\(\int \frac {\sqrt {c+d x} (A+B x+C x^2)}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx\) [71]

Optimal result

Integrand size = 38, antiderivative size = 597 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=-\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {2 \sqrt {d} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))-b^3 \left (A d e f+c \left (3 C e^2+3 B e f-2 A f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {-b c+a d} f (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 (d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

-2/3*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(3/2)-2/3*(4*a^2*C*f+b^2*
(-2*A*f+3*B*e)-a*b*(B*f+6*C*e))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^2/(-a*f+b*e)^2/(b*x+a)^(1/2)+2/3*(8*a^3*C*d*f^2-
a^2*b*f*(2*B*d*f+7*C*c*f+13*C*d*e)+a*b^2*(3*C*e*(4*c*f+d*e)+f*(-A*d*f+B*c*f+4*B*d*e))-b^3*(A*d*e*f+c*(-2*A*f^2
+3*B*e*f+3*C*e^2)))*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))*d^(1/2)
*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^3/f/(-a*f+b*e)^2/(a*d-b*c)^(1/2)/(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+
b*e))^(1/2)+2/3*(-c*f+d*e)*(4*a^2*C*d*f+b^2*(A*d*f+3*C*c*e)-a*b*(B*d*f+3*C*(c*f+d*e)))*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))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e)
)^(1/2)/b^3/f/(-a*f+b*e)/d^(1/2)/(a*d-b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1628, 155, 164, 115, 114, 122, 121} \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\frac {2 (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x} \sqrt {a d-b c} (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt {a+b x} (b e-a f)^2}+\frac {2 \sqrt {d} \sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c f (3 B e-2 A f)+A d e f+3 c C e^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt {c+d x} \sqrt {a d-b c} (b e-a f)^2 \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 (c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]

[In]

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]

[Out]

(-2*(4*a^2*C*f + b^2*(3*B*e - 2*A*f) - a*b*(6*C*e + B*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b^2*(b*e - a*f)^2*Sq
rt[a + b*x]) - (2*(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
)^(3/2)) + (2*Sqrt[d]*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f
*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]
*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])
/(3*b^3*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*(d*e - c*f)*(4*
a^2*C*d*f + b^2*(3*c*C*e + A*d*f) - a*b*(B*d*f + 3*C*(d*e + c*f)))*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))])/(3*b^3*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*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 155

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] && 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 1628

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] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \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)^{3/2}}-\frac {2 \int \frac {\sqrt {c+d x} \left (-\frac {a^2 C (3 d e+c f)+b^2 (3 B c e-2 A c f)-a b (3 c C e+3 B d e+B c f-3 A d f)}{2 b}+\frac {1}{2} \left (a B d f-\frac {4 a^2 C d f}{b}+3 a C (d e+c f)-b (3 c C e+A d f)\right ) x\right )}{(a+b x)^{3/2} \sqrt {e+f x}} \, dx}{3 (b c-a d) (b e-a f)} \\ & = -\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \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)^{3/2}}-\frac {4 \int \frac {\frac {4 a^3 C d f (d e+c f)-b^3 c e (3 c C e+3 B d e-A d f)+a b^2 \left (6 c^2 C e f+d^2 e (3 B e-2 A f)+c d \left (9 C e^2+2 B e f+A f^2\right )\right )-a^2 b \left (B d f (d e+c f)+C \left (6 d^2 e^2+11 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac {d \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) x}{4 b}}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 b (b c-a d) (b e-a f)^2} \\ & = -\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \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)^{3/2}}-\frac {\left ((d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 b^2 (b c-a d) f (b e-a f)}-\frac {\left (d \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 b^2 (b c-a d) f (b e-a f)^2} \\ & = -\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \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)^{3/2}}-\frac {\left ((d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\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}{3 b^2 (b c-a d) f (b e-a f) \sqrt {c+d x}}-\frac {\left (d \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d 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}{3 b^2 (b c-a d) f (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \\ & = -\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \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)^{3/2}}+\frac {2 \sqrt {d} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d 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 )}{3 b^3 \sqrt {-b c+a d} f (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {\left ((d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (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}}\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}{3 b^2 (b c-a d) f (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \\ & = -\frac {2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \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)^{3/2}}+\frac {2 \sqrt {d} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d 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 )}{3 b^3 \sqrt {-b c+a d} f (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 (d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (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}} 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 )}{3 b^3 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.13 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} f (c+d x) (e+f x) \left (\left (A b^2+a (-b B+a C)\right ) (b c-a d) (b e-a f)+\left (-5 a^3 C d f+b^3 (3 B c e+A d e-2 A c f)-a b^2 (6 c C e+4 B d e+B c f-A d f)+a^2 b (7 C d e+4 c C f+2 B d f)\right ) (a+b x)\right )+(a+b x) \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) (c+d x) (e+f x)+i (b c-a d) f \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+i b (b c-a d) f (d e-c f) \left (-4 a^2 C f+b^2 (-3 B e+2 A f)+a b (6 C e+B f)\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^4 \sqrt {-a+\frac {b c}{d}} (b c-a d) f (b e-a f)^2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \]

