Integrand size = 32, antiderivative size = 201 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=\frac {2 (b C d-b c D-2 a d D) \sqrt {c+d x}}{b^3 d^2}-\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) \sqrt {c+d x}}{(b c-a d) (a+b x)}+\frac {2 D (c+d x)^{3/2}}{3 b^2 d^2}-\frac {\left (b^3 (2 B c-A d)-a b^2 (4 c C+B d)-5 a^3 d D+3 a^2 b (C d+2 c D)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{7/2} (b c-a d)^{3/2}} \]
2/3*D*(d*x+c)^(3/2)/b^2/d^2-(b^3*(-A*d+2*B*c)-a*b^2*(B*d+4*C*c)-5*a^3*d*D+ 3*a^2*b*(C*d+2*D*c))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(7/ 2)/(-a*d+b*c)^(3/2)+2*(C*b*d-2*D*a*d-D*b*c)*(d*x+c)^(1/2)/b^3/d^2-(A-a*(B* b^2-C*a*b+D*a^2)/b^3)*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)
Time = 0.59 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x} \left (-15 a^3 d^2 D+a^2 b d (9 C d+8 c D-10 d D x)+b^3 \left (3 A d^2-2 c x (3 C d-2 c D+d D x)\right )+a b^2 \left (4 c^2 D-6 c d (C-D x)+d^2 \left (-3 B+6 C x+2 D x^2\right )\right )\right )}{3 b^3 d^2 (b c-a d) (a+b x)}-\frac {\left (b^3 (2 B c-A d)-a b^2 (4 c C+B d)-5 a^3 d D+3 a^2 b (C d+2 c D)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{7/2} (-b c+a d)^{3/2}} \]
-1/3*(Sqrt[c + d*x]*(-15*a^3*d^2*D + a^2*b*d*(9*C*d + 8*c*D - 10*d*D*x) + b^3*(3*A*d^2 - 2*c*x*(3*C*d - 2*c*D + d*D*x)) + a*b^2*(4*c^2*D - 6*c*d*(C - D*x) + d^2*(-3*B + 6*C*x + 2*D*x^2))))/(b^3*d^2*(b*c - a*d)*(a + b*x)) - ((b^3*(2*B*c - A*d) - a*b^2*(4*c*C + B*d) - 5*a^3*d*D + 3*a^2*b*(C*d + 2* c*D))*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(b^(7/2)*(-(b*c) + a*d)^(3/2))
Time = 0.71 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2124, 27, 1192, 1467, 2009}
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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {2 \left (c-\frac {a d}{b}\right ) D x^2+\frac {2 (b c-a d) (b C-a D) x}{b^2}+\frac {-d D a^3+b (C d+2 c D) a^2-b^2 (2 c C+B d) a+b^3 (2 B c-A d)}{b^3}}{2 (a+b x) \sqrt {c+d x}}dx}{b c-a d}-\frac {\sqrt {c+d x} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\frac {d D a^3}{b^3}+\frac {(C d+2 c D) a^2}{b^2}-\frac {(2 c C+B d) a}{b}+2 \left (c-\frac {a d}{b}\right ) D x^2+2 B c-A d+\frac {2 (b c-a d) (b C-a D) x}{b^2}}{(a+b x) \sqrt {c+d x}}dx}{2 (b c-a d)}-\frac {\sqrt {c+d x} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\int \frac {-2 D c^3+2 C d c^2-2 B d^2 c-2 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+\frac {d^3 \left (A b^3+a \left (D a^2-b C a+b^2 B\right )\right )}{b^3}-\frac {2 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{d^2 (b c-a d)}-\frac {\sqrt {c+d x} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {\int \left (\frac {2 (b c-a d) (b C d-2 a D d-b c D)}{b^3}+\frac {2 (b c-a d) D (c+d x)}{b^2}+\frac {A d^3 b^3-2 B c d^2 b^3+a B d^3 b^2+4 a c C d^2 b^2-3 a^2 C d^3 b-6 a^2 c d^2 D b+5 a^3 d^3 D}{b^3 (b c-a d-b (c+d x))}\right )d\sqrt {c+d x}}{d^2 (b c-a d)}-\frac {\sqrt {c+d x} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{b^{7/2} \sqrt {b c-a d}}+\frac {2 \sqrt {c+d x} (b c-a d) (-2 a d D-b c D+b C d)}{b^3}+\frac {2 D (c+d x)^{3/2} (b c-a d)}{3 b^2}}{d^2 (b c-a d)}-\frac {\sqrt {c+d x} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
