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

Optimal. Leaf size=450 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )-\left (b^3 \left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{8 b^3 d^2 f^2}-\frac {2 \sqrt {b c-a d} \sqrt {b e-a f} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right )}{b^4}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f} \]

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Rubi [A]  time = 1.37, antiderivative size = 453, normalized size of antiderivative = 1.01, 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 = {1615, 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 (-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)+16 a^3 C d^3 f^3-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )+b^3 \left (-\left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (\frac {(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e))}{b d f}-4 a C (c f+d e)+8 A b d f\right )}{8 b^2 d f}-\frac {2 \sqrt {b c-a d} \sqrt {b e-a f} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right )}{b^4}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d 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),x]

[Out]

((8*A*b*d*f - 4*a*C*(d*e + c*f) + ((b*d*e - b*c*f + 4*a*d*f)*(2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f)))/(b*d*f
))*Sqrt[c + d*x]*Sqrt[e + f*x])/(8*b^2*d*f) - ((2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f))*Sqrt[c + d*x]*(e + f*
x)^(3/2))/(4*b^2*d*f^2) + (C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) - ((16*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2
*(C*d*e + c*C*f + 2*B*d*f) - 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)) - b^3*(C*(d*e - c
*f)^2*(d*e + c*f) - 2*d*f*(B*(d*e - c*f)^2 - 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*S
qrt[e + f*x])])/(8*b^4*d^(5/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*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^4

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 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && 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} \, dx &=\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (\frac {3}{2} b (2 A b d f-a C (d e+c f))-\frac {3}{2} b (2 a C d f+b (C d e+c C f-2 B d f)) x\right )}{a+b x} \, dx}{3 b^2 d f}\\ &=-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\int \frac {\sqrt {e+f x} \left (\frac {3}{4} b (4 b c f (2 A b d f-a C (d e+c f))+a (d e+3 c f) (2 a C d f+b (C d e+c C f-2 B d f)))+\frac {3}{4} b (4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) x\right )}{(a+b x) \sqrt {c+d x}} \, dx}{6 b^3 d f^2}\\ &=\frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\int \frac {\frac {3}{8} b \left (16 A b^3 c d^2 e f^2-8 a^3 C d^2 f^2 (d e+c f)+2 a^2 b d f \left (4 B d f (d e+c f)+C \left (d^2 e^2+6 c d e f+c^2 f^2\right )\right )+a b^2 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (4 A d f (d e+c f)+B \left (d^2 e^2+6 c d e f+c^2 f^2\right )\right )\right )\right )-\frac {3}{8} b \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 b^4 d^2 f^2}\\ &=\frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\left (\left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a f)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{b^4}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{16 b^4 d^2 f^2}\\ &=\frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac {\left (2 \left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a 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 (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\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 )}{8 b^4 d^3 f^2}\\ &=\frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {b c-a d} \sqrt {b e-a f} \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}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\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 )}{8 b^4 d^3 f^2}\\ &=\frac {(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt {c+d x} \sqrt {e+f x}}{8 b^3 d^2 f^2}-\frac {(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt {c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac {C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac {\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{8 b^4 d^{5/2} f^{5/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {b c-a d} \sqrt {b e-a f} \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}\\ \end {align*}

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Mathematica [B]  time = 6.21, size = 1936, normalized size = 4.30

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(A*b^2 - a*b*B + a^2*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*(d*e - c*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))))^(5/2)*((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))))^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))))^(-1))/2 + (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)))])))/(32*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))))^2)))/(3*b*d^2*f*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)
*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*C*e) + b*B*f - a*C*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*b^2*d*f*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)]) - ((A*b^2 - a*b*B + a^2*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*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

