 \, c d e^{6} g^{4} - 6 \, b e^{7} g^{4}}{e^{9} g^{5}}\right )} + \frac {3 \, {\left (5 \, c e^{8} f^{2} g^{2} + 2 \, c d e^{7} f g^{3} - 6 \, b e^{8} f g^{3} + c d^{2} e^{6} g^{4} - 2 \, b d e^{7} g^{4} + 8 \, a e^{8} g^{4}\right )}}{e^{9} g^{5}}\right )} + \frac {3 \, {\left (5 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - 6 \, b e^{3} f^{2} g - c d^{2} e f g^{2} + 4 \, b d e^{2} f g^{2} + 8 \, a e^{3} f g^{2} - c d^{3} g^{3} + 2 \, b d^{2} e g^{3} - 8 \, a d e^{2} g^{3}\right )} \log \left ({\left | -\sqrt {e g} \sqrt {e x + d} + \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \right |}\right )}{\sqrt {e g} e^{2} g^{3}}\right )} e}{24 \, {\left | e \right |}} \]

[In]

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

[Out]

1/24*(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(e*x + d)*(2*(e*x + d)*(4*(e*x + d)*c/(e^3*g) - (5*c*e^7*f*g^3 +
7*c*d*e^6*g^4 - 6*b*e^7*g^4)/(e^9*g^5)) + 3*(5*c*e^8*f^2*g^2 + 2*c*d*e^7*f*g^3 - 6*b*e^8*f*g^3 + c*d^2*e^6*g^
4 - 2*b*d*e^7*g^4 + 8*a*e^8*g^4)/(e^9*g^5)) + 3*(5*c*e^3*f^3 - 3*c*d*e^2*f^2*g - 6*b*e^3*f^2*g - c*d^2*e*f*g^2
+ 4*b*d*e^2*f*g^2 + 8*a*e^3*f*g^2 - c*d^3*g^3 + 2*b*d^2*e*g^3 - 8*a*d*e^2*g^3)*log(abs(-sqrt(e*g)*sqrt(e*x +
d) + sqrt(e^2*f + (e*x + d)*e*g - d*e*g)))/(sqrt(e*g)*e^2*g^3))*e/abs(e)

Mupad [B] (verification not implemented)

