3.53 \(\int \frac{\sqrt{c+d x} (A+B x+C x^2)}{(a+b x)^4 \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=685 \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \left (-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)+8 a^4 C d^2 f^2-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{24 b^2 (a+b x) (b c-a d)^2 (b e-a f)^3}-\frac{(d e-c f) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (a^2 \left (-\left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )+a b \left (-2 c d \left (6 A f^2-7 B e f+6 C e^2\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )+b^2 \left (c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (-a^2 b (-2 B d f+7 c C f+9 C d e)+4 a^3 C d f+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{12 b^2 (a+b x)^2 (b c-a d) (b e-a f)^2}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)} \]

[Out]

((4*a^3*C*d*f - b^3*(6*B*c*e - 3*A*d*e - 5*A*c*f) + a*b^2*(12*c*C*e + 3*B*d*e + B*c*f - 8*A*d*f) - a^2*b*(9*C*
d*e + 7*c*C*f - 2*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(12*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^2) - ((8*a^
4*C*d^2*f^2 - 2*a^3*b*d*f*(13*C*d*e + 7*c*C*f - 2*B*d*f) - b^4*(3*A*d^2*e^2 - 2*c*d*e*(3*B*e - 2*A*f) - 3*c^2*
(8*C*e^2 - 6*B*e*f + 5*A*f^2)) - a*b^3*(d^2*e*(3*B*e - 10*A*f) + 3*c^2*f*(4*C*e - B*f) + 2*c*d*(30*C*e^2 - 14*
B*e*f + 13*A*f^2)) - a^2*b^2*(4*d*f*(4*B*d*e + B*c*f - 2*A*d*f) - C*(33*d^2*e^2 + 44*c*d*e*f + 3*c^2*f^2)))*Sq
rt[c + d*x]*Sqrt[e + f*x])/(24*b^2*(b*c - a*d)^2*(b*e - a*f)^3*(a + b*x)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)
^(3/2)*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) - ((d*e - c*f)*(b^2*(A*d^2*e^2 - 2*c*d*e*(B*e
- A*f) + c^2*(8*C*e^2 - 6*B*e*f + 5*A*f^2)) + a*b*(d^2*e*(B*e - 4*A*f) - c^2*f*(4*C*e - B*f) - 2*c*d*(6*C*e^2
- 7*B*e*f + 6*A*f^2)) - a^2*(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)))*ArcTanh
[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(8*(b*c - a*d)^(5/2)*(b*e - a*f)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 1.77788, antiderivative size = 685, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1613, 149, 151, 12, 93, 208} \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \left (-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)+8 a^4 C d^2 f^2-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{24 b^2 (a+b x) (b c-a d)^2 (b e-a f)^3}-\frac{(d e-c f) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (a^2 \left (-\left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )+a b \left (-2 c d \left (6 A f^2-7 B e f+6 C e^2\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )+b^2 \left (c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (-a^2 b (-2 B d f+7 c C f+9 C d e)+4 a^3 C d f+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{12 b^2 (a+b x)^2 (b c-a d) (b e-a f)^2}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^4*Sqrt[e + f*x]),x]

[Out]

((4*a^3*C*d*f - b^3*(6*B*c*e - 3*A*d*e - 5*A*c*f) + a*b^2*(12*c*C*e + 3*B*d*e + B*c*f - 8*A*d*f) - a^2*b*(9*C*
d*e + 7*c*C*f - 2*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(12*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^2) - ((8*a^
4*C*d^2*f^2 - 2*a^3*b*d*f*(13*C*d*e + 7*c*C*f - 2*B*d*f) - b^4*(3*A*d^2*e^2 - 2*c*d*e*(3*B*e - 2*A*f) - 3*c^2*
(8*C*e^2 - 6*B*e*f + 5*A*f^2)) - a*b^3*(d^2*e*(3*B*e - 10*A*f) + 3*c^2*f*(4*C*e - B*f) + 2*c*d*(30*C*e^2 - 14*
B*e*f + 13*A*f^2)) - a^2*b^2*(4*d*f*(4*B*d*e + B*c*f - 2*A*d*f) - C*(33*d^2*e^2 + 44*c*d*e*f + 3*c^2*f^2)))*Sq
rt[c + d*x]*Sqrt[e + f*x])/(24*b^2*(b*c - a*d)^2*(b*e - a*f)^3*(a + b*x)) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)
^(3/2)*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) - ((d*e - c*f)*(b^2*(A*d^2*e^2 - 2*c*d*e*(B*e
- A*f) + c^2*(8*C*e^2 - 6*B*e*f + 5*A*f^2)) + a*b*(d^2*e*(B*e - 4*A*f) - c^2*f*(4*C*e - B*f) - 2*c*d*(6*C*e^2
- 7*B*e*f + 6*A*f^2)) - a^2*(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)))*ArcTanh
[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(8*(b*c - a*d)^(5/2)*(b*e - a*f)^(7/2))

