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

Optimal. Leaf size=721 \[ -\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2+6 b d f x (6 a C d f-b (10 B d f-7 C (c f+d e)))-10 a b d f (8 B d f-5 C (c f+d e))+b^2 \left (-\left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{240 b d^3 f^3}+\frac{\sqrt{c+d x} \sqrt{e+f x} (d e-c f) \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+7 c^3 f^3+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{128 d^4 f^4}+\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+7 c^3 f^3+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{64 d^4 f^3}-\frac{(d e-c f)^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+7 c^3 f^3+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{128 d^{9/2} f^{9/2}}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f} \]

[Out]

((d*e - c*f)*(2*a*d*f*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(7*d^3*
e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c
^2*f^2))))*Sqrt[c + d*x]*Sqrt[e + f*x])/(128*d^4*f^4) + ((2*a*d*f*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d
*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*
(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(64*d^4*f^3) + (C*(a + b
*x)^2*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(5*b*d*f) - ((c + d*x)^(3/2)*(e + f*x)^(3/2)*(48*a^2*C*d^2*f^2 - 10*a*b
*d*f*(8*B*d*f - 5*C*(d*e + c*f)) - b^2*(C*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2) + 10*d*f*(8*A*d*f - 5*B*(d*e
+ c*f))) + 6*b*d*f*(6*a*C*d*f - b*(10*B*d*f - 7*C*(d*e + c*f)))*x))/(240*b*d^3*f^3) - ((d*e - c*f)^2*(2*a*d*f*
(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f +
9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))))*ArcTanh[(Sq
rt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(128*d^(9/2)*f^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.963188, antiderivative size = 719, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-6 b d f x (-6 a C d f+10 b B d f-7 b C (c f+d e))-10 a b d f (8 B d f-5 C (c f+d e))+b^2 \left (-\left (10 d f (8 A d f-5 B (c f+d e))+C \left (35 c^2 f^2+38 c d e f+35 d^2 e^2\right )\right )\right )\right )}{240 b d^3 f^3}+\frac{\sqrt{c+d x} \sqrt{e+f x} (d e-c f) \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+7 c^3 f^3+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{128 d^4 f^4}+\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+7 c^3 f^3+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{64 d^4 f^3}-\frac{(d e-c f)^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (2 a d f \left (8 d f (2 A d f-B (c f+d e))+C \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )-b \left (2 d f \left (8 A d f (c f+d e)-B \left (5 c^2 f^2+6 c d e f+5 d^2 e^2\right )\right )+C \left (9 c^2 d e f^2+7 c^3 f^3+9 c d^2 e^2 f+7 d^3 e^3\right )\right )\right )}{128 d^{9/2} f^{9/2}}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*e - c*f)*(2*a*d*f*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(7*d^3*
e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c
^2*f^2))))*Sqrt[c + d*x]*Sqrt[e + f*x])/(128*d^4*f^4) + ((2*a*d*f*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d
*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*
(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(64*d^4*f^3) + (C*(a + b
*x)^2*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(5*b*d*f) - ((c + d*x)^(3/2)*(e + f*x)^(3/2)*(48*a^2*C*d^2*f^2 - 10*a*b
*d*f*(8*B*d*f - 5*C*(d*e + c*f)) - b^2*(C*(35*d^2*e^2 + 38*c*d*e*f + 35*c^2*f^2) + 10*d*f*(8*A*d*f - 5*B*(d*e
+ c*f))) - 6*b*d*f*(10*b*B*d*f - 6*a*C*d*f - 7*b*C*(d*e + c*f))*x))/(240*b*d^3*f^3) - ((d*e - c*f)^2*(2*a*d*f*
(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f))) - b*(C*(7*d^3*e^3 + 9*c*d^2*e^2*f +
9*c^2*d*e*f^2 + 7*c^3*f^3) + 2*d*f*(8*A*d*f*(d*e + c*f) - B*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2))))*ArcTanh[(Sq
rt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(128*d^(9/2)*f^(9/2))

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]

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 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 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 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])

