3.52 \(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^3 \sqrt {e+f x}} \, dx\)

Optimal. Leaf size=484 \[ \frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b C (5 c f+7 d e)+a b^2 (-4 A d f+B c f+3 B d e+8 c C e)-b^3 (-3 A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (8 a^4 C d^2 f^2-4 a^3 b C d f (3 c f+5 d e)+3 a^2 b^2 C \left (c^2 f^2+10 c d e f+5 d^2 e^2\right )-a b^3 \left (2 c d \left (2 A f^2-B e f+12 C e^2\right )+d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)\right )-b^4 \left (-\left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )\right )-2 c d e (2 B e-A f)+A d^2 e^2\right )\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}+\frac {2 C \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {f}} \]

[Out]

-1/4*(8*a^4*C*d^2*f^2-4*a^3*b*C*d*f*(3*c*f+5*d*e)+3*a^2*b^2*C*(c^2*f^2+10*c*d*e*f+5*d^2*e^2)-a*b^3*(d^2*e*(-4*
A*f+3*B*e)+c^2*f*(-B*f+8*C*e)+2*c*d*(2*A*f^2-B*e*f+12*C*e^2))-b^4*(A*d^2*e^2-2*c*d*e*(-A*f+2*B*e)-c^2*(3*A*f^2
-4*B*e*f+8*C*e^2)))*arctanh((-a*f+b*e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b^3/(-a*d+b*c)^(3/2
)/(-a*f+b*e)^(5/2)+2*C*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))*d^(1/2)/b^3/f^(1/2)-1/2*(A*b^2-a*(
B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^2+1/4*(4*a^3*C*d*f-a^2*b*C*(5*c*f+7*d*e)
-b^3*(-3*A*c*f-A*d*e+4*B*c*e)+a*b^2*(-4*A*d*f+B*c*f+3*B*d*e+8*C*c*e))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/(-a*d+b*
c)/(-a*f+b*e)^2/(b*x+a)

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Rubi [A]  time = 1.56, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1613, 149, 157, 63, 217, 206, 93, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (3 a^2 b^2 C \left (c^2 f^2+10 c d e f+5 d^2 e^2\right )-4 a^3 b C d f (3 c f+5 d e)+8 a^4 C d^2 f^2-a b^3 \left (2 c d \left (2 A f^2-B e f+12 C e^2\right )+d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)\right )-b^4 \left (c^2 \left (-\left (3 A f^2-4 B e f+8 C e^2\right )\right )-2 c d e (2 B e-A f)+A d^2 e^2\right )\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (-a^2 b C (5 c f+7 d e)+4 a^3 C d f+a b^2 (-4 A d f+B c f+3 B d e+8 c C e)-b^3 (-3 A c f-A d e+4 B c e)\right )}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}+\frac {2 C \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {f}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^3*Sqrt[e + f*x]),x]

[Out]

((4*a^3*C*d*f - a^2*b*C*(7*d*e + 5*c*f) - b^3*(4*B*c*e - A*d*e - 3*A*c*f) + a*b^2*(8*c*C*e + 3*B*d*e + B*c*f -
4*A*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)) - ((A*b^2 - a*(b*B - a*C))
*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) + (2*C*Sqrt[d]*ArcTanh[(Sqrt[f]*Sqrt
[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^3*Sqrt[f]) - ((8*a^4*C*d^2*f^2 - 4*a^3*b*C*d*f*(5*d*e + 3*c*f) + 3*a^2
*b^2*C*(5*d^2*e^2 + 10*c*d*e*f + c^2*f^2) - a*b^3*(d^2*e*(3*B*e - 4*A*f) + c^2*f*(8*C*e - B*f) + 2*c*d*(12*C*e
^2 - B*e*f + 2*A*f^2)) - b^4*(A*d^2*e^2 - 2*c*d*e*(2*B*e - A*f) - c^2*(8*C*e^2 - 4*B*e*f + 3*A*f^2)))*ArcTanh[
(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(4*b^3*(b*c - a*d)^(3/2)*(b*e - a*f)^(5/2))

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 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 149

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

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 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 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]

