\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx\) [30]

Optimal result

Integrand size = 51, antiderivative size = 310 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {2 (3 d D e-3 C d f+2 c D f) \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f^2 \sqrt {c+d x}}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}-\frac {\sqrt {2} \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {c} d^3 (d e-c f)}-\frac {2 \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c^2-b d^2 x^2}}{\sqrt {b} \sqrt {d e+c f} \sqrt {c+d x}}\right )}{\sqrt {b} f^{5/2} (d e-c f) \sqrt {d e+c f}} \] Output:

2/3*(-3*C*d*f+2*D*c*f+3*D*d*e)*(-b*d^2*x^2+b*c^2)^(1/2)/b/d^3/f^2/(d*x+c)^ 
(1/2)-2/3*D*(d*x+c)^(1/2)*(-b*d^2*x^2+b*c^2)^(1/2)/b/d^3/f-2^(1/2)*(A*d^3- 
B*c*d^2+C*c^2*d-D*c^3)*arctanh(1/2*(-b*d^2*x^2+b*c^2)^(1/2)*2^(1/2)/b^(1/2 
)/c^(1/2)/(d*x+c)^(1/2))/b^(1/2)/c^(1/2)/d^3/(-c*f+d*e)-2*(D*e^3-f*(C*e^2- 
f*(-A*f+B*e)))*arctanh(f^(1/2)*(-b*d^2*x^2+b*c^2)^(1/2)/b^(1/2)/(c*f+d*e)^ 
(1/2)/(d*x+c)^(1/2))/b^(1/2)/f^(5/2)/(-c*f+d*e)/(c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (\frac {2 \sqrt {c^2-d^2 x^2} (c D f+d (3 D e-3 C f-D f x))}{d^3 f^2 \sqrt {c+d x}}+\frac {6 \left (D e^3-f \left (C e^2+f (-B e+A f)\right )\right ) \arctan \left (\frac {\sqrt {-d e-c f} \sqrt {c^2-d^2 x^2}}{\sqrt {f} (-c+d x) \sqrt {c+d x}}\right )}{f^{5/2} \sqrt {-d e-c f} (d e-c f)}-\frac {3 \sqrt {2} \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {c} d^3 (-d e+c f)}\right )}{3 \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*((2*Sqrt[c^2 - d^2*x^2]*(c*D*f + d*(3*D*e - 3*C*f - D 
*f*x)))/(d^3*f^2*Sqrt[c + d*x]) + (6*(D*e^3 - f*(C*e^2 + f*(-(B*e) + A*f)) 
)*ArcTan[(Sqrt[-(d*e) - c*f]*Sqrt[c^2 - d^2*x^2])/(Sqrt[f]*(-c + d*x)*Sqrt 
[c + d*x])])/(f^(5/2)*Sqrt[-(d*e) - c*f]*(d*e - c*f)) - (3*Sqrt[2]*(-(c^2* 
C*d) + B*c*d^2 - A*d^3 + c^3*D)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sq 
rt[c^2 - d^2*x^2]])/(Sqrt[c]*d^3*(-(d*e) + c*f))))/(3*Sqrt[b*(c^2 - d^2*x^ 
2)])
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.33, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2349, 718, 97, 73, 221, 2170, 27, 600, 458, 471, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx\)

\(\Big \downarrow \) 2349

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx-\frac {\left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \int \frac {1}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}}dx}{f^3}\)

\(\Big \downarrow \) 718

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \int \frac {1}{(c+d x) \sqrt {b c-b d x} (e+f x)}dx}{f^3 \sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 97

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {d \int \frac {1}{(c+d x) \sqrt {b c-b d x}}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b c-b d x} (e+f x)}dx}{d e-c f}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 f \int \frac {1}{e+\frac {c f}{d}-\frac {f (b c-b d x)}{b d}}d\sqrt {b c-b d x}}{b d (d e-c f)}-\frac {2 \int \frac {1}{2 c-\frac {b c-b d x}{b}}d\sqrt {b c-b d x}}{b (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \int \frac {\frac {D e^2}{f^3}-\frac {C e}{f^2}+\frac {D x^2}{f}+\left (\frac {C}{f}-\frac {D e}{f^2}\right ) x+\frac {B}{f}}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}\)