-(((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*Sqrt[c + d*x])/((b*c - a*d)*(a + b*x))) + ((2*(b*c - a*d)*(b*C*d - b*c*D - 2*a*d*D)*Sqrt[c + d*x])/b^3 + (2 *(b*c - a*d)*D*(c + d*x)^(3/2))/(3*b^2) - (d^2*(b^3*(2*B*c - A*d) - a*b^2* (4*c*C + B*d) - 5*a^3*d*D + 3*a^2*b*(C*d + 2*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(7/2)*Sqrt[b*c - a*d]))/(d^2*(b*c - a*d))
3.1.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : > With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px , a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - a*d))), x] + Simp[1/((m + 1)*(b*c - a*d)) Int[(a + b*x)^(m + 1)*(c + d*x )^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] || ! ILtQ[n, -1])
Time = 1.77 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (\frac {D \left (d x +c \right )^{\frac {3}{2}} b}{3}+d b C \sqrt {d x +c}-2 D a d \sqrt {d x +c}-D c b \sqrt {d x +c}\right )}{b^{3}}+\frac {2 d^{2} \left (\frac {d \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (A \,b^{3} d +B a \,b^{2} d -2 B \,b^{3} c -3 C \,a^{2} b d +4 C a \,b^{2} c +5 a^{3} d D-6 D a^{2} b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{b^{3}}}{d^{2}}\) | \(226\) |
| default | \(\frac {\frac {2 \left (\frac {D \left (d x +c \right )^{\frac {3}{2}} b}{3}+d b C \sqrt {d x +c}-2 D a d \sqrt {d x +c}-D c b \sqrt {d x +c}\right )}{b^{3}}+\frac {2 d^{2} \left (\frac {d \left (b^{3} A -a \,b^{2} B +C \,a^{2} b -D a^{3}\right ) \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (A \,b^{3} d +B a \,b^{2} d -2 B \,b^{3} c -3 C \,a^{2} b d +4 C a \,b^{2} c +5 a^{3} d D-6 D a^{2} b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{b^{3}}}{d^{2}}\) | \(226\) |
| pseudoelliptic | \(\frac {\left (\left (A d -2 B c \right ) b^{3}+a \,b^{2} \left (B d +4 C c \right )-3 a^{2} b \left (C d +2 D c \right )+5 a^{3} d D\right ) \left (b x +a \right ) d^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\, \left (\left (A \,d^{2}-2 x c \left (\frac {D x}{3}+C \right ) d +\frac {4 D c^{2} x}{3}\right ) b^{3}-a \left (\left (-\frac {2}{3} D x^{2}-2 C x +B \right ) d^{2}+2 c \left (-D x +C \right ) d -\frac {4 D c^{2}}{3}\right ) b^{2}+3 a^{2} \left (\left (-\frac {10 D x}{9}+C \right ) d +\frac {8 D c}{9}\right ) d b -5 a^{3} d^{2} D\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{2} b^{3} \left (a d -b c \right ) \left (b x +a \right )}\) | \(233\) |
2/d^2*(1/b^3*(1/3*D*(d*x+c)^(3/2)*b+d*b*C*(d*x+c)^(1/2)-2*D*a*d*(d*x+c)^(1 /2)-D*c*b*(d*x+c)^(1/2))+d^2/b^3*(1/2*d*(A*b^3-B*a*b^2+C*a^2*b-D*a^3)/(a*d -b*c)*(d*x+c)^(1/2)/((d*x+c)*b+a*d-b*c)+1/2*(A*b^3*d+B*a*b^2*d-2*B*b^3*c-3 *C*a^2*b*d+4*C*a*b^2*c+5*D*a^3*d-6*D*a^2*b*c)/(a*d-b*c)/((a*d-b*c)*b)^(1/2 )*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (182) = 364\).