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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\int \frac {\sqrt {c+d x} \left (-\frac {a^2 C (3 d e+c f)+b^2 (4 B c e-A d e-3 A c f)-a b (4 c C e+3 B d e+B c f-4 A d f)}{2 b}-\frac {2 C (b c-a d) (b e-a f) x}{b}\right )}{(a+b x)^2 \sqrt {e+f x}} \, dx}{2 (b c-a d) (b e-a f)}\\ &=\frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\int \frac {\frac {4 a^3 C d f (d e+c f)-a^2 b C \left (7 d^2 e^2+14 c d e f+3 c^2 f^2\right )+a b^2 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (8 C e^2-B e f+2 A f^2\right )\right )+b^3 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )}{4 b}-\frac {2 C d (b c-a d) (b e-a f)^2 x}{b}}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 b (b c-a d) (b e-a f)^2}\\ &=\frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {(C d) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{b^3}+\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{8 b^3 (b c-a d) (b e-a f)^2}\\ &=\frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {(2 C) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b^3}+\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\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 )}{4 b^3 (b c-a d) (b e-a f)^2}\\ &=\frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}+\frac {(2 C) \operatorname {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{b^3}\\ &=\frac {\left (4 a^3 C d f-a^2 b C (7 d e+5 c f)-b^3 (4 B c e-A d e-3 A c f)+a b^2 (8 c C e+3 B d e+B c f-4 A d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b^2 (b c-a d) (b e-a f)^2 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {2 C \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{b^3 \sqrt {f}}-\frac {\left (8 a^4 C d^2 f^2-4 a^3 b C d f (5 d e+3 c f)+3 a^2 b^2 C \left (5 d^2 e^2+10 c d e f+c^2 f^2\right )-a b^3 \left (d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)+2 c d \left (12 C e^2-B e f+2 A f^2\right )\right )-b^4 \left (A d^2 e^2-2 c d e (2 B e-A f)-c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 5.67, size = 523, normalized size = 1.08 \[ -\frac {\frac {2 b^2 (c+d x)^{3/2} \sqrt {e+f x} \left (a (a C-b B)+A b^2\right )}{(a+b x)^2 (b c-a d) (b e-a f)}+\frac {b \left (a (a C-b B)+A b^2\right ) (-4 a d f+3 b c f+b d e) \left (\sqrt {c+d x} \sqrt {e+f x} \sqrt {a d-b c} \sqrt {a f-b e}-(a+b x) (d e-c f) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )\right )}{(a+b x) (a d-b c)^{3/2} (a f-b e)^{5/2}}+\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (b B-2 a C)}{(a+b x) (b e-a f)}-\frac {4 b (b B-2 a C) (c f-d e) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{\sqrt {a d-b c} (a f-b e)^{3/2}}+\frac {8 C \sqrt {a d-b c} \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{\sqrt {a f-b e}}-\frac {8 C \sqrt {d e-c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sinh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} \sqrt {e+f x}}}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^3*Sqrt[e + f*x]),x]

[Out]

-1/4*((4*b*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) + (2*b^2*(A*b^2 + a*(-(b*B) + a*
C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (8*C*Sqrt[d*e - c*f]*Sqrt[(d*(e + f
*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[e + f*x]) + (8*C*Sqrt[-(b*c)
+ a*d]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/Sqrt[-(b*e) + a*f] - (
4*b*(b*B - 2*a*C)*(-(d*e) + c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])
])/(Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2)) + (b*(A*b^2 + a*(-(b*B) + a*C))*(b*d*e + 3*b*c*f - 4*a*d*f)*(Sqrt
[-(b*c) + a*d]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqrt[e + f*x] - (d*e - c*f)*(a + b*x)*ArcTanh[(Sqrt[-(b*e) + a
*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])]))/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(5/2)*(a + b*x))
)/b^3

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 134.87, size = 8004, normalized size = 16.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

