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

Optimal. Leaf size=540 \[ -\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f} \]

[Out]

1/64*(-c*f+d*e)*(8*a*d*f*(2*d*f*(-4*A*d*f+B*c*f+3*B*d*e)-C*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))+b*(C*(5*c^3*f^3+9*c^
2*d*e*f^2+15*c*d^2*e^2*f+35*d^3*e^3)+8*d*f*(2*A*d*f*(c*f+3*d*e)-B*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))))*arctanh(f^(
1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2))/d^(7/2)/f^(9/2)+1/4*C*(b*x+a)^2*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/d/f-1/
96*(d*x+c)^(3/2)*(24*a^2*C*d^2*f^2+8*a*b*d*f*(-6*B*d*f+3*C*c*f+5*C*d*e)+b^2*(8*d*f*(-6*A*d*f+3*B*c*f+5*B*d*e)-
C*(15*c^2*f^2+22*c*d*e*f+35*d^2*e^2))+4*b*d*f*(4*a*C*d*f+b*(-8*B*d*f+5*C*c*f+7*C*d*e))*x)*(f*x+e)^(1/2)/b/d^3/
f^3-1/64*(8*a*d*f*(2*d*f*(-4*A*d*f+B*c*f+3*B*d*e)-C*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))+b*(C*(5*c^3*f^3+9*c^2*d*e*f
^2+15*c*d^2*e^2*f+35*d^3*e^3)+8*d*f*(2*A*d*f*(c*f+3*d*e)-B*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))))*(d*x+c)^(1/2)*(f*x
+e)^(1/2)/d^3/f^4

________________________________________________________________________________________

Rubi [A]  time = 0.71, antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1615, 147, 50, 63, 217, 206} \[ -\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+5 c^3 f^3+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+5 c^3 f^3+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((8*a*d*f*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + b*(C*(35*d^3*e^3 + 15*c
*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)))
)*Sqrt[c + d*x]*Sqrt[e + f*x])/(64*d^3*f^4) + (C*(a + b*x)^2*(c + d*x)^(3/2)*Sqrt[e + f*x])/(4*b*d*f) - ((c +
d*x)^(3/2)*Sqrt[e + f*x]*(24*a^2*C*d^2*f^2 + 8*a*b*d*f*(5*C*d*e + 3*c*C*f - 6*B*d*f) + b^2*(8*d*f*(5*B*d*e + 3
*B*c*f - 6*A*d*f) - C*(35*d^2*e^2 + 22*c*d*e*f + 15*c^2*f^2)) + 4*b*d*f*(4*a*C*d*f + b*(7*C*d*e + 5*c*C*f - 8*
B*d*f))*x))/(96*b*d^3*f^3) + ((d*e - c*f)*(8*a*d*f*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e
*f + c^2*f^2)) + b*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f)
- B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(64*d^(7/2
)*f^(9/2))

Rule 50

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

Rule 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 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 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 {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx &=\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}+\frac {\int \frac {(a+b x) \sqrt {c+d x} \left (-\frac {1}{2} b (4 b c C e+3 a C d e+a c C f-8 A b d f)-\frac {1}{2} b (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{\sqrt {e+f x}} \, dx}{4 b^2 d f}\\ &=\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}} \, dx}{64 d^3 f^3}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {\left ((d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{128 d^3 f^4}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {\left ((d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\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 )}{64 d^4 f^4}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {\left ((d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\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 )}{64 d^4 f^4}\\ &=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {(d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{7/2} f^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 3.54, size = 478, normalized size = 0.89 \[ \frac {d \sqrt {f} \sqrt {c+d x} (e+f x) \left (8 a d f \left (6 d f (4 A d f+B (c f-3 d e+2 d f x))+C \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+b \left (8 d f \left (6 A d f (c f-3 d e+2 d f x)+B \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+C \left (15 c^3 f^3+c^2 d f^2 (17 e-10 f x)+c d^2 f \left (25 e^2-12 e f x+8 f^2 x^2\right )+d^3 \left (-105 e^3+70 e^2 f x-56 e f^2 x^2+48 f^3 x^3\right )\right )\right )\right )+3 (d e-c f)^{3/2} \sqrt {\frac {d (e+f x)}{d e-c f}} \sinh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right ) \left (b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-8 a d f \left (2 d f (4 A d f-B c f-3 B d e)+C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )}{192 d^4 f^{9/2} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(8*a*d*f*(6*d*f*(4*A*d*f + B*(-3*d*e + c*f + 2*d*f*x)) + C*(-3*c^2*f^2 + 2*
c*d*f*(-2*e + f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2))) + b*(C*(15*c^3*f^3 + c^2*d*f^2*(17*e - 10*f*x) + c*
d^2*f*(25*e^2 - 12*e*f*x + 8*f^2*x^2) + d^3*(-105*e^3 + 70*e^2*f*x - 56*e*f^2*x^2 + 48*f^3*x^3)) + 8*d*f*(6*A*
d*f*(-3*d*e + c*f + 2*d*f*x) + B*(-3*c^2*f^2 + 2*c*d*f*(-2*e + f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2)))))
+ 3*(d*e - c*f)^(3/2)*(-8*a*d*f*(2*d*f*(-3*B*d*e - B*c*f + 4*A*d*f) + C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + b
*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 +
2*c*d*e*f + c^2*f^2))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(192
*d^4*f^(9/2)*Sqrt[e + f*x])