Rubi steps

\begin{align*} \int (a+b x) \sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right ) \, dx &=\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}+\frac{\int (a+b x) \sqrt{c+d x} \sqrt{e+f x} \left (-\frac{1}{2} b (4 b c C e+3 a C d e+3 a c C f-10 A b d f)+\frac{1}{2} b (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right ) \, dx}{5 b^2 d f}\\ &=\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-10 a b d f (8 B d f-5 C (d e+c f))-b^2 \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )-6 b d f (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right )}{240 b d^3 f^3}+\frac{\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \int \sqrt{c+d x} \sqrt{e+f x} \, dx}{32 d^3 f^3}\\ &=\frac{\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt{e+f x}}{64 d^4 f^3}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-10 a b d f (8 B d f-5 C (d e+c f))-b^2 \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )-6 b d f (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right )}{240 b d^3 f^3}+\frac{\left ((d e-c f) \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right )\right ) \int \frac{\sqrt{c+d x}}{\sqrt{e+f x}} \, dx}{128 d^4 f^3}\\ &=\frac{(d e-c f) \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^4}+\frac{\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt{e+f x}}{64 d^4 f^3}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-10 a b d f (8 B d f-5 C (d e+c f))-b^2 \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )-6 b d f (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right )}{240 b d^3 f^3}-\frac{\left ((d e-c f)^2 \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{256 d^4 f^4}\\ &=\frac{(d e-c f) \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^4}+\frac{\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt{e+f x}}{64 d^4 f^3}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-10 a b d f (8 B d f-5 C (d e+c f))-b^2 \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )-6 b d f (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right )}{240 b d^3 f^3}-\frac{\left ((d e-c f)^2 \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 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 )}{128 d^5 f^4}\\ &=\frac{(d e-c f) \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^4}+\frac{\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt{e+f x}}{64 d^4 f^3}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-10 a b d f (8 B d f-5 C (d e+c f))-b^2 \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )-6 b d f (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right )}{240 b d^3 f^3}-\frac{\left ((d e-c f)^2 \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 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 )}{128 d^5 f^4}\\ &=\frac{(d e-c f) \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^4}+\frac{\left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt{e+f x}}{64 d^4 f^3}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left (48 a^2 C d^2 f^2-10 a b d f (8 B d f-5 C (d e+c f))-b^2 \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )-6 b d f (10 b B d f-6 a C d f-7 b C (d e+c f)) x\right )}{240 b d^3 f^3}-\frac{(d e-c f)^2 \left (2 a d f \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-b \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 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 )}{128 d^{9/2} f^{9/2}}\\ \end{align*}

Mathematica [B]  time = 6.48294, size = 2722, normalized size = 3.78 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*C*(d*e - c*f)^3*(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))))^(9/2)*((3*(35/(64*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e -
c*f))))^4) + 35/(48*(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) + 7/(8
*(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)))/10 + (21*(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)))])))/(512*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))))^4)))/(3*d^4*f^3*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(
d*e - c*f)))^(7/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)^2*(-3*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))))^(7/2)*((3*(5
/(8*(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) + 5/(6*(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)))/8 + (15*(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)))])))/(256*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)))/(3*d^3*f^3*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(5/2)
*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)*(3*b*C*e^2 - 2*b*B*e*f - 2*a*C*e*f + A*b*f^2 + a*B*f^2)*(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*d^2*f^3*(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*e) + a*f)*(C*e^2 - B*e*f + A*f^2)*(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*d*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)])

________________________________________________________________________________________

Maple [B]  time = 0.021, size = 3571, normalized size = 5. \begin{align*} \text{output too large to display} \end{align*}

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]