-1/4*(3*sqrt(d*f)*C*a^2*b^2*c^2*d^2*f^2 + sqrt(d*f)*B*a*b^3*c^2*d^2*f^2 + 3*sqrt(d*f)*A*b^4*c^2*d^2*f^2 - 12*s
qrt(d*f)*C*a^3*b*c*d^3*f^2 - 4*sqrt(d*f)*A*a*b^3*c*d^3*f^2 + 8*sqrt(d*f)*C*a^4*d^4*f^2 - 8*sqrt(d*f)*C*a*b^3*c
^2*d^2*f*e - 4*sqrt(d*f)*B*b^4*c^2*d^2*f*e + 30*sqrt(d*f)*C*a^2*b^2*c*d^3*f*e + 2*sqrt(d*f)*B*a*b^3*c*d^3*f*e
- 2*sqrt(d*f)*A*b^4*c*d^3*f*e - 20*sqrt(d*f)*C*a^3*b*d^4*f*e + 4*sqrt(d*f)*A*a*b^3*d^4*f*e + 8*sqrt(d*f)*C*b^4
*c^2*d^2*e^2 - 24*sqrt(d*f)*C*a*b^3*c*d^3*e^2 + 4*sqrt(d*f)*B*b^4*c*d^3*e^2 + 15*sqrt(d*f)*C*a^2*b^2*d^4*e^2 -
3*sqrt(d*f)*B*a*b^3*d^4*e^2 - sqrt(d*f)*A*b^4*d^4*e^2)*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))/((a^2*b^4*c*f^2*abs(d) - a^3*b^3*d*f^2*abs(d) - 2*a*b^5*c*f*abs(d)*e + 2*a^2*b^4*d*f*abs(d)*e +
b^6*c*abs(d)*e^2 - a*b^5*d*abs(d)*e^2)*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(
d*f)*C*d*log((sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2)/(b^3*f*abs(d)) - 1/2*(5*sqrt(d
*f)*C*a^2*b^3*c^5*d^5*f^5 - sqrt(d*f)*B*a*b^4*c^5*d^5*f^5 - 3*sqrt(d*f)*A*b^5*c^5*d^5*f^5 - 6*sqrt(d*f)*C*a^3*
b^2*c^4*d^6*f^5 + 2*sqrt(d*f)*B*a^2*b^3*c^4*d^6*f^5 + 2*sqrt(d*f)*A*a*b^4*c^4*d^6*f^5 - 8*sqrt(d*f)*C*a*b^4*c^
5*d^5*f^4*e + 4*sqrt(d*f)*B*b^5*c^5*d^5*f^4*e - 11*sqrt(d*f)*C*a^2*b^3*c^4*d^6*f^4*e - sqrt(d*f)*B*a*b^4*c^4*d
^6*f^4*e + 13*sqrt(d*f)*A*b^5*c^4*d^6*f^4*e + 24*sqrt(d*f)*C*a^3*b^2*c^3*d^7*f^4*e - 8*sqrt(d*f)*B*a^2*b^3*c^3
*d^7*f^4*e - 8*sqrt(d*f)*A*a*b^4*c^3*d^7*f^4*e - 15*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^3*c^4*d^4*f^4 + 3*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^4*c^4*d^4*f^4 + 9*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^5*c^4*d^4*f^4 + 44*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*b^2*c
^3*d^5*f^4 - 8*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^3*c^3*d^5*f
^4 - 28*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^4*c^3*d^5*f^4 - 32*s
qrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^4*b*c^2*d^6*f^4 + 8*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^3*b^2*c^2*d^6*f^4 + 16*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^2*b^3*c^2*d^6*f^4 + 32*sqrt(d*f)*C*a*b^4*c^4*d^6
*f^3*e^2 - 16*sqrt(d*f)*B*b^5*c^4*d^6*f^3*e^2 - 6*sqrt(d*f)*C*a^2*b^3*c^3*d^7*f^3*e^2 + 14*sqrt(d*f)*B*a*b^4*c
^3*d^7*f^3*e^2 - 22*sqrt(d*f)*A*b^5*c^3*d^7*f^3*e^2 - 36*sqrt(d*f)*C*a^3*b^2*c^2*d^8*f^3*e^2 + 12*sqrt(d*f)*B*
a^2*b^3*c^2*d^8*f^3*e^2 + 12*sqrt(d*f)*A*a*b^4*c^2*d^8*f^3*e^2 + 24*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*b^4*c^4*d^4*f^3*e - 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x +
c)*d*f - c*d*f + d^2*e))^2*B*b^5*c^4*d^4*f^3*e - 56*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^3*c^3*d^5*f^3*e + 32*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^4*c^3*d^5*f^3*e - 8*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^5*c^3*d^5*f^3*e - 20*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*b^2*c^2*d^6*f^3*e - 16*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^3
*c^2*d^6*f^3*e + 52*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^4*c^2*d^
6*f^3*e + 64*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^4*b*c*d^7*f^3*e -
16*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^3*b^2*c*d^7*f^3*e - 32*sqr
t(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*a^2*b^3*c*d^7*f^3*e + 15*sqrt(d*f)*
(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^2*b^3*c^3*d^3*f^3 - 3*sqrt(d*f)*(sqrt(d*
f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*c^3*d^3*f^3 - 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*
x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*c^3*d^3*f^3 - 58*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sq
rt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^3*b^2*c^2*d^4*f^3 + 14*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x
+ c)*d*f - c*d*f + d^2*e))^4*B*a^2*b^3*c^2*d^4*f^3 + 30*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d