Time = 0.28 (sec) , antiderivative size = 1004, normalized size of antiderivative = 5.00 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, D a^{4} - 3 \, C a^{3} b + B a^{2} b^{2} + A a b^{3}\right )} d^{3} - 2 \, {\left (3 \, D a^{3} b c - {\left (2 \, C a^{2} b^{2} - B a b^{3}\right )} c\right )} d^{2} + {\left ({\left (5 \, D a^{3} b - 3 \, C a^{2} b^{2} + B a b^{3} + A b^{4}\right )} d^{3} - 2 \, {\left (3 \, D a^{2} b^{2} c - {\left (2 \, C a b^{3} - B b^{4}\right )} c\right )} d^{2}\right )} x\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (4 \, D a b^{4} c^{3} + 3 \, {\left (5 \, D a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} d^{3} - {\left (23 \, D a^{3} b^{2} c - 3 \, {\left (5 \, C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c\right )} d^{2} - 2 \, {\left (D b^{5} c^{2} d - 2 \, D a b^{4} c d^{2} + D a^{2} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (2 \, D a^{2} b^{3} c^{2} - 3 \, C a b^{4} c^{2}\right )} d + 2 \, {\left (2 \, D b^{5} c^{3} + {\left (5 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3}\right )} d^{3} - 2 \, {\left (4 \, D a^{2} b^{3} c - 3 \, C a b^{4} c\right )} d^{2} + {\left (D a b^{4} c^{2} - 3 \, C b^{5} c^{2}\right )} d\right )} x\right )} \sqrt {d x + c}}{6 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x\right )}}, -\frac {3 \, {\left ({\left (5 \, D a^{4} - 3 \, C a^{3} b + B a^{2} b^{2} + A a b^{3}\right )} d^{3} - 2 \, {\left (3 \, D a^{3} b c - {\left (2 \, C a^{2} b^{2} - B a b^{3}\right )} c\right )} d^{2} + {\left ({\left (5 \, D a^{3} b - 3 \, C a^{2} b^{2} + B a b^{3} + A b^{4}\right )} d^{3} - 2 \, {\left (3 \, D a^{2} b^{2} c - {\left (2 \, C a b^{3} - B b^{4}\right )} c\right )} d^{2}\right )} x\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (4 \, D a b^{4} c^{3} + 3 \, {\left (5 \, D a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} d^{3} - {\left (23 \, D a^{3} b^{2} c - 3 \, {\left (5 \, C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c\right )} d^{2} - 2 \, {\left (D b^{5} c^{2} d - 2 \, D a b^{4} c d^{2} + D a^{2} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (2 \, D a^{2} b^{3} c^{2} - 3 \, C a b^{4} c^{2}\right )} d + 2 \, {\left (2 \, D b^{5} c^{3} + {\left (5 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3}\right )} d^{3} - 2 \, {\left (4 \, D a^{2} b^{3} c - 3 \, C a b^{4} c\right )} d^{2} + {\left (D a b^{4} c^{2} - 3 \, C b^{5} c^{2}\right )} d\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (a b^{6} c^{2} d^{2} - 2 \, a^{2} b^{5} c d^{3} + a^{3} b^{4} d^{4} + {\left (b^{7} c^{2} d^{2} - 2 \, a b^{6} c d^{3} + a^{2} b^{5} d^{4}\right )} x\right )}}\right ] \]