-1/3840*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-105*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+
d*e)/(d*f)^(1/2))*b*d^5*e^5-120*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)
^(1/2))*b*c^3*d^2*e*f^4+30*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2
))*b*c^3*d^2*e^2*f^3+30*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*
b*c^2*d^3*e^3*f^2+75*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c
*d^4*e^4*f-120*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c*d^4*e
^3*f^2+75*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^4*d*e*f^4-
768*C*x^4*b*d^4*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-960*B*x^3*b*d^4*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x
+d*e*x+c*e)^(1/2)-960*C*x^3*a*d^4*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-1280*A*x^2*b*d^4*f^4*(d*f)^(
1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-1280*B*x^2*a*d^4*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-60*B*ln(
1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^2*d^3*e^2*f^3+480*B*(d*f)
^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*c^2*d^2*f^4+480*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*d^4*e
^2*f^2-300*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c^3*d*f^4-300*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*
e)^(1/2)*b*d^4*e^3*f-960*A*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*c*d^3*f^4-960*A*(d*f)^(1/2)*(d*f*x^2+
c*f*x+d*e*x+c*e)^(1/2)*a*d^4*e*f^3+480*A*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c^2*d^2*f^4+480*A*(d*f)
^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*d^4*e^2*f^2-960*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(
d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c*d^4*e*f^4+240*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(
1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^2*d^3*e*f^4+240*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)
+c*f+d*e)/(d*f)^(1/2))*b*c*d^4*e^2*f^3-1920*A*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*a*d^4*f^4-300*C*(d
*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*c^3*d*f^4-300*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*d^4*
e^3*f-120*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c*d^4*e^3*f^
2-120*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^3*d^2*e*f^4-60
*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^2*d^3*e^2*f^3-80*B*
(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b*c*d^3*e*f^3-80*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x
*a*c*d^3*e*f^3+44*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b*c^2*d^2*e*f^3+44*C*(d*f)^(1/2)*(d*f*x^2+c*
f*x+d*e*x+c*e)^(1/2)*x*b*c*d^3*e^2*f^2-32*C*x^2*b*c*d^3*e*f^3*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)+150*
B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^4*d*f^5-160*C*x^2*a*
c*d^3*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-160*C*x^2*a*d^4*e*f^3*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c
*e)^(1/2)+112*C*x^2*b*c^2*d^2*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)+112*C*x^2*b*d^4*e^2*f^2*(d*f)^(1
/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-105*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*
e)/(d*f)^(1/2))*b*c^5*f^5+480*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(
1/2))*a*c^2*d^3*f^5+480*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*
a*d^5*e^2*f^3-240*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*c^3*
d^2*f^5+210*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c^4*f^4+210*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e
)^(1/2)*b*d^4*e^4+240*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*
c^2*d^3*e*f^4+240*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c*d^
4*e^2*f^3-80*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c*d^3*e^3*f+140*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*
x+c*e)^(1/2)*a*c^2*d^2*e*f^3+140*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*c*d^3*e^2*f^2+140*B*(d*f)^(1/
2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c^2*d^2*e*f^3+140*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c*d^3*e
^2*f^2+200*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*a*d^4*e^2*f^2-140*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*
x+c*e)^(1/2)*x*b*c^3*d*f^4-140*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b*d^4*e^3*f-320*A*(d*f)^(1/2)*(
d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c*d^3*e*f^3-320*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*c*d^3*e*f^3+2
00*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*a*c^2*d^2*f^4+200*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(
1/2)*x*b*c^2*d^2*f^4+200*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b*d^4*e^2*f^2-320*A*(d*f)^(1/2)*(d*f*
x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b*c*d^3*f^4-320*A*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b*d^4*e*f^3-320*B
*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*a*c*d^3*f^4-320*B*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x
*a*d^4*e*f^3-96*C*x^3*b*c*d^3*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-96*C*x^3*b*d^4*e*f^3*(d*f)^(1/2)
*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-160*B*x^2*b*c*d^3*f^4*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-160*B*x^2*b
*d^4*e*f^3*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)-80*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c^3*
d*e*f^3-68*C*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b*c^2*d^2*e^2*f^2+150*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*
f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b*d^5*e^4*f+150*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*
x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^4*d*f^5+150*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(
1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*d^5*e^4*f-240*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*
f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*c^3*d^2*f^5-240*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/
2)+c*f+d*e)/(d*f)^(1/2))*a*d^5*e^3*f^2-240*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f
+d*e)/(d*f)^(1/2))*b*d^5*e^3*f^2)/(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)/d^4/f^4/(d*f)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 3.78587, size = 3633, normalized size = 5.04 \begin{align*} \text{result too large to display} \end{align*}

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]

