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

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

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Rubi [A]  time = 1.70, antiderivative size = 521, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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 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 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} \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*}

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Mathematica [B]  time = 6.37, size = 2532, normalized size = 4.86 \begin {gather*} \text {Result too large to show} \end {gather*}

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

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IntegrateAlgebraic [A]  time = 2.76, size = 942, normalized size = 1.81 \begin {gather*} \frac {(d e-c f) \sqrt {e+f x} \left (-\frac {a b C e (e+f x)^2 d^3}{(c+d x)^2}+\frac {4 A b^2 f (e+f x)^2 d^3}{(c+d x)^2}-\frac {8 a b B f (e+f x)^2 d^3}{(c+d x)^2}+\frac {12 a^2 C f (e+f x)^2 d^3}{(c+d x)^2}+\frac {b^2 c C e (e+f x)^2 d^2}{(c+d x)^2}+\frac {4 b^2 B c f (e+f x)^2 d^2}{(c+d x)^2}-\frac {7 a b c C f (e+f x)^2 d^2}{(c+d x)^2}-\frac {b^2 C e^2 (e+f x) d^2}{c+d x}-\frac {8 A b^2 f^2 (e+f x) d^2}{c+d x}+\frac {16 a b B f^2 (e+f x) d^2}{c+d x}-\frac {24 a^2 C f^2 (e+f x) d^2}{c+d x}-\frac {4 b^2 B e f (e+f x) d^2}{c+d x}+\frac {8 a b C e f (e+f x) d^2}{c+d x}+4 A b^2 f^3 d-8 a b B f^3 d+12 a^2 C f^3 d+4 b^2 B e f^2 d-7 a b C e f^2 d-\frac {b^2 c^2 C f (e+f x)^2 d}{(c+d x)^2}-b^2 C e^2 f d-\frac {4 b^2 B c f^2 (e+f x) d}{c+d x}+\frac {8 a b c C f^2 (e+f x) d}{c+d x}+\frac {2 b^2 c C e f (e+f x) d}{c+d x}-a b c C f^3+b^2 c C e f^2-\frac {b^2 c^2 C f^2 (e+f x)}{c+d x}\right )}{4 b^3 d f \sqrt {c+d x} \left (\frac {d (e+f x)}{c+d x}-f\right )^2 \left (-b e+a f+\frac {b c (e+f x)}{c+d x}-\frac {a d (e+f x)}{c+d x}\right )}+\frac {\left (6 C d f a^3-5 b C d e a^2-5 b c C f a^2-4 b B d f a^2+4 b^2 c C e a+3 b^2 B d e a+3 b^2 B c f a+2 A b^2 d f a-2 b^3 B c e-A b^3 d e-A b^3 c f\right ) \tan ^{-1}\left (\frac {\sqrt {b c-a d} \sqrt {a f-b e} \sqrt {e+f x}}{(b e-a f) \sqrt {c+d x}}\right )}{b^4 \sqrt {b c-a d} \sqrt {a f-b e}}+\frac {\left (-C d^2 e^2 b^2+8 A d^2 f^2 b^2-c^2 C f^2 b^2+4 B c d f^2 b^2+4 B d^2 e f b^2+2 c C d e f b^2-16 a B d^2 f^2 b-8 a c C d f^2 b-8 a C d^2 e f b+24 a^2 C d^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{4 b^4 d^{3/2} f^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*e - c*f)*Sqrt[e + f*x]*(-(b^2*C*d*e^2*f) + b^2*c*C*e*f^2 + 4*b^2*B*d*e*f^2 - 7*a*b*C*d*e*f^2 - a*b*c*C*f^3
+ 4*A*b^2*d*f^3 - 8*a*b*B*d*f^3 + 12*a^2*C*d*f^3 - (b^2*C*d^2*e^2*(e + f*x))/(c + d*x) + (2*b^2*c*C*d*e*f*(e
+ f*x))/(c + d*x) - (4*b^2*B*d^2*e*f*(e + f*x))/(c + d*x) + (8*a*b*C*d^2*e*f*(e + f*x))/(c + d*x) - (b^2*c^2*C
*f^2*(e + f*x))/(c + d*x) - (4*b^2*B*c*d*f^2*(e + f*x))/(c + d*x) + (8*a*b*c*C*d*f^2*(e + f*x))/(c + d*x) - (8
*A*b^2*d^2*f^2*(e + f*x))/(c + d*x) + (16*a*b*B*d^2*f^2*(e + f*x))/(c + d*x) - (24*a^2*C*d^2*f^2*(e + f*x))/(c
+ d*x) + (b^2*c*C*d^2*e*(e + f*x)^2)/(c + d*x)^2 - (a*b*C*d^3*e*(e + f*x)^2)/(c + d*x)^2 - (b^2*c^2*C*d*f*(e
+ f*x)^2)/(c + d*x)^2 + (4*b^2*B*c*d^2*f*(e + f*x)^2)/(c + d*x)^2 - (7*a*b*c*C*d^2*f*(e + f*x)^2)/(c + d*x)^2
+ (4*A*b^2*d^3*f*(e + f*x)^2)/(c + d*x)^2 - (8*a*b*B*d^3*f*(e + f*x)^2)/(c + d*x)^2 + (12*a^2*C*d^3*f*(e + f*x
)^2)/(c + d*x)^2))/(4*b^3*d*f*Sqrt[c + d*x]*(-f + (d*(e + f*x))/(c + d*x))^2*(-(b*e) + a*f + (b*c*(e + f*x))/(
c + d*x) - (a*d*(e + f*x))/(c + d*x))) + ((-2*b^3*B*c*e + 4*a*b^2*c*C*e - A*b^3*d*e + 3*a*b^2*B*d*e - 5*a^2*b*
C*d*e - A*b^3*c*f + 3*a*b^2*B*c*f - 5*a^2*b*c*C*f + 2*a*A*b^2*d*f - 4*a^2*b*B*d*f + 6*a^3*C*d*f)*ArcTan[(Sqrt[
b*c - a*d]*Sqrt[-(b*e) + a*f]*Sqrt[e + f*x])/((b*e - a*f)*Sqrt[c + d*x])])/(b^4*Sqrt[b*c - a*d]*Sqrt[-(b*e) +
a*f]) + ((-(b^2*C*d^2*e^2) + 2*b^2*c*C*d*e*f + 4*b^2*B*d^2*e*f - 8*a*b*C*d^2*e*f - b^2*c^2*C*f^2 + 4*b^2*B*c*d
*f^2 - 8*a*b*c*C*d*f^2 + 8*A*b^2*d^2*f^2 - 16*a*b*B*d^2*f^2 + 24*a^2*C*d^2*f^2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x]
)/(Sqrt[f]*Sqrt[c + d*x])])/(4*b^4*d^(3/2)*f^(3/2))

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fricas [F(-1)]  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((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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giac [B]  time = 13.12, size = 1585, normalized size = 3.04

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)*(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)

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maple [B]  time = 0.05, size = 5051, normalized size = 9.69 \begin {gather*} \text {output too large to display} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> 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*c*f>0)', see `assume
?` for more details)Is 2*a*d*f-b*c*f                                        -b*d*e    zero or nonzero?

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

[Out]

\text{Hanged}

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sympy [F(-1)]  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((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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