________________________________________________________________________________________

fricas [A]  time = 3.38, size = 1114, normalized size = 2.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(3*(35*C*b*d^4*e^4 - 20*(C*b*c*d^3 + 2*(C*a + B*b)*d^4)*e^3*f - 6*(C*b*c^2*d^2 - 4*(C*a + B*b)*c*d^3 -
8*(B*a + A*b)*d^4)*e^2*f^2 - 4*(C*b*c^3*d + 16*A*a*d^4 - 2*(C*a + B*b)*c^2*d^2 + 8*(B*a + A*b)*c*d^3)*e*f^3 -
(5*C*b*c^4 - 64*A*a*c*d^3 - 8*(C*a + B*b)*c^3*d + 16*(B*a + A*b)*c^2*d^2)*f^4)*sqrt(d*f)*log(8*d^2*f^2*x^2 + d
^2*e^2 + 6*c*d*e*f + c^2*f^2 + 4*(2*d*f*x + d*e + c*f)*sqrt(d*f)*sqrt(d*x + c)*sqrt(f*x + e) + 8*(d^2*e*f + c*
d*f^2)*x) + 4*(48*C*b*d^4*f^4*x^3 - 105*C*b*d^4*e^3*f + 5*(5*C*b*c*d^3 + 24*(C*a + B*b)*d^4)*e^2*f^2 + (17*C*b
*c^2*d^2 - 32*(C*a + B*b)*c*d^3 - 144*(B*a + A*b)*d^4)*e*f^3 + 3*(5*C*b*c^3*d + 64*A*a*d^4 - 8*(C*a + B*b)*c^2
*d^2 + 16*(B*a + A*b)*c*d^3)*f^4 - 8*(7*C*b*d^4*e*f^3 - (C*b*c*d^3 + 8*(C*a + B*b)*d^4)*f^4)*x^2 + 2*(35*C*b*d
^4*e^2*f^2 - 2*(3*C*b*c*d^3 + 20*(C*a + B*b)*d^4)*e*f^3 - (5*C*b*c^2*d^2 - 8*(C*a + B*b)*c*d^3 - 48*(B*a + A*b
)*d^4)*f^4)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^4*f^5), -1/384*(3*(35*C*b*d^4*e^4 - 20*(C*b*c*d^3 + 2*(C*a + B*
b)*d^4)*e^3*f - 6*(C*b*c^2*d^2 - 4*(C*a + B*b)*c*d^3 - 8*(B*a + A*b)*d^4)*e^2*f^2 - 4*(C*b*c^3*d + 16*A*a*d^4
- 2*(C*a + B*b)*c^2*d^2 + 8*(B*a + A*b)*c*d^3)*e*f^3 - (5*C*b*c^4 - 64*A*a*c*d^3 - 8*(C*a + B*b)*c^3*d + 16*(B
*a + A*b)*c^2*d^2)*f^4)*sqrt(-d*f)*arctan(1/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)*sqrt(d*x + c)*sqrt(f*x + e)/(d^
2*f^2*x^2 + c*d*e*f + (d^2*e*f + c*d*f^2)*x)) - 2*(48*C*b*d^4*f^4*x^3 - 105*C*b*d^4*e^3*f + 5*(5*C*b*c*d^3 + 2
4*(C*a + B*b)*d^4)*e^2*f^2 + (17*C*b*c^2*d^2 - 32*(C*a + B*b)*c*d^3 - 144*(B*a + A*b)*d^4)*e*f^3 + 3*(5*C*b*c^
3*d + 64*A*a*d^4 - 8*(C*a + B*b)*c^2*d^2 + 16*(B*a + A*b)*c*d^3)*f^4 - 8*(7*C*b*d^4*e*f^3 - (C*b*c*d^3 + 8*(C*
a + B*b)*d^4)*f^4)*x^2 + 2*(35*C*b*d^4*e^2*f^2 - 2*(3*C*b*c*d^3 + 20*(C*a + B*b)*d^4)*e*f^3 - (5*C*b*c^2*d^2 -
8*(C*a + B*b)*c*d^3 - 48*(B*a + A*b)*d^4)*f^4)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^4*f^5)]