[In]

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]

[Out]

(-2*(b^2*Sqrt[-a + (b*c)/d]*f*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)*(b*e - a*f) + (-5*a^
3*C*d*f + b^3*(3*B*c*e + A*d*e - 2*A*c*f) - a*b^2*(6*c*C*e + 4*B*d*e + B*c*f - A*d*f) + a^2*b*(7*C*d*e + 4*c*C
*f + 2*B*d*f))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2
*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f -
A*d*f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(
3*c*C*e^2 + A*d*e*f + c*f*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)))*(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*(d*e - c*f)*(-4*a^2*C*f + b^2*(-3*
B*e + 2*A*f) + a*b*(6*C*e + B*f))*(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)])))/(3*b^4*Sqrt
[-a + (b*c)/d]*(b*c - a*d)*f*(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1268\) vs. \(2(543)=1086\).

Time = 4.20 (sec) , antiderivative size = 1269, normalized size of antiderivative = 2.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1269\)
default	\(\text {Expression too large to display}\)	\(15367\)

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(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)*(2/3*(A*b^2-B*a*b+C*a^2)/b^4/(a*f-b*
e)*(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)/(x+a/b)^2+2/3*(b*d*f*x^2+b*c*
f*x+b*d*e*x+b*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^3*(A*a*b^2*d*f-2*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*
b^2*c*f-4*B*a*b^2*d*e+3*B*b^3*c*e-5*C*a^3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(a*f-b*e)/((x+a/b)*(b
*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e))^(1/2)+2*((B*b*d-2*C*a*d+C*b*c)/b^3+1/3*(A*b^2-B*a*b+C*a^2)/b^3*d*f/(a*f-b*e)-
1/3/b^3*(a*d*f-b*c*f-b*d*e)*(A*a*b^2*d*f-2*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-4*B*a*b^2*d*e+3*B*b^3
*c*e-5*C*a^3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/(a*f-b*e)-1/3*(b
*c*f+b*d*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^3*(A*a*b^2*d*f-2*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f
-4*B*a*b^2*d*e+3*B*b^3*c*e-5*C*a^3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(a*f-b*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*
(C*d/b^2-1/3*d*f/b^2*(A*a*b^2*d*f-2*A*b^3*c*f+A*b^3*d*e+2*B*a^2*b*d*f-B*a*b^2*c*f-4*B*a*b^2*d*e+3*B*b^3*c*e-5*
C*a^3*d*f+4*C*a^2*b*c*f+7*C*a^2*b*d*e-6*C*a*b^2*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/(a*f-b*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 = 0.31 (sec) , antiderivative size = 2429, normalized size of antiderivative = 4.07 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\text {Too large to display} \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

