\(\int \frac {(a+b x)^{3/2} (A+B x+C x^2)}{\sqrt {c+d x} \sqrt {e+f x}} \, dx\) [73]

Optimal result

Integrand size = 38, antiderivative size = 838 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f)))) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^3 f^3}-\frac {2 (2 a C d f-b (7 B d f-6 C (d e+c f))) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 b c e+a d e+a c f) (2 a C d f-b (7 B d f-6 C (d e+c f))))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f))))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

-2/35*(2*a*C*d*f-b*(7*B*d*f-6*C*(c*f+d*e)))*(b*x+a)^(3/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^2/f^2+2/7*C*(b*x+a)^
(5/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d/f-2/105*(5*b*d*f*(-7*A*b*d*f+C*a*c*f+C*a*d*e+5*C*b*c*e)+(3*a*d*f-4*b*(c*
f+d*e))*(2*a*C*d*f-b*(7*B*d*f-6*C*(c*f+d*e))))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^3/f^3-2/105*(3*b*
d*f*(5*a*d*f*(-7*A*b*d*f+C*a*c*f+C*a*d*e+5*C*b*c*e)-(a*c*f+a*d*e+3*b*c*e)*(2*a*C*d*f-b*(7*B*d*f-6*C*(c*f+d*e))
))+2*(1/2*a*d*f-b*(c*f+d*e))*(5*b*d*f*(-7*A*b*d*f+C*a*c*f+C*a*d*e+5*C*b*c*e)+(3*a*d*f-4*b*(c*f+d*e))*(2*a*C*d*
f-b*(7*B*d*f-6*C*(c*f+d*e)))))*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/
2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^2/d^(7/2)/f^4/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+
b*e))^(1/2)-2/105*(-a*f+b*e)*(3*a^2*C*d^2*f^2*(-c*f+d*e)-3*a*b*d*f*(7*d*f*(-5*A*d*f+2*B*c*f+3*B*d*e)-C*(11*c^2
*f^2+8*c*d*e*f+16*d^2*e^2))-b^2*(C*(24*c^3*f^3+17*c^2*d*e*f^2+16*c*d^2*e^2*f+48*d^3*e^3)+7*d*f*(5*A*d*f*(c*f+2
*d*e)-B*(4*c^2*f^2+3*c*d*e*f+8*d^2*e^2))))*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a
*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^2/d^(7/2)/f^4/(d*x
+c)^(1/2)/(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 831, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1629, 159, 164, 115, 114, 122, 121} \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 C \sqrt {c+d x} \sqrt {e+f x} (a+b x)^{5/2}}{7 b d f}+\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) \sqrt {c+d x} \sqrt {e+f x} (a+b x)^{3/2}}{35 b d^2 f^2}-\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt {c+d x} \sqrt {e+f x} \sqrt {a+b x}}{105 b d^3 f^3}-\frac {2 \sqrt {a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a d-b c} (b e-a f) \left (-\left (\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2\right )-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \]

[In]

Int[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*a*d*f - 4*b*(d*e + c*f))*(7*b*B*d*f - 2*a*C*d*f
- 6*b*C*(d*e + c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b*d^3*f^3) + (2*(7*b*B*d*f - 2*a*C*d*f -
6*b*C*(d*e + c*f))*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(5/2)*Sqrt[c
+ d*x]*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A
*b*d*f) + (3*b*c*e + a*d*e + a*c*f)*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f))) + 2*((a*d*f)/2 - b*(d*e + c*f
))*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*a*d*f - 4*b*(d*e + c*f))*(7*b*B*d*f - 2*a*C*d*f -
6*b*C*(d*e + c*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b
*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(3*B*d*e + 2*B
*c*f - 5*A*d*f) - C*(16*d^2*e^2 + 8*c*d*e*f + 11*c^2*f^2)) - b^2*(C*(48*d^3*e^3 + 16*c*d^2*e^2*f + 17*c^2*d*e*
f^2 + 24*c^3*f^3) + 7*d*f*(5*A*d*f*(2*d*e + c*f) - B*(8*d^2*e^2 + 3*c*d*e*f + 4*c^2*f^2))))*Sqrt[(b*(c + d*x))
/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((
b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && Si
mplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

