3.45 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} (A+B x+C x^2)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=521 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)+b^2 \left (-\left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{4 b^3 d f (b e-a f)}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left (3 a^2 C d f-a b (2 B d f+c C f+C d e)+b^2 (2 A d f+c C e)\right )}{2 b^2 f (b c-a d) (b e-a f)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (-a^2 b (4 B d f+5 C (c f+d e))+6 a^3 C d f+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right )}{b^4 \sqrt{b c-a d} \sqrt{b e-a f}}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)} \]

[Out]

((12*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 8*B*d*f) + b^2*(4*d*f*(B*e + A*f) - C*e*(d*e - c*f)))*Sqrt[c + d*x
]*Sqrt[e + f*x])/(4*b^3*d*f*(b*e - a*f)) + ((3*a^2*C*d*f + b^2*(c*C*e + 2*A*d*f) - a*b*(C*d*e + c*C*f + 2*B*d*
f))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(2*b^2*(b*c - a*d)*f*(b*e - a*f)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2
)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((24*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 2*B
*d*f) - b^2*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt
[e + f*x])])/(4*b^4*d^(3/2)*f^(3/2)) + ((6*a^3*C*d*f - b^3*(2*B*c*e + A*d*e + A*c*f) + a*b^2*(4*c*C*e + 3*B*d*
e + 3*B*c*f + 2*A*d*f) - a^2*b*(4*B*d*f + 5*C*(d*e + c*f)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c
- a*d]*Sqrt[e + f*x])])/(b^4*Sqrt[b*c - a*d]*Sqrt[b*e - a*f])

________________________________________________________________________________________

Rubi [A]  time = 1.69576, antiderivative size = 521, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1613, 154, 157, 63, 217, 206, 93, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)+b^2 \left (-\left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )\right )\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right )}{4 b^3 d f (b e-a f)}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left (3 a^2 C d f-a b (2 B d f+c C f+C d e)+b^2 (2 A d f+c C e)\right )}{2 b^2 f (b c-a d) (b e-a f)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (-a^2 b (4 B d f+5 C (c f+d e))+6 a^3 C d f+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right )}{b^4 \sqrt{b c-a d} \sqrt{b e-a f}}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 8*B*d*f) + b^2*(4*d*f*(B*e + A*f) - C*e*(d*e - c*f)))*Sqrt[c + d*x
]*Sqrt[e + f*x])/(4*b^3*d*f*(b*e - a*f)) + ((3*a^2*C*d*f + b^2*(c*C*e + 2*A*d*f) - a*b*(C*d*e + c*C*f + 2*B*d*
f))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(2*b^2*(b*c - a*d)*f*(b*e - a*f)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2
)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((24*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 2*B
*d*f) - b^2*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt
[e + f*x])])/(4*b^4*d^(3/2)*f^(3/2)) + ((6*a^3*C*d*f - b^3*(2*B*c*e + A*d*e + A*c*f) + a*b^2*(4*c*C*e + 3*B*d*
e + 3*B*c*f + 2*A*d*f) - a^2*b*(4*B*d*f + 5*C*(d*e + c*f)))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c
- a*d]*Sqrt[e + f*x])])/(b^4*Sqrt[b*c - a*d]*Sqrt[b*e - a*f])