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {b d^2 \left (3 \left (D e^2-f (C e-B f)\right ) d^2-f (3 d D e-3 C d f+2 c D f) x d+c^2 D f^2\right )}{2 f^3 \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx}{3 b d^4}-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 \left (D e^2-f (C e-B f)\right ) d^2-f (3 d D e-3 C d f+2 c D f) x d+c^2 D f^2}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx}{3 d^2 f^3}-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {3 \left (d^2 \left (D e^2-f (C e-B f)\right )+c^2 D f^2+c d f (D e-C f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx-f (2 c D f-3 C d f+3 d D e) \int \frac {\sqrt {c+d x}}{\sqrt {b c^2-b d^2 x^2}}dx}{3 d^2 f^3}-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}\)

\(\Big \downarrow \) 458

\(\displaystyle \frac {3 \left (d^2 \left (D e^2-f (C e-B f)\right )+c^2 D f^2+c d f (D e-C f)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}dx+\frac {2 f \sqrt {b c^2-b d^2 x^2} (2 c D f-3 C d f+3 d D e)}{b d \sqrt {c+d x}}}{3 d^2 f^3}-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {6 d \left (d^2 \left (D e^2-f (C e-B f)\right )+c^2 D f^2+c d f (D e-C f)\right ) \int \frac {1}{\frac {d^2 \left (b c^2-b d^2 x^2\right )}{c+d x}-2 b c d^2}d\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {c+d x}}+\frac {2 f \sqrt {b c^2-b d^2 x^2} (2 c D f-3 C d f+3 d D e)}{b d \sqrt {c+d x}}}{3 d^2 f^3}-\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (D e^3-f \left (C e^2-f (B e-A f)\right )\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{f^3 \sqrt {b c^2-b d^2 x^2}}+\frac {\frac {2 f \sqrt {b c^2-b d^2 x^2} (2 c D f-3 C d f+3 d D e)}{b d \sqrt {c+d x}}-\frac {3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right ) \left (d^2 \left (D e^2-f (C e-B f)\right )+c^2 D f^2+c d f (D e-C f)\right )}{\sqrt {b} \sqrt {c} d}}{3 d^2 f^3}-\frac {2 D \sqrt {c+d x} \sqrt {b c^2-b d^2 x^2}}{3 b d^3 f}\)

Input:

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

Output:

(-2*D*Sqrt[c + d*x]*Sqrt[b*c^2 - b*d^2*x^2])/(3*b*d^3*f) - ((D*e^3 - f*(C* 
e^2 - f*(B*e - A*f)))*Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*(-((Sqrt[2]*ArcTanh[ 
Sqrt[b*c - b*d*x]/(Sqrt[2]*Sqrt[b]*Sqrt[c])])/(Sqrt[b]*Sqrt[c]*(d*e - c*f) 
)) + (2*Sqrt[f]*ArcTanh[(Sqrt[f]*Sqrt[b*c - b*d*x])/(Sqrt[b]*Sqrt[d*e + c* 
f])])/(Sqrt[b]*(d*e - c*f)*Sqrt[d*e + c*f])))/(f^3*Sqrt[b*c^2 - b*d^2*x^2] 
) + ((2*f*(3*d*D*e - 3*C*d*f + 2*c*D*f)*Sqrt[b*c^2 - b*d^2*x^2])/(b*d*Sqrt 
[c + d*x]) - (3*Sqrt[2]*(c^2*D*f^2 + c*d*f*(D*e - C*f) + d^2*(D*e^2 - f*(C 
*e - B*f)))*ArcTanh[Sqrt[b*c^2 - b*d^2*x^2]/(Sqrt[2]*Sqrt[b]*Sqrt[c]*Sqrt[ 
c + d*x])])/(Sqrt[b]*Sqrt[c]*d))/(3*d^2*f^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 221

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

rule 458

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c 
, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
 

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 718

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]* 
(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/ 
e)*x)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 
 0]
 

rule 2170

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + 
 p + q)*(d + e*x)^(q - 2)*(a*e - b*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 
 0]] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 
0] &&  !IGtQ[m, 0]
 

rule 2349

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d 
*x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c 
+ d*x, x]   Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] &&  !IntegerQ[n 
] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(668\) vs. \(2(267)=534\).