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IntegrateAlgebraic [B]  time = 1.79, size = 950, normalized size = 2.11 \begin {gather*} \frac {(d e-c f) \sqrt {e+f x} \left (-\frac {3 b^2 C e^2 (e+f x)^2 d^4}{(c+d x)^2}+\frac {24 A b^2 f^2 (e+f x)^2 d^4}{(c+d x)^2}-\frac {24 a b B f^2 (e+f x)^2 d^4}{(c+d x)^2}+\frac {24 a^2 C f^2 (e+f x)^2 d^4}{(c+d x)^2}+\frac {6 b^2 B e f (e+f x)^2 d^4}{(c+d x)^2}-\frac {6 a b C e f (e+f x)^2 d^4}{(c+d x)^2}-\frac {6 b^2 B c f^2 (e+f x)^2 d^3}{(c+d x)^2}+\frac {6 a b c C f^2 (e+f x)^2 d^3}{(c+d x)^2}-\frac {48 A b^2 f^3 (e+f x) d^3}{c+d x}+\frac {48 a b B f^3 (e+f x) d^3}{c+d x}-\frac {48 a^2 C f^3 (e+f x) d^3}{c+d x}+\frac {8 b^2 C e^2 f (e+f x) d^3}{c+d x}+24 A b^2 f^4 d^2-24 a b B f^4 d^2+24 a^2 C f^4 d^2-6 b^2 B e f^3 d^2+6 a b C e f^3 d^2+3 b^2 C e^2 f^2 d^2+\frac {3 b^2 c^2 C f^2 (e+f x)^2 d^2}{(c+d x)^2}-\frac {16 b^2 c C e f^2 (e+f x) d^2}{c+d x}+6 b^2 B c f^4 d-6 a b c C f^4 d+\frac {8 b^2 c^2 C f^3 (e+f x) d}{c+d x}-3 b^2 c^2 C f^4\right )}{24 b^3 d^2 f^2 \sqrt {c+d x} \left (\frac {d (e+f x)}{c+d x}-f\right )^3}+\frac {2 \left (C a^2-b B a+A b^2\right ) \sqrt {b c-a d} \sqrt {a f-b e} \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}+\frac {\left (C d^3 e^3 b^3+8 A c d^2 f^3 b^3+c^3 C f^3 b^3-2 B c^2 d f^3 b^3+8 A d^3 e f^2 b^3+4 B c d^2 e f^2 b^3-c^2 C d e f^2 b^3-2 B d^3 e^2 f b^3-c C d^2 e^2 f b^3-16 a A d^3 f^3 b^2-8 a B c d^2 f^3 b^2+2 a c^2 C d f^3 b^2-8 a B d^3 e f^2 b^2-4 a c C d^2 e f^2 b^2+2 a C d^3 e^2 f b^2+16 a^2 B d^3 f^3 b+8 a^2 c C d^2 f^3 b+8 a^2 C d^3 e f^2 b-16 a^3 C d^3 f^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{8 b^4 d^{5/2} f^{5/2}} \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),x]

[Out]

((d*e - c*f)*Sqrt[e + f*x]*(3*b^2*C*d^2*e^2*f^2 - 6*b^2*B*d^2*e*f^3 + 6*a*b*C*d^2*e*f^3 - 3*b^2*c^2*C*f^4 + 6*
b^2*B*c*d*f^4 - 6*a*b*c*C*d*f^4 + 24*A*b^2*d^2*f^4 - 24*a*b*B*d^2*f^4 + 24*a^2*C*d^2*f^4 + (8*b^2*C*d^3*e^2*f*
(e + f*x))/(c + d*x) - (16*b^2*c*C*d^2*e*f^2*(e + f*x))/(c + d*x) + (8*b^2*c^2*C*d*f^3*(e + f*x))/(c + d*x) -
(48*A*b^2*d^3*f^3*(e + f*x))/(c + d*x) + (48*a*b*B*d^3*f^3*(e + f*x))/(c + d*x) - (48*a^2*C*d^3*f^3*(e + f*x))
/(c + d*x) - (3*b^2*C*d^4*e^2*(e + f*x)^2)/(c + d*x)^2 + (6*b^2*B*d^4*e*f*(e + f*x)^2)/(c + d*x)^2 - (6*a*b*C*
d^4*e*f*(e + f*x)^2)/(c + d*x)^2 + (3*b^2*c^2*C*d^2*f^2*(e + f*x)^2)/(c + d*x)^2 - (6*b^2*B*c*d^3*f^2*(e + f*x
)^2)/(c + d*x)^2 + (6*a*b*c*C*d^3*f^2*(e + f*x)^2)/(c + d*x)^2 + (24*A*b^2*d^4*f^2*(e + f*x)^2)/(c + d*x)^2 -
(24*a*b*B*d^4*f^2*(e + f*x)^2)/(c + d*x)^2 + (24*a^2*C*d^4*f^2*(e + f*x)^2)/(c + d*x)^2))/(24*b^3*d^2*f^2*Sqrt
[c + d*x]*(-f + (d*(e + f*x))/(c + d*x))^3) + (2*(A*b^2 - a*b*B + a^2*C)*Sqrt[b*c - a*d]*Sqrt[-(b*e) + a*f]*Ar
cTan[(Sqrt[b*c - a*d]*Sqrt[-(b*e) + a*f]*Sqrt[e + f*x])/((b*e - a*f)*Sqrt[c + d*x])])/b^4 + ((b^3*C*d^3*e^3 -
b^3*c*C*d^2*e^2*f - 2*b^3*B*d^3*e^2*f + 2*a*b^2*C*d^3*e^2*f - b^3*c^2*C*d*e*f^2 + 4*b^3*B*c*d^2*e*f^2 - 4*a*b^
2*c*C*d^2*e*f^2 + 8*A*b^3*d^3*e*f^2 - 8*a*b^2*B*d^3*e*f^2 + 8*a^2*b*C*d^3*e*f^2 + b^3*c^3*C*f^3 - 2*b^3*B*c^2*
d*f^3 + 2*a*b^2*c^2*C*d*f^3 + 8*A*b^3*c*d^2*f^3 - 8*a*b^2*B*c*d^2*f^3 + 8*a^2*b*c*C*d^2*f^3 - 16*a*A*b^2*d^3*f
^3 + 16*a^2*b*B*d^3*f^3 - 16*a^3*C*d^3*f^3)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(8*b^4*d
^(5/2)*f^(5/2))