Time = 109.82 (sec) , antiderivative size = 1832, normalized size of antiderivative = 7.45 \[ \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(((2*a*d*g + 2*a*e*f)*((d + e*x)^(1/2) - d^(1/2))^3)/(g^2*((f + g*x)^(1/2) - f^(1/2))^3) + ((2*a*e^2*f + 2*a*d
*e*g)*((d + e*x)^(1/2) - d^(1/2)))/(g^3*((f + g*x)^(1/2) - f^(1/2))) - (8*a*d^(1/2)*e*f^(1/2)*((d + e*x)^(1/2)
- d^(1/2))^2)/(g^2*((f + g*x)^(1/2) - f^(1/2))^2))/(((d + e*x)^(1/2) - d^(1/2))^4/((f + g*x)^(1/2) - f^(1/2))
^4 + e^2/g^2 - (2*e*((d + e*x)^(1/2) - d^(1/2))^2)/(g*((f + g*x)^(1/2) - f^(1/2))^2)) - ((((d + e*x)^(1/2) - d
^(1/2))*((c*d^3*e^3*g^3)/4 - (5*c*e^6*f^3)/4 + (3*c*d*e^5*f^2*g)/4 + (c*d^2*e^4*f*g^2)/4))/(g^9*((f + g*x)^(1/
2) - f^(1/2))) - (((d + e*x)^(1/2) - d^(1/2))^5*((33*c*e^4*f^3)/2 + (19*c*d^3*e*g^3)/2 + (313*c*d*e^3*f^2*g)/2
+ (275*c*d^2*e^2*f*g^2)/2))/(g^7*((f + g*x)^(1/2) - f^(1/2))^5) - (((d + e*x)^(1/2) - d^(1/2))^7*((19*c*d^3*g
^3)/2 + (33*c*e^3*f^3)/2 + (313*c*d*e^2*f^2*g)/2 + (275*c*d^2*e*f*g^2)/2))/(g^6*((f + g*x)^(1/2) - f^(1/2))^7)
- (((d + e*x)^(1/2) - d^(1/2))^3*((17*c*d^3*e^2*g^3)/12 - (85*c*e^5*f^3)/12 + (17*c*d*e^4*f^2*g)/4 + (91*c*d^
2*e^3*f*g^2)/4))/(g^8*((f + g*x)^(1/2) - f^(1/2))^3) + (((d + e*x)^(1/2) - d^(1/2))^11*((c*d^3*g^3)/4 - (5*c*e
^3*f^3)/4 + (3*c*d*e^2*f^2*g)/4 + (c*d^2*e*f*g^2)/4))/(e^2*g^4*((f + g*x)^(1/2) - f^(1/2))^11) - (((d + e*x)^(
1/2) - d^(1/2))^9*((17*c*d^3*g^3)/12 - (85*c*e^3*f^3)/12 + (17*c*d*e^2*f^2*g)/4 + (91*c*d^2*e*f*g^2)/4))/(e*g^
5*((f + g*x)^(1/2) - f^(1/2))^9) + (d^(1/2)*f^(1/2)*((d + e*x)^(1/2) - d^(1/2))^6*(128*c*e^3*f^2 + 64*c*d^2*e*
g^2 + (704*c*d*e^2*f*g)/3))/(g^6*((f + g*x)^(1/2) - f^(1/2))^6) + (d^(1/2)*f^(1/2)*(32*c*d^2*g + 96*c*d*e*f)*(
(d + e*x)^(1/2) - d^(1/2))^8)/(g^4*((f + g*x)^(1/2) - f^(1/2))^8) + (d^(1/2)*f^(1/2)*(96*c*d*e^3*f + 32*c*d^2*
e^2*g)*((d + e*x)^(1/2) - d^(1/2))^4)/(g^6*((f + g*x)^(1/2) - f^(1/2))^4))/(((d + e*x)^(1/2) - d^(1/2))^12/((f
+ g*x)^(1/2) - f^(1/2))^12 + e^6/g^6 - (6*e*((d + e*x)^(1/2) - d^(1/2))^10)/(g*((f + g*x)^(1/2) - f^(1/2))^10
) - (6*e^5*((d + e*x)^(1/2) - d^(1/2))^2)/(g^5*((f + g*x)^(1/2) - f^(1/2))^2) + (15*e^4*((d + e*x)^(1/2) - d^(
1/2))^4)/(g^4*((f + g*x)^(1/2) - f^(1/2))^4) - (20*e^3*((d + e*x)^(1/2) - d^(1/2))^6)/(g^3*((f + g*x)^(1/2) -
f^(1/2))^6) + (15*e^2*((d + e*x)^(1/2) - d^(1/2))^8)/(g^2*((f + g*x)^(1/2) - f^(1/2))^8)) + ((((d + e*x)^(1/2)
- d^(1/2))*((b*d^2*e^2*g^2)/2 - (3*b*e^4*f^2)/2 + b*d*e^3*f*g))/(g^6*((f + g*x)^(1/2) - f^(1/2))) + (((d + e*
x)^(1/2) - d^(1/2))^3*((11*b*e^3*f^2)/2 + (7*b*d^2*e*g^2)/2 + 23*b*d*e^2*f*g))/(g^5*((f + g*x)^(1/2) - f^(1/2)
)^3) + (((d + e*x)^(1/2) - d^(1/2))^5*((7*b*d^2*g^2)/2 + (11*b*e^2*f^2)/2 + 23*b*d*e*f*g))/(g^4*((f + g*x)^(1/
2) - f^(1/2))^5) + (((d + e*x)^(1/2) - d^(1/2))^7*((b*d^2*g^2)/2 - (3*b*e^2*f^2)/2 + b*d*e*f*g))/(e*g^3*((f +
g*x)^(1/2) - f^(1/2))^7) - (d^(1/2)*f^(1/2)*(32*b*e^2*f + 16*b*d*e*g)*((d + e*x)^(1/2) - d^(1/2))^4)/(g^4*((f
+ g*x)^(1/2) - f^(1/2))^4) - (8*b*d^(3/2)*f^(1/2)*((d + e*x)^(1/2) - d^(1/2))^6)/(g^2*((f + g*x)^(1/2) - f^(1/
2))^6) - (8*b*d^(3/2)*e^2*f^(1/2)*((d + e*x)^(1/2) - d^(1/2))^2)/(g^4*((f + g*x)^(1/2) - f^(1/2))^2))/(((d + e
*x)^(1/2) - d^(1/2))^8/((f + g*x)^(1/2) - f^(1/2))^8 + e^4/g^4 - (4*e*((d + e*x)^(1/2) - d^(1/2))^6)/(g*((f +
g*x)^(1/2) - f^(1/2))^6) - (4*e^3*((d + e*x)^(1/2) - d^(1/2))^2)/(g^3*((f + g*x)^(1/2) - f^(1/2))^2) + (6*e^2*
((d + e*x)^(1/2) - d^(1/2))^4)/(g^2*((f + g*x)^(1/2) - f^(1/2))^4)) + (2*a*atanh((g^(1/2)*((d + e*x)^(1/2) - d
^(1/2)))/(e^(1/2)*((f + g*x)^(1/2) - f^(1/2))))*(d*g - e*f))/(e^(1/2)*g^(3/2)) - (b*atanh((g^(1/2)*((d + e*x)^
(1/2) - d^(1/2)))/(e^(1/2)*((f + g*x)^(1/2) - f^(1/2))))*(d*g - e*f)*(d*g + 3*e*f))/(2*e^(3/2)*g^(5/2)) + (c*a
tanh((g^(1/2)*((d + e*x)^(1/2) - d^(1/2)))/(e^(1/2)*((f + g*x)^(1/2) - f^(1/2))))*(d*g - e*f)*(d^2*g^2 + 5*e^2
*f^2 + 2*d*e*f*g))/(4*e^(5/2)*g^(7/2))