*f - c*d*f + d^2*e))^4*A*a*b^4*c^2*d^4*f^3 + 88*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*
f + d^2*e))^4*C*a^4*b*c*d^5*f^3 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))
^4*B*a^3*b^2*c*d^5*f^3 - 40*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a^2*
b^3*c*d^5*f^3 - 48*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^5*d^6*f^3 +
16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^4*b*d^6*f^3 + 16*sqrt(d*f)
*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a^3*b^2*d^6*f^3 - 48*sqrt(d*f)*C*a*b^4*c^
3*d^7*f^2*e^3 + 24*sqrt(d*f)*B*b^5*c^3*d^7*f^2*e^3 + 34*sqrt(d*f)*C*a^2*b^3*c^2*d^8*f^2*e^3 - 26*sqrt(d*f)*B*a
*b^4*c^2*d^8*f^2*e^3 + 18*sqrt(d*f)*A*b^5*c^2*d^8*f^2*e^3 + 24*sqrt(d*f)*C*a^3*b^2*c*d^9*f^2*e^3 - 8*sqrt(d*f)
*B*a^2*b^3*c*d^9*f^2*e^3 - 8*sqrt(d*f)*A*a*b^4*c*d^9*f^2*e^3 - 24*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*b^4*c^3*d^5*f^2*e^2 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x +
c)*d*f - c*d*f + d^2*e))^2*B*b^5*c^3*d^5*f^2*e^2 + 130*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^3*c^2*d^6*f^2*e^2 - 58*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^4*c^2*d^6*f^2*e^2 - 14*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^5*c^2*d^6*f^2*e^2 - 92*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*b^2*c*d^7*f^2*e^2 + 56*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^3*c*d^7*f^2*e^2 - 20*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^4*c*d^7*f^2*e^2 - 32*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^4*b*d^8
*f^2*e^2 + 8*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^3*b^2*d^8*f^2*e^2
+ 16*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^2*b^3*d^8*f^2*e^2 - 24*s
qrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a*b^4*c^3*d^3*f^2*e + 12*sqrt(d*f
)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*b^5*c^3*d^3*f^2*e + 101*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^2*b^3*c^2*d^4*f^2*e - 49*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*c^2*d^4*f^2*e - 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x
+ c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*c^2*d^4*f^2*e - 188*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^3*b^2*c*d^5*f^2*e + 84*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d
*x + c)*d*f - c*d*f + d^2*e))^4*B*a^2*b^3*c*d^5*f^2*e + 20*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)
*d*f - c*d*f + d^2*e))^4*A*a*b^4*c*d^5*f^2*e + 120*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c
*d*f + d^2*e))^4*C*a^4*b*d^6*f^2*e - 56*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*
e))^4*B*a^3*b^2*d^6*f^2*e - 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a^
2*b^3*d^6*f^2*e - 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^2*b^3*c^2*
d^2*f^2 + sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a*b^4*c^2*d^2*f^2 + 3*
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*b^5*c^2*d^2*f^2 + 20*sqrt(d*f)*(
sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^3*b^2*c*d^3*f^2 - 8*sqrt(d*f)*(sqrt(d*f)*
sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a^2*b^3*c*d^3*f^2 - 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*a*b^4*c*d^3*f^2 - 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt(
(d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^4*b*d^4*f^2 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f
- c*d*f + d^2*e))^6*B*a^3*b^2*d^4*f^2 + 32*sqrt(d*f)*C*a*b^4*c^2*d^8*f*e^4 - 16*sqrt(d*f)*B*b^5*c^2*d^8*f*e^4
- 31*sqrt(d*f)*C*a^2*b^3*c*d^9*f*e^4 + 19*sqrt(d*f)*B*a*b^4*c*d^9*f*e^4 - 7*sqrt(d*f)*A*b^5*c*d^9*f*e^4 - 6*sq
rt(d*f)*C*a^3*b^2*d^10*f*e^4 + 2*sqrt(d*f)*B*a^2*b^3*d^10*f*e^4 + 2*sqrt(d*f)*A*a*b^4*d^10*f*e^4 - 24*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*b^4*c^2*d^6*f*e^3 + 12*sqrt(d*f)*(sqrt
(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*b^5*c^2*d^6*f*e^3 - 32*sqrt(d*f)*(sqrt(d*f)*sqr