[-1/6*(3*((5*D*a^4 - 3*C*a^3*b + B*a^2*b^2 + A*a*b^3)*d^3 - 2*(3*D*a^3*b*c - (2*C*a^2*b^2 - B*a*b^3)*c)*d^2 + ((5*D*a^3*b - 3*C*a^2*b^2 + B*a*b^3 + A*b^4)*d^3 - 2*(3*D*a^2*b^2*c - (2*C*a*b^3 - B*b^4)*c)*d^2)*x)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/( b*x + a)) + 2*(4*D*a*b^4*c^3 + 3*(5*D*a^4*b - 3*C*a^3*b^2 + B*a^2*b^3 - A* a*b^4)*d^3 - (23*D*a^3*b^2*c - 3*(5*C*a^2*b^3 - B*a*b^4 + A*b^5)*c)*d^2 - 2*(D*b^5*c^2*d - 2*D*a*b^4*c*d^2 + D*a^2*b^3*d^3)*x^2 + 2*(2*D*a^2*b^3*c^2 - 3*C*a*b^4*c^2)*d + 2*(2*D*b^5*c^3 + (5*D*a^3*b^2 - 3*C*a^2*b^3)*d^3 - 2 *(4*D*a^2*b^3*c - 3*C*a*b^4*c)*d^2 + (D*a*b^4*c^2 - 3*C*b^5*c^2)*d)*x)*sqr t(d*x + c))/(a*b^6*c^2*d^2 - 2*a^2*b^5*c*d^3 + a^3*b^4*d^4 + (b^7*c^2*d^2 - 2*a*b^6*c*d^3 + a^2*b^5*d^4)*x), -1/3*(3*((5*D*a^4 - 3*C*a^3*b + B*a^2*b ^2 + A*a*b^3)*d^3 - 2*(3*D*a^3*b*c - (2*C*a^2*b^2 - B*a*b^3)*c)*d^2 + ((5* D*a^3*b - 3*C*a^2*b^2 + B*a*b^3 + A*b^4)*d^3 - 2*(3*D*a^2*b^2*c - (2*C*a*b ^3 - B*b^4)*c)*d^2)*x)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sq rt(d*x + c)/(b*d*x + b*c)) + (4*D*a*b^4*c^3 + 3*(5*D*a^4*b - 3*C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*d^3 - (23*D*a^3*b^2*c - 3*(5*C*a^2*b^3 - B*a*b^4 + A *b^5)*c)*d^2 - 2*(D*b^5*c^2*d - 2*D*a*b^4*c*d^2 + D*a^2*b^3*d^3)*x^2 + 2*( 2*D*a^2*b^3*c^2 - 3*C*a*b^4*c^2)*d + 2*(2*D*b^5*c^3 + (5*D*a^3*b^2 - 3*C*a ^2*b^3)*d^3 - 2*(4*D*a^2*b^3*c - 3*C*a*b^4*c)*d^2 + (D*a*b^4*c^2 - 3*C*b^5 *c^2)*d)*x)*sqrt(d*x + c))/(a*b^6*c^2*d^2 - 2*a^2*b^5*c*d^3 + a^3*b^4*d...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.35 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=\frac {{\left (6 \, D a^{2} b c - 4 \, C a b^{2} c + 2 \, B b^{3} c - 5 \, D a^{3} d + 3 \, C a^{2} b d - B a b^{2} d - A b^{3} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{4} c - a b^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {\sqrt {d x + c} D a^{3} d - \sqrt {d x + c} C a^{2} b d + \sqrt {d x + c} B a b^{2} d - \sqrt {d x + c} A b^{3} d}{{\left (b^{4} c - a b^{3} d\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} D b^{4} d^{4} - 3 \, \sqrt {d x + c} D b^{4} c d^{4} - 6 \, \sqrt {d x + c} D a b^{3} d^{5} + 3 \, \sqrt {d x + c} C b^{4} d^{5}\right )}}{3 \, b^{6} d^{6}} \]
(6*D*a^2*b*c - 4*C*a*b^2*c + 2*B*b^3*c - 5*D*a^3*d + 3*C*a^2*b*d - B*a*b^2 *d - A*b^3*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c - a*b^3 *d)*sqrt(-b^2*c + a*b*d)) + (sqrt(d*x + c)*D*a^3*d - sqrt(d*x + c)*C*a^2*b *d + sqrt(d*x + c)*B*a*b^2*d - sqrt(d*x + c)*A*b^3*d)/((b^4*c - a*b^3*d)*( (d*x + c)*b - b*c + a*d)) + 2/3*((d*x + c)^(3/2)*D*b^4*d^4 - 3*sqrt(d*x + c)*D*b^4*c*d^4 - 6*sqrt(d*x + c)*D*a*b^3*d^5 + 3*sqrt(d*x + c)*C*b^4*d^5)/ (b^6*d^6)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^2 \sqrt {c+d x}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^2\,\sqrt {c+d\,x}} \,d x \]