Time = 0.20 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.16

method	result	size
default	\(-\frac {\sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (6 A \,\operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) \sqrt {b c}\, b \,d^{3} f^{3}-3 A \sqrt {b \left (c f +d e \right ) f}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) b \,d^{3} f^{2}-6 B \,\operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) \sqrt {b c}\, b \,d^{3} e \,f^{2}+3 B \sqrt {b \left (c f +d e \right ) f}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) b c \,d^{2} f^{2}+6 C \,\operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) \sqrt {b c}\, b \,d^{3} e^{2} f -3 C \sqrt {b \left (c f +d e \right ) f}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) b \,c^{2} d \,f^{2}-6 D \,\operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) \sqrt {b c}\, b \,d^{3} e^{3}+3 D \sqrt {b \left (c f +d e \right ) f}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) b \,c^{3} f^{2}+2 D c d \,f^{2} x \sqrt {b c}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {\left (-d x +c \right ) b}-2 D d^{2} e f x \sqrt {b c}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {\left (-d x +c \right ) b}+6 C \sqrt {\left (-d x +c \right ) b}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {b c}\, c d \,f^{2}-6 C \sqrt {\left (-d x +c \right ) b}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {b c}\, d^{2} e f -2 D c^{2} f^{2} \sqrt {b c}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {\left (-d x +c \right ) b}-4 D \sqrt {\left (-d x +c \right ) b}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {b c}\, c d e f +6 D \sqrt {\left (-d x +c \right ) b}\, \sqrt {b \left (c f +d e \right ) f}\, \sqrt {b c}\, d^{2} e^{2}\right )}{3 b \sqrt {d x +c}\, \sqrt {\left (-d x +c \right ) b}\, d^{3} f^{2} \left (c f -d e \right ) \sqrt {b \left (c f +d e \right ) f}\, \sqrt {b c}}\)	\(669\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(-b*d^2*x^2+b*c^2)^(1/2),x,m 
ethod=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-1/3*(b*(-d^2*x^2+c^2))^(1/2)/b*(6*A*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+ 
d*e)*f)^(1/2))*(b*c)^(1/2)*b*d^3*f^3-3*A*(b*(c*f+d*e)*f)^(1/2)*2^(1/2)*arc 
tanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*b*d^3*f^2-6*B*arctanh(f*( 
(-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*d^3*e*f^2+3*B*(b*(c 
*f+d*e)*f)^(1/2)*2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2 
))*b*c*d^2*f^2+6*C*arctanh(f*((-d*x+c)*b)^(1/2)/(b*(c*f+d*e)*f)^(1/2))*(b* 
c)^(1/2)*b*d^3*e^2*f-3*C*(b*(c*f+d*e)*f)^(1/2)*2^(1/2)*arctanh(1/2*((-d*x+ 
c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*b*c^2*d*f^2-6*D*arctanh(f*((-d*x+c)*b)^(1 
/2)/(b*(c*f+d*e)*f)^(1/2))*(b*c)^(1/2)*b*d^3*e^3+3*D*(b*(c*f+d*e)*f)^(1/2) 
*2^(1/2)*arctanh(1/2*((-d*x+c)*b)^(1/2)*2^(1/2)/(b*c)^(1/2))*b*c^3*f^2+2*D 
*c*d*f^2*x*(b*c)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*((-d*x+c)*b)^(1/2)-2*D*d^2*e* 
f*x*(b*c)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*((-d*x+c)*b)^(1/2)+6*C*((-d*x+c)*b)^ 
(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*c*d*f^2-6*C*((-d*x+c)*b)^(1/2)*(b* 
(c*f+d*e)*f)^(1/2)*(b*c)^(1/2)*d^2*e*f-2*D*c^2*f^2*(b*c)^(1/2)*(b*(c*f+d*e 
)*f)^(1/2)*((-d*x+c)*b)^(1/2)-4*D*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2) 
*(b*c)^(1/2)*c*d*e*f+6*D*((-d*x+c)*b)^(1/2)*(b*(c*f+d*e)*f)^(1/2)*(b*c)^(1 
/2)*d^2*e^2)/(d*x+c)^(1/2)/((-d*x+c)*b)^(1/2)/d^3/f^2/(c*f-d*e)/(b*(c*f+d* 
e)*f)^(1/2)/(b*c)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (266) = 532\).