[-1/7680*(15*(7*C*b*d^5*e^5 - 5*(C*b*c*d^4 + 2*(C*a + B*b)*d^5)*e^4*f - 2*(C*b*c^2*d^3 - 4*(C*a + B*b)*c*d^4 -
8*(B*a + A*b)*d^5)*e^3*f^2 - 2*(C*b*c^3*d^2 + 16*A*a*d^5 - 2*(C*a + B*b)*c^2*d^3 + 8*(B*a + A*b)*c*d^4)*e^2*f
^3 - (5*C*b*c^4*d - 64*A*a*c*d^4 - 8*(C*a + B*b)*c^3*d^2 + 16*(B*a + A*b)*c^2*d^3)*e*f^4 + (7*C*b*c^5 - 32*A*a
*c^2*d^3 - 10*(C*a + B*b)*c^4*d + 16*(B*a + A*b)*c^3*d^2)*f^5)*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*(3
84*C*b*d^5*f^5*x^4 - 105*C*b*d^5*e^4*f + 10*(4*C*b*c*d^4 + 15*(C*a + B*b)*d^5)*e^3*f^2 + 2*(17*C*b*c^2*d^3 - 3
5*(C*a + B*b)*c*d^4 - 120*(B*a + A*b)*d^5)*e^2*f^3 + 10*(4*C*b*c^3*d^2 + 48*A*a*d^5 - 7*(C*a + B*b)*c^2*d^3 +
16*(B*a + A*b)*c*d^4)*e*f^4 - 15*(7*C*b*c^4*d - 32*A*a*c*d^4 - 10*(C*a + B*b)*c^3*d^2 + 16*(B*a + A*b)*c^2*d^3
)*f^5 + 48*(C*b*d^5*e*f^4 + (C*b*c*d^4 + 10*(C*a + B*b)*d^5)*f^5)*x^3 - 8*(7*C*b*d^5*e^2*f^3 - 2*(C*b*c*d^4 +
5*(C*a + B*b)*d^5)*e*f^4 + (7*C*b*c^2*d^3 - 10*(C*a + B*b)*c*d^4 - 80*(B*a + A*b)*d^5)*f^5)*x^2 + 2*(35*C*b*d^
5*e^3*f^2 - (11*C*b*c*d^4 + 50*(C*a + B*b)*d^5)*e^2*f^3 - (11*C*b*c^2*d^3 - 20*(C*a + B*b)*c*d^4 - 80*(B*a + A
*b)*d^5)*e*f^4 + 5*(7*C*b*c^3*d^2 + 96*A*a*d^5 - 10*(C*a + B*b)*c^2*d^3 + 16*(B*a + A*b)*c*d^4)*f^5)*x)*sqrt(d
*x + c)*sqrt(f*x + e))/(d^5*f^5), -1/3840*(15*(7*C*b*d^5*e^5 - 5*(C*b*c*d^4 + 2*(C*a + B*b)*d^5)*e^4*f - 2*(C*
b*c^2*d^3 - 4*(C*a + B*b)*c*d^4 - 8*(B*a + A*b)*d^5)*e^3*f^2 - 2*(C*b*c^3*d^2 + 16*A*a*d^5 - 2*(C*a + B*b)*c^2
*d^3 + 8*(B*a + A*b)*c*d^4)*e^2*f^3 - (5*C*b*c^4*d - 64*A*a*c*d^4 - 8*(C*a + B*b)*c^3*d^2 + 16*(B*a + A*b)*c^2
*d^3)*e*f^4 + (7*C*b*c^5 - 32*A*a*c^2*d^3 - 10*(C*a + B*b)*c^4*d + 16*(B*a + A*b)*c^3*d^2)*f^5)*sqrt(-d*f)*arc
tan(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*(384*C*b*d^5*f^5*x^4 - 105*C*b*d^5*e^4*f + 10*(4*C*b*c*d^4 + 15*(C*a + B*b)*d^5)*e^3*f^2 + 2*(17*C
*b*c^2*d^3 - 35*(C*a + B*b)*c*d^4 - 120*(B*a + A*b)*d^5)*e^2*f^3 + 10*(4*C*b*c^3*d^2 + 48*A*a*d^5 - 7*(C*a + B
*b)*c^2*d^3 + 16*(B*a + A*b)*c*d^4)*e*f^4 - 15*(7*C*b*c^4*d - 32*A*a*c*d^4 - 10*(C*a + B*b)*c^3*d^2 + 16*(B*a
+ A*b)*c^2*d^3)*f^5 + 48*(C*b*d^5*e*f^4 + (C*b*c*d^4 + 10*(C*a + B*b)*d^5)*f^5)*x^3 - 8*(7*C*b*d^5*e^2*f^3 - 2
*(C*b*c*d^4 + 5*(C*a + B*b)*d^5)*e*f^4 + (7*C*b*c^2*d^3 - 10*(C*a + B*b)*c*d^4 - 80*(B*a + A*b)*d^5)*f^5)*x^2
+ 2*(35*C*b*d^5*e^3*f^2 - (11*C*b*c*d^4 + 50*(C*a + B*b)*d^5)*e^2*f^3 - (11*C*b*c^2*d^3 - 20*(C*a + B*b)*c*d^4
- 80*(B*a + A*b)*d^5)*e*f^4 + 5*(7*C*b*c^3*d^2 + 96*A*a*d^5 - 10*(C*a + B*b)*c^2*d^3 + 16*(B*a + A*b)*c*d^4)*
f^5)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^5*f^5)]