2/9*(3*(((5*C*a^2*b^4 - 2*B*a*b^5 - A*b^6)*c*d - 3*(2*C*a^3*b^3 - B*a^2*b^4)*d^2)*e*f^2 - (3*(C*a^3*b^3 - A*a*
b^5)*c*d - (4*C*a^4*b^2 - B*a^3*b^3 - 2*A*a^2*b^4)*d^2)*f^3 + ((3*(2*C*a*b^5 - B*b^6)*c*d - (7*C*a^2*b^4 - 4*B
*a*b^5 + A*b^6)*d^2)*e*f^2 - ((4*C*a^2*b^4 - B*a*b^5 - 2*A*b^6)*c*d - (5*C*a^3*b^3 - 2*B*a^2*b^4 - A*a*b^5)*d^
2)*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) - (3*(C*a^2*b^4*c*d - C*a^3*b^3*d^2)*e^3 - (6*C*a^2*b^4*c
^2 - 3*(5*C*a^3*b^3 - 2*B*a^2*b^4)*c*d + (8*C*a^4*b^2 - 5*B*a^3*b^3 - A*a^2*b^4)*d^2)*e^2*f + (3*(2*C*a^3*b^3
+ B*a^2*b^4)*c^2 - (25*C*a^4*b^2 - 4*B*a^3*b^3 - 2*A*a^2*b^4)*c*d + (17*C*a^5*b - 5*B*a^4*b^2 - 4*A*a^3*b^3)*d
^2)*e*f^2 - ((2*C*a^4*b^2 + B*a^3*b^3 + 2*A*a^2*b^4)*c^2 - (11*C*a^5*b - 2*B*a^4*b^2 + 2*A*a^3*b^3)*c*d + (8*C
*a^6 - 2*B*a^5*b - A*a^4*b^2)*d^2)*f^3 + (3*(C*b^6*c*d - C*a*b^5*d^2)*e^3 - (6*C*b^6*c^2 - 3*(5*C*a*b^5 - 2*B*
b^6)*c*d + (8*C*a^2*b^4 - 5*B*a*b^5 - A*b^6)*d^2)*e^2*f + (3*(2*C*a*b^5 + B*b^6)*c^2 - (25*C*a^2*b^4 - 4*B*a*b
^5 - 2*A*b^6)*c*d + (17*C*a^3*b^3 - 5*B*a^2*b^4 - 4*A*a*b^5)*d^2)*e*f^2 - ((2*C*a^2*b^4 + B*a*b^5 + 2*A*b^6)*c
^2 - (11*C*a^3*b^3 - 2*B*a^2*b^4 + 2*A*a*b^5)*c*d + (8*C*a^4*b^2 - 2*B*a^3*b^3 - A*a^2*b^4)*d^2)*f^3)*x^2 + 2*
(3*(C*a*b^5*c*d - C*a^2*b^4*d^2)*e^3 - (6*C*a*b^5*c^2 - 3*(5*C*a^2*b^4 - 2*B*a*b^5)*c*d + (8*C*a^3*b^3 - 5*B*a
^2*b^4 - A*a*b^5)*d^2)*e^2*f + (3*(2*C*a^2*b^4 + B*a*b^5)*c^2 - (25*C*a^3*b^3 - 4*B*a^2*b^4 - 2*A*a*b^5)*c*d +
(17*C*a^4*b^2 - 5*B*a^3*b^3 - 4*A*a^2*b^4)*d^2)*e*f^2 - ((2*C*a^3*b^3 + B*a^2*b^4 + 2*A*a*b^5)*c^2 - (11*C*a^
4*b^2 - 2*B*a^3*b^3 + 2*A*a^2*b^4)*c*d + (8*C*a^5*b - 2*B*a^4*b^2 - A*a^3*b^3)*d^2)*f^3)*x)*sqrt(b*d*f)*weiers
trassPInverse(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*(3*(C*a^2*b^4*c*d - C*a^3*b^3*d^2)*e^2*f - (3*(4*C*a^3*b^3 - B*a^2*b^4)*c*d - (13*C*a^4*b^2 - 4
*B*a^3*b^3 + A*a^2*b^4)*d^2)*e*f^2 + ((7*C*a^4*b^2 - B*a^3*b^3 - 2*A*a^2*b^4)*c*d - (8*C*a^5*b - 2*B*a^4*b^2 -
A*a^3*b^3)*d^2)*f^3 + (3*(C*b^6*c*d - C*a*b^5*d^2)*e^2*f - (3*(4*C*a*b^5 - B*b^6)*c*d - (13*C*a^2*b^4 - 4*B*a
*b^5 + A*b^6)*d^2)*e*f^2 + ((7*C*a^2*b^4 - B*a*b^5 - 2*A*b^6)*c*d - (8*C*a^3*b^3 - 2*B*a^2*b^4 - A*a*b^5)*d^2)
*f^3)*x^2 + 2*(3*(C*a*b^5*c*d - C*a^2*b^4*d^2)*e^2*f - (3*(4*C*a^2*b^4 - B*a*b^5)*c*d - (13*C*a^3*b^3 - 4*B*a^
2*b^4 + A*a*b^5)*d^2)*e*f^2 + ((7*C*a^3*b^3 - B*a^2*b^4 - 2*A*a*b^5)*c*d - (8*C*a^4*b^2 - 2*B*a^3*b^3 - A*a^2*
b^4)*d^2)*f^3)*x)*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), weiers
trassPInverse(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))))/((a^2*b^7*c*d - a^3*b^6*d^2)*e^2*f^2 - 2*(a^3*b^6*c*d - a^4*b^5*d^2)*e*f^3 + (a^4*b^5*c*d - a^5*b
^4*d^2)*f^4 + ((b^9*c*d - a*b^8*d^2)*e^2*f^2 - 2*(a*b^8*c*d - a^2*b^7*d^2)*e*f^3 + (a^2*b^7*c*d - a^3*b^6*d^2)
*f^4)*x^2 + 2*((a*b^8*c*d - a^2*b^7*d^2)*e^2*f^2 - 2*(a^2*b^7*c*d - a^3*b^6*d^2)*e*f^3 + (a^3*b^6*c*d - a^4*b^
5*d^2)*f^4)*x)

Sympy [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {e + f x}}\, dx \]

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(5/2)/(f*x+e)**(1/2),x)

[Out]

Integral(sqrt(c + d*x)*(A + B*x + C*x**2)/((a + b*x)**(5/2)*sqrt(e + f*x)), x)

Maxima [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {f x + e}} \,d x } \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)), x)

Giac [F]

\[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {f x + e}} \,d x } \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{5/2}} \,d x \]

[In]

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^(5/2)),x)

[Out]

int(((c + d*x)^(1/2)*(A + B*x + C*x^2))/((e + f*x)^(1/2)*(a + b*x)^(5/2)), x)