Time = 18.25 (sec) , antiderivative size = 2365, normalized size of antiderivative = 7.63 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/6*(3*sqrt(2)*((D*b*c^4*d - C*b*c^3*d^2 + B*b*c^2*d^3 - A*b*c*d^4)*e*f^ 
3 + (D*b*c^5 - C*b*c^4*d + B*b*c^3*d^2 - A*b*c^2*d^3)*f^4 + ((D*b*c^3*d^2 
- C*b*c^2*d^3 + B*b*c*d^4 - A*b*d^5)*e*f^3 + (D*b*c^4*d - C*b*c^3*d^2 + B* 
b*c^2*d^3 - A*b*c*d^4)*f^4)*x)*sqrt(1/(b*c))*log(-(d^2*x^2 - 2*c*d*x + 2*s 
qrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*c*sqrt(1/(b*c)) - 3*c^2)/(d^ 
2*x^2 + 2*c*d*x + c^2)) - 6*(D*c*d^3*e^3 - C*c*d^3*e^2*f + B*c*d^3*e*f^2 - 
 A*c*d^3*f^3 + (D*d^4*e^3 - C*d^4*e^2*f + B*d^4*e*f^2 - A*d^4*f^3)*x)*sqrt 
(b*d*e*f + b*c*f^2)*log(-(b*d^2*f*x^2 - b*c*d*e - 2*b*c^2*f - (b*d^2*e + b 
*c*d*f)*x + 2*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(b*d*e*f + b*c*f^2)*sqrt(d*x + 
c))/(d*f*x^2 + c*e + (d*e + c*f)*x)) - 4*(3*D*d^3*e^3*f - 3*D*c^2*d*e*f^3 
+ (D*c*d^2 - 3*C*d^3)*e^2*f^2 - (D*c^3 - 3*C*c^2*d)*f^4 - (D*d^3*e^2*f^2 - 
 D*c^2*d*f^4)*x)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c))/(b*c*d^5*e^2*f^3 
- b*c^3*d^3*f^5 + (b*d^6*e^2*f^3 - b*c^2*d^4*f^5)*x), 1/3*(3*sqrt(2)*((D*b 
*c^4*d - C*b*c^3*d^2 + B*b*c^2*d^3 - A*b*c*d^4)*e*f^3 + (D*b*c^5 - C*b*c^4 
*d + B*b*c^3*d^2 - A*b*c^2*d^3)*f^4 + ((D*b*c^3*d^2 - C*b*c^2*d^3 + B*b*c* 
d^4 - A*b*d^5)*e*f^3 + (D*b*c^4*d - C*b*c^3*d^2 + B*b*c^2*d^3 - A*b*c*d^4) 
*f^4)*x)*sqrt(-1/(b*c))*arctan(sqrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + 
 c)*c*sqrt(-1/(b*c))/(d^2*x^2 - c^2)) + 3*(D*c*d^3*e^3 - C*c*d^3*e^2*f + B 
*c*d^3*e*f^2 - A*c*d^3*f^3 + (D*d^4*e^3 - C*d^4*e^2*f + B*d^4*e*f^2 - A*d^ 
4*f^3)*x)*sqrt(b*d*e*f + b*c*f^2)*log(-(b*d^2*f*x^2 - b*c*d*e - 2*b*c^2...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\frac {3 \, \sqrt {2} {\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{{\left (d^{3} e - c d^{2} f\right )} \sqrt {-b c}} - \frac {6 \, {\left (D d e^{3} - C d e^{2} f + B d e f^{2} - A d f^{3}\right )} \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c} f}{\sqrt {-b d e f - b c f^{2}}}\right )}{\sqrt {-b d e f - b c f^{2}} {\left (d e f^{2} - c f^{3}\right )}} - \frac {2 \, {\left (3 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c} D b^{5} d^{5} e f - 3 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c} C b^{5} d^{5} f^{2} + {\left (-{\left (d x + c\right )} b + 2 \, b c\right )}^{\frac {3}{2}} D b^{4} d^{4} f^{2}\right )}}{b^{6} d^{6} f^{3}}}{3 \, d} \] Input:

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

Output:

-1/3*(3*sqrt(2)*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)*arctan(1/2*sqrt(2)*sqr 
t(-(d*x + c)*b + 2*b*c)/sqrt(-b*c))/((d^3*e - c*d^2*f)*sqrt(-b*c)) - 6*(D* 
d*e^3 - C*d*e^2*f + B*d*e*f^2 - A*d*f^3)*arctan(sqrt(-(d*x + c)*b + 2*b*c) 
*f/sqrt(-b*d*e*f - b*c*f^2))/(sqrt(-b*d*e*f - b*c*f^2)*(d*e*f^2 - c*f^3)) 
- 2*(3*sqrt(-(d*x + c)*b + 2*b*c)*D*b^5*d^5*e*f - 3*sqrt(-(d*x + c)*b + 2* 
b*c)*C*b^5*d^5*f^2 + (-(d*x + c)*b + 2*b*c)^(3/2)*D*b^4*d^4*f^2)/(b^6*d^6* 
f^3))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {b}\, \left (12 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) a c \,d^{2} f^{3} i -12 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) b c \,d^{2} e \,f^{2} i +12 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) c^{2} d^{2} e^{2} f i -12 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) c \,d^{3} e^{3} i -8 \sqrt {-d x +c}\, c^{4} f^{4}+12 \sqrt {-d x +c}\, c^{3} d e \,f^{3}-4 \sqrt {-d x +c}\, c^{3} d \,f^{4} x +8 \sqrt {-d x +c}\, c^{2} d^{2} e^{2} f^{2}-12 \sqrt {-d x +c}\, c \,d^{3} e^{3} f +4 \sqrt {-d x +c}\, c \,d^{3} e^{2} f^{2} x -3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a c \,d^{2} f^{4}-3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a \,d^{3} e \,f^{3}+3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d \,f^{4}+3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b c \,d^{2} e \,f^{3}+3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a c \,d^{2} f^{4}+3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a \,d^{3} e \,f^{3}-3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d \,f^{4}-3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b c \,d^{2} e \,f^{3}\right )}{6 b c \,d^{2} f^{3} \left (c^{2} f^{2}-d^{2} e^{2}\right )} \] Input:

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

Output:

(sqrt(b)*(12*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqr 
t(c*f + d*e)))*a*c*d**2*f**3*i - 12*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - 
 d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c*d**2*e*f**2*i + 12*sqrt(f)*sqrt( 
c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*c**2*d**2*e 
**2*f*i - 12*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqr 
t(c*f + d*e)))*c*d**3*e**3*i - 8*sqrt(c - d*x)*c**4*f**4 + 12*sqrt(c - d*x 
)*c**3*d*e*f**3 - 4*sqrt(c - d*x)*c**3*d*f**4*x + 8*sqrt(c - d*x)*c**2*d** 
2*e**2*f**2 - 12*sqrt(c - d*x)*c*d**3*e**3*f + 4*sqrt(c - d*x)*c*d**3*e**2 
*f**2*x - 3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*c*d**2* 
f**4 - 3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*d**3*e*f** 
3 + 3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**2*d*f**4 + 
 3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c*d**2*e*f**3 + 
3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c*d**2*f**4 + 3*s 
qrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*d**3*e*f**3 - 3*sqrt 
(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**2*d*f**4 - 3*sqrt(c) 
*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c*d**2*e*f**3))/(6*b*c*d** 
2*f**3*(c**2*f**2 - d**2*e**2))