Rule 1613

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt{e+f x}} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac{\int \frac{\sqrt{c+d x} \left (-\frac{a^2 C (3 d e+c f)+b^2 (6 B c e-3 A d e-5 A c f)-a b (6 c C e+3 B d e+B c f-6 A d f)}{2 b}+\left (-3 b c C e+3 a C d e+3 a c C f+A b d f-a B d f-\frac{2 a^2 C d f}{b}\right ) x\right )}{(a+b x)^3 \sqrt{e+f x}} \, dx}{3 (b c-a d) (b e-a f)}\\ &=\frac{\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac{\int \frac{\frac{4 a^3 C d f (d e+c f)+a b^2 \left (d^2 e (3 B e-8 A f)+3 c^2 f (4 C e-B f)+4 c d \left (9 C e^2-4 B e f+4 A f^2\right )\right )+b^3 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a^2 b \left (2 B d f (d e+c f)-C \left (9 d^2 e^2+20 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac{d \left (4 a^3 C d f^2-a^2 b f (11 C d e+5 c C f-2 B d f)-b^3 \left (12 c C e^2-A d e f-c f (6 B e-5 A f)\right )+a b^2 (12 C e (d e+c f)-f (7 B d e+B c f-4 A d f))\right ) x}{2 b}}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x}} \, dx}{6 b (b c-a d) (b e-a f)^2}\\ &=\frac{\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac{\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\int \frac{3 b (d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \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 )\right )}{8 (a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{6 b (b c-a d)^2 (b e-a f)^3}\\ &=\frac{\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac{\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left ((d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \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 )\right )\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{16 (b c-a d)^2 (b e-a f)^3}\\ &=\frac{\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac{\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left ((d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \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 )\right )\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 )}{8 (b c-a d)^2 (b e-a f)^3}\\ &=\frac{\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac{\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac{\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac{(d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \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 )\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 6.33094, size = 739, normalized size = 1.08 \[ -\frac{\left (a^2 C-a b B+A b^2\right ) \left (-\frac{3 \left (8 a^2 d^2 f^2-4 a b d f (3 c f+d e)+b^2 \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )\right ) \left (\frac{\sqrt{c+d x} \sqrt{e+f x}}{(a+b x) (a f-b e)}-\frac{(d e-c f) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{\sqrt{a d-b c} \sqrt{b e-a f} (a f-b e)}\right )}{8 (b c-a d) (b e-a f)}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (\frac{1}{2} b (-6 a d f+5 b c f+3 b d e)-a b d f\right )}{2 (a+b x)^2 (b c-a d) (b e-a f)}\right )}{3 b^2 (b c-a d) (b e-a f)}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}+\frac{(b B-2 a C) (-4 a d f+3 b c f+b d e) \left (\frac{\sqrt{c+d x} \sqrt{e+f x}}{(a+b x) (b e-a f)}-\frac{(d e-c f) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{\sqrt{a d-b c} (b e-a f)^{3/2}}\right )}{4 b^2 (b c-a d) (b e-a f)}-\frac{C \sqrt{c+d x} \sqrt{e+f x}}{b^2 (a+b x) (b e-a f)}+\frac{C (d e-c f) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{b^2 \sqrt{a d-b c} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} (b B-2 a C)}{2 b (a+b x)^2 (b c-a d) (b e-a f)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^4*Sqrt[e + f*x]),x]

[Out]

-((C*Sqrt[c + d*x]*Sqrt[e + f*x])/(b^2*(b*e - a*f)*(a + b*x))) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt
[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) - ((b*B - 2*a*C)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*b*(b*c
- a*d)*(b*e - a*f)*(a + b*x)^2) + (C*(d*e - c*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*S
qrt[e + f*x])])/(b^2*Sqrt[-(b*c) + a*d]*(b*e - a*f)^(3/2)) + ((b*B - 2*a*C)*(b*d*e + 3*b*c*f - 4*a*d*f)*((Sqrt
[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) - ((d*e - c*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-
(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*(b*e - a*f)^(3/2))))/(4*b^2*(b*c - a*d)*(b*e - a*f)) - ((A*b
^2 - a*b*B + a^2*C)*(-((-(a*b*d*f) + (b*(3*b*d*e + 5*b*c*f - 6*a*d*f))/2)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*(b
*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (3*(8*a^2*d^2*f^2 - 4*a*b*d*f*(d*e + 3*c*f) + b^2*(d^2*e^2 + 2*c*d*e*f +
5*c^2*f^2))*((Sqrt[c + d*x]*Sqrt[e + f*x])/((-(b*e) + a*f)*(a + b*x)) - ((d*e - c*f)*ArcTan[(Sqrt[b*e - a*f]*S
qrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*Sqrt[b*e - a*f]*(-(b*e) + a*f))))/(8*(b
*c - a*d)*(b*e - a*f))))/(3*b^2*(b*c - a*d)*(b*e - a*f))

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Maple [B]  time = 0.148, size = 15990, normalized size = 23.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^4/(f*x+e)^(1/2),x)

[Out]

result too large to display

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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((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**4/(f*x+e)**(1/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

Timed out