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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),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.1

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maple [B]  time = 0.05, size = 4227, normalized size = 9.39 \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),x)

[Out]

-1/48*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(48*C*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e
+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^4*d^3*f^3-3*C*ln(1/2*(2
*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)*b^4*c^3*f^3-3*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))
*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*d^3*e^3+48*B*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(
(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a
^2*b^2*c*d^2*f^3-48*A*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/
2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*c*d^2*f^3-48*A*(d*f)^(1/2)*ln((-2*a*d
*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a
*d*e+2*b*c*e)/(b*x+a))*a*b^3*d^3*e*f^2+48*B*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b
*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*d^3*e*f^2-24*
B*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^4*d^2*f^2+24*B
*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e
+b^2*c*e)/b^2)^(1/2)*a*b^3*c*d^2*f^3+24*B*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2
))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*d^3*e*f^2-16*C*x^2*b^4*d^2*f^2*(d*f*x^2+c*
f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)-24*C*ln(1/2*(2*d*f*x+c*f+d*e+2*
(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a^2*b^
2*d^3*e*f^2-6*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-
a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*c^2*d*f^3+48*B*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(
1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*b^3*d^2*f^2-12*B*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1
/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*c*d*f^2-12*B*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*d^2*e*f-48*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d
*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a^2*b^2*d^2*f^2-6*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d
*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*d^3*e^2*f+3*C*ln(1/2*(2*d*f*x+c*f+
d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*
b^4*c^2*d*e*f^2+3*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*
d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c*d^2*e^2*f-12*B*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*
e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c*d^2*e*f^2-48*C*(d*f)^(1
/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(
1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^3*b*c*d^2*f^3-48*C*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^3*b*d
^3*e*f^2-24*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*
b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a^2*b^2*c*d^2*f^3+48*A*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*
f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*
d^3*f^3+48*A*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b
*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*d^3*f^3-24*A*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*
(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c*d^2*f^3-24*A*ln(1/2*(2*d*f*x+c*f
+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
*b^4*d^3*e*f^2-48*B*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^3*b*d^3*f^3-48*B*ln(1/2*(2*d*f*x+c*f+d*e+2*
(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a^2*b^
2*d^3*f^3+6*B*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*
b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c^2*d*f^3+6*B*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*
(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*d^3*e^2*f+48*A*(d*f)^(1/2)*ln((-2*
a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*
f-a*d*e+2*b*c*e)/(b*x+a))*b^4*c*d^2*e*f^2+48*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)
^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a^3*b*d^3*f^3-48*A*(d*f)^(1/2)*((a^2*d*f-a*
b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*d^2*f^2+6*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*
f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*c^2*f^2+6*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*
b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*d^2*e^2-4*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*
e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*c*d*e*f-4*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^
2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^4*c*d*f^2-4*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^4*d^2*e*f+12*C*ln(1/2*(2*d*f*x+c*f+d*e+2*(d*f*x^2+c*f*x+d*
e*x+c*e)^(1/2)*(d*f)^(1/2))/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*c*d^2*e*f^2+12*C*
(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*b^3*c*d*f^2+12*C*(
d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*b^3*d^2*e*f+48*C*(d
*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+
c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*c*d^2*e*f^2+24*C*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*a*b^3*d^2*f^2-48*B*(d*f)^(1/2)*ln((-2*a*d*f*x+b*c*f*x+b*d*e*
x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+
a))*a*b^3*c*d^2*e*f^2)/(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)/b^5/d^2/f^2/(d*f)^(1/2)/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c
*e)/b^2)^(1/2)

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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),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),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )}{a + b x}\, dx \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),x)

[Out]

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

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