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

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]

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

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Giac [B]  time = 3.57262, size = 2006, normalized size = 2.78 \begin{align*} \text{result too large to display} \end{align*}

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]

1/1920*(20*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)/(d^4*f^2) - (c*f^2 - d*f*e)/(d^4*f^
4)) + (c^2*f^2 - 2*c*d*f*e + d^2*e^2)*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^3))*A*a*abs(d)/d^2 + 10*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d
*x + c)/d^2 - (17*c*d^6*f^6 - d^7*f^5*e)/(d^8*f^6)) + (59*c^2*d^6*f^6 - 6*c*d^7*f^5*e - 5*d^8*f^4*e^2)/(d^8*f^
6)) - 3*(5*c^3*d^6*f^6 + c^2*d^7*f^5*e - c*d^8*f^4*e^2 - 5*d^9*f^3*e^3)/(d^8*f^6))*sqrt(d*x + c) + 3*(5*c^4*f^
4 - 4*c^3*d*f^3*e - 2*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 + 5*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*f^3))*C*a*abs(d)/d^2 + 10*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(
d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^2 - (17*c*d^6*f^6 - d^7*f^5*e)/(d^8*f^6)) + (59*c^2*d^6*f^6 - 6*c*d^7*f^5
*e - 5*d^8*f^4*e^2)/(d^8*f^6)) - 3*(5*c^3*d^6*f^6 + c^2*d^7*f^5*e - c*d^8*f^4*e^2 - 5*d^9*f^3*e^3)/(d^8*f^6))*
sqrt(d*x + c) + 3*(5*c^4*f^4 - 4*c^3*d*f^3*e - 2*c^2*d^2*f^2*e^2 - 4*c*d^3*f*e^3 + 5*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*f^3))*B*b*abs(d)/d^2 + (sqrt((d*x + c)*d
*f - c*d*f + d^2*e)*(2*(4*(d*x + c)*(6*(d*x + c)*(8*(d*x + c)/d^3 - (31*c*d^12*f^8 - d^13*f^7*e)/(d^15*f^8)) +
(263*c^2*d^12*f^8 - 16*c*d^13*f^7*e - 7*d^14*f^6*e^2)/(d^15*f^8)) - 5*(121*c^3*d^12*f^8 - 9*c^2*d^13*f^7*e -
9*c*d^14*f^6*e^2 - 7*d^15*f^5*e^3)/(d^15*f^8))*(d*x + c) + 15*(7*c^4*d^12*f^8 + 2*c^3*d^13*f^7*e - 2*c*d^15*f^
5*e^3 - 7*d^16*f^4*e^4)/(d^15*f^8))*sqrt(d*x + c) - 15*(7*c^5*f^5 - 5*c^4*d*f^4*e - 2*c^3*d^2*f^3*e^2 - 2*c^2*
d^3*f^2*e^3 - 5*c*d^4*f*e^4 + 7*d^5*e^5)*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^2*f^4))*C*b*abs(d)/d^2 + (sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(
d*x + c)/(d^6*f^2) - (7*c*f^4 - d*f^3*e)/(d^6*f^6)) + 3*(c^2*f^4 - d^2*f^2*e^2)/(d^6*f^6)) - 3*(c^3*f^3 - c^2*
d*f^2*e - c*d^2*f*e^2 + d^3*e^3)*log(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqr
t(d*f)*d^5*f^4))*B*a*abs(d)/d^3 + (sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)
/(d^6*f^2) - (7*c*f^4 - d*f^3*e)/(d^6*f^6)) + 3*(c^2*f^4 - d^2*f^2*e^2)/(d^6*f^6)) - 3*(c^3*f^3 - c^2*d*f^2*e
- c*d^2*f*e^2 + d^3*e^3)*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
^5*f^4))*A*b*abs(d)/d^3)/d