________________________________________________________________________________________

giac [A]  time = 1.82, size = 736, normalized size = 1.36 \[ \frac {{\left (\sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )} C b}{d^{4} f} - \frac {17 \, C b c d^{12} f^{6} - 8 \, C a d^{13} f^{6} - 8 \, B b d^{13} f^{6} + 7 \, C b d^{13} f^{5} e}{d^{16} f^{7}}\right )} + \frac {59 \, C b c^{2} d^{12} f^{6} - 56 \, C a c d^{13} f^{6} - 56 \, B b c d^{13} f^{6} + 48 \, B a d^{14} f^{6} + 48 \, A b d^{14} f^{6} + 50 \, C b c d^{13} f^{5} e - 40 \, C a d^{14} f^{5} e - 40 \, B b d^{14} f^{5} e + 35 \, C b d^{14} f^{4} e^{2}}{d^{16} f^{7}}\right )} - \frac {3 \, {\left (5 \, C b c^{3} d^{12} f^{6} - 8 \, C a c^{2} d^{13} f^{6} - 8 \, B b c^{2} d^{13} f^{6} + 16 \, B a c d^{14} f^{6} + 16 \, A b c d^{14} f^{6} - 64 \, A a d^{15} f^{6} + 9 \, C b c^{2} d^{13} f^{5} e - 16 \, C a c d^{14} f^{5} e - 16 \, B b c d^{14} f^{5} e + 48 \, B a d^{15} f^{5} e + 48 \, A b d^{15} f^{5} e + 15 \, C b c d^{14} f^{4} e^{2} - 40 \, C a d^{15} f^{4} e^{2} - 40 \, B b d^{15} f^{4} e^{2} + 35 \, C b d^{15} f^{3} e^{3}\right )}}{d^{16} f^{7}}\right )} \sqrt {d x + c} + \frac {3 \, {\left (5 \, C b c^{4} f^{4} - 8 \, C a c^{3} d f^{4} - 8 \, B b c^{3} d f^{4} + 16 \, B a c^{2} d^{2} f^{4} + 16 \, A b c^{2} d^{2} f^{4} - 64 \, A a c d^{3} f^{4} + 4 \, C b c^{3} d f^{3} e - 8 \, C a c^{2} d^{2} f^{3} e - 8 \, B b c^{2} d^{2} f^{3} e + 32 \, B a c d^{3} f^{3} e + 32 \, A b c d^{3} f^{3} e + 64 \, A a d^{4} f^{3} e + 6 \, C b c^{2} d^{2} f^{2} e^{2} - 24 \, C a c d^{3} f^{2} e^{2} - 24 \, B b c d^{3} f^{2} e^{2} - 48 \, B a d^{4} f^{2} e^{2} - 48 \, A b d^{4} f^{2} e^{2} + 20 \, C b c d^{3} f e^{3} + 40 \, C a d^{4} f e^{3} + 40 \, B b d^{4} f e^{3} - 35 \, C b d^{4} e^{4}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt {d f} d^{3} f^{4}}\right )} d}{192 \, {\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)*C*b/(d^4*f) - (17*C*b*c*d^12
*f^6 - 8*C*a*d^13*f^6 - 8*B*b*d^13*f^6 + 7*C*b*d^13*f^5*e)/(d^16*f^7)) + (59*C*b*c^2*d^12*f^6 - 56*C*a*c*d^13*
f^6 - 56*B*b*c*d^13*f^6 + 48*B*a*d^14*f^6 + 48*A*b*d^14*f^6 + 50*C*b*c*d^13*f^5*e - 40*C*a*d^14*f^5*e - 40*B*b
*d^14*f^5*e + 35*C*b*d^14*f^4*e^2)/(d^16*f^7)) - 3*(5*C*b*c^3*d^12*f^6 - 8*C*a*c^2*d^13*f^6 - 8*B*b*c^2*d^13*f
^6 + 16*B*a*c*d^14*f^6 + 16*A*b*c*d^14*f^6 - 64*A*a*d^15*f^6 + 9*C*b*c^2*d^13*f^5*e - 16*C*a*c*d^14*f^5*e - 16
*B*b*c*d^14*f^5*e + 48*B*a*d^15*f^5*e + 48*A*b*d^15*f^5*e + 15*C*b*c*d^14*f^4*e^2 - 40*C*a*d^15*f^4*e^2 - 40*B
*b*d^15*f^4*e^2 + 35*C*b*d^15*f^3*e^3)/(d^16*f^7))*sqrt(d*x + c) + 3*(5*C*b*c^4*f^4 - 8*C*a*c^3*d*f^4 - 8*B*b*
c^3*d*f^4 + 16*B*a*c^2*d^2*f^4 + 16*A*b*c^2*d^2*f^4 - 64*A*a*c*d^3*f^4 + 4*C*b*c^3*d*f^3*e - 8*C*a*c^2*d^2*f^3
*e - 8*B*b*c^2*d^2*f^3*e + 32*B*a*c*d^3*f^3*e + 32*A*b*c*d^3*f^3*e + 64*A*a*d^4*f^3*e + 6*C*b*c^2*d^2*f^2*e^2
- 24*C*a*c*d^3*f^2*e^2 - 24*B*b*c*d^3*f^2*e^2 - 48*B*a*d^4*f^2*e^2 - 48*A*b*d^4*f^2*e^2 + 20*C*b*c*d^3*f*e^3 +
40*C*a*d^4*f*e^3 + 40*B*b*d^4*f*e^3 - 35*C*b*d^4*e^4)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f -
c*d*f + d^2*e)))/(sqrt(d*f)*d^3*f^4))*d/abs(d)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 2002, normalized size = 3.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(192*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^
(1/2))*a*c*d^3*f^4+105*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*d^4*e^4
-24*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b*c*d^2*e*f^2-15*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)
