\(\int \frac {(a+b x) \sqrt {c+d x} (A+B x+C x^2)}{\sqrt {e+f x}} \, dx\) [48]

Optimal result

Integrand size = 34, antiderivative size = 540 \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=-\frac {\left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+8 a b d f (5 C d e+3 c C f-6 B d f)+b^2 \left (8 d f (5 B d e+3 B c f-6 A d f)-C \left (35 d^2 e^2+22 c d e f+15 c^2 f^2\right )\right )+4 b d f (4 a C d f+b (7 C d e+5 c C f-8 B d f)) x\right )}{96 b d^3 f^3}+\frac {(d e-c f) \left (8 a d f \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{7/2} f^{9/2}} \]

[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] (verified)

Time = 0.45 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1629, 152, 52, 65, 223, 212} \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac {C (a+b x)^2 (c+d x)^{3/2} \sqrt {e+f x}}{4 b d f} \]

[In]

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

[Out]

-1/64*((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])/(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 52

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 65

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 152

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 212

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

Rule 223

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 1629

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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] (verified)

Time = 8.63 (sec) , antiderivative size = 474, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\frac {\sqrt {d} \sqrt {f} \sqrt {c+d x} \sqrt {e+f x} \left (8 a d f \left (6 d f (4 A d f+B (-3 d e+c f+2 d f x))+C \left (-3 c^2 f^2+2 c d f (-2 e+f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+b \left (C \left (15 c^3 f^3+c^2 d f^2 (17 e-10 f x)+c d^2 f \left (25 e^2-12 e f x+8 f^2 x^2\right )+d^3 \left (-105 e^3+70 e^2 f x-56 e f^2 x^2+48 f^3 x^3\right )\right )+8 d f \left (6 A d f (-3 d e+c f+2 d f x)+B \left (-3 c^2 f^2+2 c d f (-2 e+f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )\right )-6 (d e-c f) \left (-8 a d f \left (2 d f (-3 B d e-B c f+4 A d f)+C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )+b \left (C \left (35 d^3 e^3+15 c d^2 e^2 f+9 c^2 d e f^2+5 c^3 f^3\right )+8 d f \left (2 A d f (3 d e+c f)-B \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \left (\sqrt {c-\frac {d e}{f}}-\sqrt {c+d x}\right )}\right )}{192 d^{7/2} f^{9/2}} \]

[In]

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

[Out]

(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[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))))) - 6*(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[d]*Sqrt[e + f*x])/(Sqrt[f]*(Sqrt[c - (d*e)/f] - Sqrt[c + d*x]))])/(1
92*d^(7/2)*f^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2001\) vs. \(2(508)=1016\).

Time = 1.67 (sec) , antiderivative size = 2002, normalized size of antiderivative = 3.71

method	result	size
default	\(\text {Expression too large to display}\)	\(2002\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.92 (sec) , antiderivative size = 1114, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Too large to display} \]

[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)]

Sympy [F]

\[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\int \frac {\left (a + b x\right ) \sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\sqrt {e + f x}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]

[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 detail

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\frac {{\left (\sqrt {d^{2} e + {\left (d x + c\right )} d f - c d f} {\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 {7 \, C b d^{13} e f^{5} + 17 \, C b c d^{12} f^{6} - 8 \, C a d^{13} f^{6} - 8 \, B b d^{13} f^{6}}{d^{16} f^{7}}\right )} + \frac {35 \, C b d^{14} e^{2} f^{4} + 50 \, C b c d^{13} e f^{5} - 40 \, C a d^{14} e f^{5} - 40 \, B b d^{14} e f^{5} + 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}}{d^{16} f^{7}}\right )} - \frac {3 \, {\left (35 \, C b d^{15} e^{3} f^{3} + 15 \, C b c d^{14} e^{2} f^{4} - 40 \, C a d^{15} e^{2} f^{4} - 40 \, B b d^{15} e^{2} f^{4} + 9 \, C b c^{2} d^{13} e f^{5} - 16 \, C a c d^{14} e f^{5} - 16 \, B b c d^{14} e f^{5} + 48 \, B a d^{15} e f^{5} + 48 \, A b d^{15} e f^{5} + 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}\right )}}{d^{16} f^{7}}\right )} \sqrt {d x + c} - \frac {3 \, {\left (35 \, C b d^{4} e^{4} - 20 \, C b c d^{3} e^{3} f - 40 \, C a d^{4} e^{3} f - 40 \, B b d^{4} e^{3} f - 6 \, C b c^{2} d^{2} e^{2} f^{2} + 24 \, C a c d^{3} e^{2} f^{2} + 24 \, B b c d^{3} e^{2} f^{2} + 48 \, B a d^{4} e^{2} f^{2} + 48 \, A b d^{4} e^{2} f^{2} - 4 \, C b c^{3} d e f^{3} + 8 \, C a c^{2} d^{2} e f^{3} + 8 \, B b c^{2} d^{2} e f^{3} - 32 \, B a c d^{3} e f^{3} - 32 \, A b c d^{3} e f^{3} - 64 \, A a d^{4} e f^{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}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {d^{2} e + {\left (d x + c\right )} d f - c d f} \right |}\right )}{\sqrt {d f} d^{3} f^{4}}\right )} d}{192 \, {\left | d \right |}} \]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x) \sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {e+f x}} \, dx=\text {Hanged} \]

[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}