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 {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}+\frac {2 \int \frac {(a+b x)^{3/2} \left (-\frac {1}{2} b (5 b c C e+a C d e+a c C f-7 A b d f)+\frac {1}{2} b (7 b B d f-2 a C d f-6 b C (d e+c f)) x\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{7 b^2 d f} \\ & = \frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}+\frac {4 \int \frac {\sqrt {a+b x} \left (-\frac {1}{4} b (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))-\frac {1}{4} b (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) x\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{35 b^2 d^2 f^2} \\ & = -\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^3 f^3}+\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}+\frac {8 \int \frac {-\frac {1}{8} b (3 a d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))-(b c e+a d e+a c f) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))))-\frac {1}{8} b \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{105 b^2 d^3 f^3} \\ & = -\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^3 f^3}+\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{105 b d^3 f^4}-\frac {\left ((b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{105 b d^3 f^4} \\ & = -\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^3 f^3}+\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {\left ((b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}} \, dx}{105 b d^3 f^4 \sqrt {c+d x}}-\frac {\left (\left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{105 b d^3 f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \\ & = -\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^3 f^3}+\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {\left ((b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}} \, dx}{105 b d^3 f^4 \sqrt {c+d x} \sqrt {e+f x}} \\ & = -\frac {2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{105 b d^3 f^3}+\frac {2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}{35 b d^2 f^2}+\frac {2 C (a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}}{7 b d f}-\frac {2 \sqrt {-b c+a d} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac {a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.12 (sec) , antiderivative size = 1000, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \left (-b^2 \sqrt {-a+\frac {b c}{d}} \left (6 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (-7 B d f+4 C (d e+c f))-a b^2 d f \left (C \left (72 d^2 e^2+62 c d e f+72 c^2 f^2\right )+7 d f (20 A d f-13 B (d e+c f))\right )+b^3 \left (8 C \left (6 d^3 e^3+5 c d^2 e^2 f+5 c^2 d e f^2+6 c^3 f^3\right )+7 d f \left (10 A d f (d e+c f)-B \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )\right )\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (a+b x) (c+d x) (e+f x) \left (3 a^2 C d^2 f^2+3 a b d f (14 B d f+C (-11 d e-11 c f+8 d f x))+b^2 \left (7 d f (5 A d f+B (-4 d e-4 c f+3 d f x))+C \left (24 c^2 f^2+c d f (23 e-18 f x)+3 d^2 \left (8 e^2-6 e f x+5 f^2 x^2\right )\right )\right )\right )-i (b c-a d) f \left (6 a^3 C d^3 f^3+3 a^2 b d^2 f^2 (-7 B d f+4 C (d e+c f))-a b^2 d f \left (C \left (72 d^2 e^2+62 c d e f+72 c^2 f^2\right )+7 d f (20 A d f-13 B (d e+c f))\right )+b^3 \left (8 C \left (6 d^3 e^3+5 c d^2 e^2 f+5 c^2 d e f^2+6 c^3 f^3\right )+7 d f \left (10 A d f (d e+c f)-B \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+i b (b c-a d) f \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (-2 B d e-3 B c f+5 A d f)+C \left (11 d^2 e^2+8 c d e f+16 c^2 f^2\right )\right )+b^2 \left (C \left (24 d^3 e^3+17 c d^2 e^2 f+16 c^2 d e f^2+48 c^3 f^3\right )+7 d f \left (5 A d f (d e+2 c f)-B \left (4 d^2 e^2+3 c d e f+8 c^2 f^2\right )\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{105 b^3 \sqrt {-a+\frac {b c}{d}} d^4 f^4 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \]

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(2*(-(b^2*Sqrt[-a + (b*c)/d]*(6*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(7
2*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 7*d*f*(20*A*d*f - 13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^
2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e + c*f) - B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(c +
d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(3*a^2*C*d^2*f^2 + 3*a*b*d*f*(14*B
*d*f + C*(-11*d*e - 11*c*f + 8*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e - 4*c*f + 3*d*f*x)) + C*(24*c^2*f^2 +
c*d*f*(23*e - 18*f*x) + 3*d^2*(8*e^2 - 6*e*f*x + 5*f^2*x^2)))) - I*(b*c - a*d)*f*(6*a^3*C*d^3*f^3 + 3*a^2*b*d
^2*f^2*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(72*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 7*d*f*(20*A*d*f -
13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e +
c*f) - B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e
+ f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]
+ I*b*(b*c - a*d)*f*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(-2*B*d*e - 3*B*c*f + 5*A*d*f) + C*(11*d^
2*e^2 + 8*c*d*e*f + 16*c^2*f^2)) + b^2*(C*(24*d^3*e^3 + 17*c*d^2*e^2*f + 16*c^2*d*e*f^2 + 48*c^3*f^3) + 7*d*f*
(5*A*d*f*(d*e + 2*c*f) - B*(4*d^2*e^2 + 3*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b
*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)
/(b*c*f - a*d*f)]))/(105*b^3*Sqrt[-a + (b*c)/d]*d^4*f^4*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])

Maple [A] (verified)

Time = 2.92 (sec) , antiderivative size = 1233, normalized size of antiderivative = 1.47

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1233\)
default	\(\text {Expression too large to display}\)	\(9580\)

[In]