Rule 1613

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 63

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 217

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 93

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right )}{(a+b x)^2} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac{\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (-\frac{3 a^2 C (d e+c f)+b^2 (2 B c e+A d e+A c f)-a b (2 c C e+3 B d e+3 B c f-2 A d f)}{2 b}+\left (-\frac{3 a^2 C d f}{b}-b (c C e+2 A d f)+a (C d e+c C f+2 B d f)\right ) x\right )}{a+b x} \, dx}{(b c-a d) (b e-a f)}\\ &=\frac{\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt{c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac{\int \frac{\sqrt{e+f x} \left (-\frac{(b c-a d) \left (3 a^2 C f (d e+3 c f)+2 b^2 f (2 B c e+A d e+A c f)-a b (2 B f (d e+3 c f)+C e (d e+7 c f))\right )}{2 b}-\frac{(b c-a d) \left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) x}{2 b}\right )}{(a+b x) \sqrt{c+d x}} \, dx}{2 b (b c-a d) f (b e-a f)}\\ &=\frac{\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 d f (b e-a f)}+\frac{\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt{c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac{\int \frac{-\frac{(b c-a d) (b e-a f) \left (12 a^2 C d f (d e+c f)+4 b^2 d f (2 B c e+A d e+A c f)-a b \left (8 B d f (d e+c f)+C \left (d^2 e^2+14 c d e f+c^2 f^2\right )\right )\right )}{4 b}-\frac{(b c-a d) (b e-a f) \left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) x}{4 b}}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 b^2 d (b c-a d) f (b e-a f)}\\ &=\frac{\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 d f (b e-a f)}+\frac{\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt{c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac{\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 b^4}+\frac{\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{8 b^4 d f}\\ &=\frac{\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 d f (b e-a f)}+\frac{\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt{c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}-\frac{\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \operatorname{Subst}\left (\int \frac{1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{b^4}+\frac{\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e-\frac{c f}{d}+\frac{f x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{4 b^4 d^2 f}\\ &=\frac{\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 d f (b e-a f)}+\frac{\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt{c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}+\frac{\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{b^4 \sqrt{b c-a d} \sqrt{b e-a f}}+\frac{\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{f x^2}{d}} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{4 b^4 d^2 f}\\ &=\frac{\left (12 a^2 C d f^2-a b f (7 C d e+c C f+8 B d f)+b^2 (4 d f (B e+A f)-C e (d e-c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{4 b^3 d f (b e-a f)}+\frac{\left (3 a^2 C d f+b^2 (c C e+2 A d f)-a b (C d e+c C f+2 B d f)\right ) \sqrt{c+d x} (e+f x)^{3/2}}{2 b^2 (b c-a d) f (b e-a f)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) (a+b x)}+\frac{\left (24 a^2 C d^2 f^2-8 a b d f (C d e+c C f+2 B d f)-b^2 \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{4 b^4 d^{3/2} f^{3/2}}+\frac{\left (6 a^3 C d f-b^3 (2 B c e+A d e+A c f)+a b^2 (4 c C e+3 B d e+3 B c f+2 A d f)-a^2 b (4 B d f+5 C (d e+c f))\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{b^4 \sqrt{b c-a d} \sqrt{b e-a f}}\\ \end{align*}

Mathematica [B]  time = 6.27427, size = 2665, normalized size = 5.12 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x))) + (2*(b*B -
2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*
f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e
- c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*
f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((
d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b^3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(
d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*C*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e
- c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d
*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(
c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh
[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e
- c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e -
c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f
) - (c*d*f)/(d*e - c*f)))))))/(3*b^2*d*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/
(d*e - c*f)]) - ((b*B - 2*a*C)*(-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f
)/(d*e - c*f))]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[
d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(b*d^(3/2)*Sqrt[
e + f*x]) - (2*(-(b*e) + a*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*S
qrt[-(b*c) + a*d]*Sqrt[b*e - a*f])))/b^3 - ((A*b^2 - a*b*B + a^2*C)*((-4*f*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 +
(d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((
d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d
*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]
*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*
e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e -
c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*
f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqr
t[(d*(e + f*x))/(d*e - c*f)]) + ((2*a*b*d*f + (b*(-2*a*d*f - b*(d*e + c*f)))/2)*((2*Sqrt[c + d*x]*Sqrt[e + f*x
]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d
*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) -
(c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d
*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) -
(c*d*f)/(d*e - c*f))))^(3/2))))/(b*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e
- c*f)]) - ((-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sq
rt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c
+ d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(b*d^(3/2)*Sqrt[e + f*x]) - (2*(-(
b*e) + a*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d]*
Sqrt[b*e - a*f])))/b))/b))/(b^2*(b*c - a*d)*(b*e - a*f))