t(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b^3*c*d^7*f*e^3 + 8*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^4*c*d^7*f*e^3 + 16*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^5*c*d^7*f*e^3 + 68*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*b^2*d^8*f*e^3 - 32*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^3*d^8*f*e^3 - 4*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^4*d^8*f*e^3 - 16*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a*b^
4*c^2*d^4*f*e^2 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*b^5*c^2*d^4*
f*e^2 + 97*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^2*b^3*c*d^5*f*e^2 -
45*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*c*d^5*f*e^2 - 7*sqrt(d
*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*c*d^5*f*e^2 - 90*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*C*a^3*b^2*d^6*f*e^2 + 46*sqrt(d*f)*(sqrt(d*f)*sqrt(
d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a^2*b^3*d^6*f*e^2 - 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) -
sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*a*b^4*d^6*f*e^2 + 8*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x +
c)*d*f - c*d*f + d^2*e))^6*C*a*b^4*c^2*d^2*f*e - 4*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f -
c*d*f + d^2*e))^6*B*b^5*c^2*d^2*f*e - 34*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2
*e))^6*C*a^2*b^3*c*d^3*f*e + 18*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*
a*b^4*c*d^3*f*e - 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*b^5*c*d^3*f*
e + 28*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^3*b^2*d^4*f*e - 16*sqrt
(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*a^2*b^3*d^4*f*e + 4*sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*A*a*b^4*d^4*f*e - 8*sqrt(d*f)*C*a*b^4*c*d^9*e^5 +
4*sqrt(d*f)*B*b^5*c*d^9*e^5 + 9*sqrt(d*f)*C*a^2*b^3*d^10*e^5 - 5*sqrt(d*f)*B*a*b^4*d^10*e^5 + sqrt(d*f)*A*b^5*
d^10*e^5 + 24*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*b^4*c*d^7*e^4 -
12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*b^5*c*d^7*e^4 - 27*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^3*d^8*e^4 + 15*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^4*d^8*e^4 - 3*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^5*d^8*e^4 - 24*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c
)*d*f - c*d*f + d^2*e))^4*C*a*b^4*c*d^5*e^3 + 12*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d
*f + d^2*e))^4*B*b^5*c*d^5*e^3 + 27*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^
4*C*a^2*b^3*d^6*e^3 - 15*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*B*a*b^4*d
^6*e^3 + 3*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*A*b^5*d^6*e^3 + 8*sqrt(
d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a*b^4*c*d^3*e^2 - 4*sqrt(d*f)*(sqrt(d
*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*B*b^5*c*d^3*e^2 - 9*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^6*C*a^2*b^3*d^4*e^2 + 5*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((
d*x + c)*d*f - c*d*f + d^2*e))^6*B*a*b^4*d^4*e^2 - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c
*d*f + d^2*e))^6*A*b^5*d^4*e^2)/((a^2*b^4*c*f^2*abs(d) - a^3*b^3*d*f^2*abs(d) - 2*a*b^5*c*f*abs(d)*e + 2*a^2*b
^4*d*f*abs(d)*e + b^6*c*abs(d)*e^2 - a*b^5*d*abs(d)*e^2)*(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)^2)

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maple [B]  time = 0.10, size = 9100, normalized size = 18.80 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(1/2),x)

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^3/(f*x+e)^(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(((-(2*a*d*f)/b^2)>0)', see `as
sume?` for more details)Is ((-(2*a*d*f)/b^2)    +(c*f)/b    +(d*e)/b)    ^2    -(4*d*f       *((a^2*d*f)/b^2
-(a*c*f)/b        -(a*d*e)/b        +c*e))     /b^2 zero or nonzero?

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^3),x)

[Out]

\text{Hanged}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**3/(f*x+e)**(1/2),x)

[Out]

Timed out

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