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((b*x+a)*(d*x+c)*(f*x+e))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)*(2/7*b*C/d/f*x^2*(b*d*f*x^3+a*d*f*x^
2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)+2/5*(B*b^2+2*C*a*b-2/7*b*C/d/f*(3*a*d*f+3*b*c*f+3*b
*d*e))/b/d/f*x*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)+2/3*(b^2*A+2*a*b*
B+C*a^2-2/7*b*C/d/f*(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(B*b^2+2*C*a*b-2/7*b*C/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/
b/d/f*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)
^(1/2)+2*(a^2*A-2/5*(B*b^2+2*C*a*b-2/7*b*C/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/b/d/f*a*c*e-2/3*(b^2*A+2*a*b*B+C*a^2
-2/7*b*C/d/f*(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(B*b^2+2*C*a*b-2/7*b*C/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/b/d/f*(
2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(1/2*a*c*f+1/2*a*d*e+1/2*b*c*e))*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/b)/
(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+
a*c*e)^(1/2)*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))+2*(2*a*b*A+a^2*B-4/7*b*C/d/f*a
*c*e-2/5*(B*b^2+2*C*a*b-2/7*b*C/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/b/d/f*(3/2*a*c*f+3/2*a*d*e+3/2*b*c*e)-2/3*(b^2*
A+2*a*b*B+C*a^2-2/7*b*C/d/f*(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(B*b^2+2*C*a*b-2/7*b*C/d/f*(3*a*d*f+3*b*c*f+3*
b*d*e))/b/d/f*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(a*d*f+b*c*f+b*d*e))*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/
b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e
*x+a*c*e)^(1/2)*((-e/f+a/b)*EllipticE(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))-a/b*EllipticF((
(x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 1388, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

2/315*(3*(15*C*b^4*d^4*f^4*x^2 + 24*C*b^4*d^4*e^2*f^2 + (23*C*b^4*c*d^3 - (33*C*a*b^3 + 28*B*b^4)*d^4)*e*f^3 +
(24*C*b^4*c^2*d^2 - (33*C*a*b^3 + 28*B*b^4)*c*d^3 + (3*C*a^2*b^2 + 42*B*a*b^3 + 35*A*b^4)*d^4)*f^4 - 3*(6*C*b
^4*d^4*e*f^3 + (6*C*b^4*c*d^3 - (8*C*a*b^3 + 7*B*b^4)*d^4)*f^4)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) +
(48*C*b^4*d^4*e^4 + 8*(2*C*b^4*c*d^3 - (12*C*a*b^3 + 7*B*b^4)*d^4)*e^3*f + (11*C*b^4*c^2*d^2 - 7*(4*C*a*b^3 +
3*B*b^4)*c*d^3 + (39*C*a^2*b^2 + 119*B*a*b^3 + 70*A*b^4)*d^4)*e^2*f^2 + (16*C*b^4*c^3*d - 7*(4*C*a*b^3 + 3*B*
b^4)*c^2*d^2 + 7*(C*a^2*b^2 + 7*B*a*b^3 + 5*A*b^4)*c*d^3 + (9*C*a^3*b - 56*B*a^2*b^2 - 175*A*a*b^3)*d^4)*e*f^3
+ (48*C*b^4*c^4 - 8*(12*C*a*b^3 + 7*B*b^4)*c^3*d + (39*C*a^2*b^2 + 119*B*a*b^3 + 70*A*b^4)*c^2*d^2 + (9*C*a^3
*b - 56*B*a^2*b^2 - 175*A*a*b^3)*c*d^3 + (6*C*a^4 - 21*B*a^3*b + 175*A*a^2*b^2)*d^4)*f^4)*sqrt(b*d*f)*weierstr
assPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/
27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3
*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/
(b*d*f)) + 3*(48*C*b^4*d^4*e^3*f + 8*(5*C*b^4*c*d^3 - (9*C*a*b^3 + 7*B*b^4)*d^4)*e^2*f^2 + (40*C*b^4*c^2*d^2 -
(62*C*a*b^3 + 49*B*b^4)*c*d^3 + (12*C*a^2*b^2 + 91*B*a*b^3 + 70*A*b^4)*d^4)*e*f^3 + (48*C*b^4*c^3*d - 8*(9*C*
a*b^3 + 7*B*b^4)*c^2*d^2 + (12*C*a^2*b^2 + 91*B*a*b^3 + 70*A*b^4)*c*d^3 + (6*C*a^3*b - 21*B*a^2*b^2 - 140*A*a*
b^3)*d^4)*f^4)*sqrt(b*d*f)*weierstrassZeta(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a
^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*
d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), weierstra
ssPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/2
7*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*
c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(
b*d*f))))/(b^3*d^5*f^5)

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{\sqrt {c + d x} \sqrt {e + f x}}\, dx \]

[In]

integrate((b*x+a)**(3/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Integral((a + b*x)**(3/2)*(A + B*x + C*x**2)/(sqrt(c + d*x)*sqrt(e + f*x)), x)

Maxima [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e}} \,d x } \]

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)), x)

Giac [F]

\[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} \sqrt {f x + e}} \,d x } \]

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,\sqrt {c+d\,x}} \,d x \]

[In]

int(((a + b*x)^(3/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(c + d*x)^(1/2)),x)

[Out]

int(((a + b*x)^(3/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(c + d*x)^(1/2)), x)