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Maple [B]  time = 0.042, size = 5051, normalized size = 9.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 13.1467, size = 2140, normalized size = 4.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*C*abs(d)/(b^2*d^3) - (C*b^7*c*d^3*f^2*abs(d
) + 8*C*a*b^6*d^4*f^2*abs(d) - 4*B*b^7*d^4*f^2*abs(d) - C*b^7*d^4*f*abs(d)*e)/(b^9*d^6*f^2)) - (5*sqrt(d*f)*C*
a^2*b*c*f*abs(d) - 3*sqrt(d*f)*B*a*b^2*c*f*abs(d) + sqrt(d*f)*A*b^3*c*f*abs(d) - 6*sqrt(d*f)*C*a^3*d*f*abs(d)
+ 4*sqrt(d*f)*B*a^2*b*d*f*abs(d) - 2*sqrt(d*f)*A*a*b^2*d*f*abs(d) - 4*sqrt(d*f)*C*a*b^2*c*abs(d)*e + 2*sqrt(d*
f)*B*b^3*c*abs(d)*e + 5*sqrt(d*f)*C*a^2*b*d*abs(d)*e - 3*sqrt(d*f)*B*a*b^2*d*abs(d)*e + sqrt(d*f)*A*b^3*d*abs(
d)*e)*arctan(-1/2*(b*c*d*f - 2*a*d^2*f + b*d^2*e - (sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2
*e))^2*b)/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d))/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b
^2*c*d*f*e + a*b*d^2*f*e)*b^4*d) - 2*(sqrt(d*f)*C*a^2*b*c^2*d*f^2*abs(d) - sqrt(d*f)*B*a*b^2*c^2*d*f^2*abs(d)
+ sqrt(d*f)*A*b^3*c^2*d*f^2*abs(d) - 2*sqrt(d*f)*C*a^2*b*c*d^2*f*abs(d)*e + 2*sqrt(d*f)*B*a*b^2*c*d^2*f*abs(d)
*e - 2*sqrt(d*f)*A*b^3*c*d^2*f*abs(d)*e - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^
2*e))^2*C*a^2*b*c*f*abs(d) + sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b
^2*c*f*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*b^3*c*f*abs(d) +
2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^3*d*f*abs(d) - 2*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a^2*b*d*f*abs(d) + 2*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a*b^2*d*f*abs(d) + sqrt(d*f)*C*a^2*b*d^3*abs(d)*e^2 -
sqrt(d*f)*B*a*b^2*d^3*abs(d)*e^2 + sqrt(d*f)*A*b^3*d^3*abs(d)*e^2 - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt
((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b*d*abs(d)*e + sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*
f - c*d*f + d^2*e))^2*B*a*b^2*d*abs(d)*e - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d
^2*e))^2*A*b^3*d*abs(d)*e)/((b*c^2*d^2*f^2 - 2*b*c*d^3*f*e - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f -
c*d*f + d^2*e))^2*b*c*d*f + 4*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*a*d^2*f + b*d
^4*e^2 - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b*d^2*e + (sqrt(d*f)*sqrt(d*x + c
) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*b)*b^4) + 1/8*(sqrt(d*f)*C*b^2*c^2*f^2*abs(d) + 8*sqrt(d*f)*C*a*b*c
*d*f^2*abs(d) - 4*sqrt(d*f)*B*b^2*c*d*f^2*abs(d) - 24*sqrt(d*f)*C*a^2*d^2*f^2*abs(d) + 16*sqrt(d*f)*B*a*b*d^2*
f^2*abs(d) - 8*sqrt(d*f)*A*b^2*d^2*f^2*abs(d) - 2*sqrt(d*f)*C*b^2*c*d*f*abs(d)*e + 8*sqrt(d*f)*C*a*b*d^2*f*abs
(d)*e - 4*sqrt(d*f)*B*b^2*d^2*f*abs(d)*e + sqrt(d*f)*C*b^2*d^2*abs(d)*e^2)*log((sqrt(d*f)*sqrt(d*x + c) - sqrt
((d*x + c)*d*f - c*d*f + d^2*e))^2)/(b^4*d^3*f^2)