^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^4*f^4-96*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*
f)^(1/2))*a*c*d^3*e*f^3+72*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c*d
^3*e^2*f^2+72*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c*d^3*e^2*f^2-60
*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c*d^3*e^3*f-288*A*(d*f)^(1/2)
*((d*x+c)*(f*x+e))^(1/2)*b*d^3*e*f^2-96*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)
^(1/2))*b*c*d^3*e*f^3+24*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^2*d
^2*e*f^3+24*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^2*d^2*e*f^3-12*C
*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^3*d*e*f^3-192*A*ln(1/2*(2*d*f
*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*d^4*e*f^3+144*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f
*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*d^4*e^2*f^2-120*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(
d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*d^4*e^3*f-120*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d
*e)/(d*f)^(1/2))*a*d^4*e^3*f+384*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*d^3*f^3-210*C*(d*f)^(1/2)*((d*x+c)*(f
*x+e))^(1/2)*b*d^3*e^3-48*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^2*
d^2*f^4+24*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^3*d*f^4+24*B*ln(1
/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^3*d*f^4-48*B*ln(1/2*(2*d*f*x+2*((d
*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^2*d^2*f^4+144*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))
^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*d^4*e^2*f^2+30*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c^3*f^3+192*
B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*d^3*f^3+96*A*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c*d^2*f^3+192*A*(
d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b*d^3*f^3+240*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*d^3*e^2*f+96*C*x^3*
b*d^3*f^3*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)+128*B*x^2*b*d^3*f^3*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)+128*C*x^
2*a*d^3*f^3*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)-18*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f
+d*e)/(d*f)^(1/2))*b*c^2*d^2*e^2*f^2-288*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*d^3*e*f^2+240*B*(d*f)^(1/2)*(
(d*x+c)*(f*x+e))^(1/2)*b*d^3*e^2*f-48*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*c^2*d*f^3+96*B*(d*f)^(1/2)*((d*x
+c)*(f*x+e))^(1/2)*a*c*d^2*f^3-48*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c^2*d*f^3-64*C*(d*f)^(1/2)*((d*x+c)*
(f*x+e))^(1/2)*a*c*d^2*e*f^2+34*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c^2*d*e*f^2+50*C*(d*f)^(1/2)*((d*x+c)*
(f*x+e))^(1/2)*b*c*d^2*e^2*f+32*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b*c*d^2*f^3-160*B*(d*f)^(1/2)*((d*x+c)
*(f*x+e))^(1/2)*x*b*d^3*e*f^2+32*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*a*c*d^2*f^3-160*C*(d*f)^(1/2)*((d*x+c
)*(f*x+e))^(1/2)*x*a*d^3*e*f^2-20*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*x*b*c^2*d*f^3+140*C*(d*f)^(1/2)*((d*x+
c)*(f*x+e))^(1/2)*x*b*d^3*e^2*f-64*B*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b*c*d^2*e*f^2+16*C*x^2*b*c*d^2*f^3*(d
*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)-112*C*x^2*b*d^3*e*f^2*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2))/f^4/((d*x+c)*(f*x
+e))^(1/2)/d^3/(d*f)^(1/2)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

[Out]

\text{Hanged}

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________