\(\int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx\) [86]

Optimal result

Integrand size = 32, antiderivative size = 375 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^3 (b c-a d) (a+b x)^3}-\frac {\left (b^3 (6 B c-5 A d)-a b^2 (12 c C+B d)-13 a^3 d D+a^2 b (7 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^3 (b c-a d)^2 (a+b x)^2}-\frac {\left (b^3 \left (8 c^2 C-6 B c d+5 A d^2\right )-11 a^3 d^2 D+a^2 b d (C d+30 c D)-a b^2 \left (4 c C d-B d^2+24 c^2 D\right )\right ) \sqrt {c+d x}}{8 b^3 (b c-a d)^3 (a+b x)}+\frac {\left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (4 c C d-B d^2-24 c^2 D\right )+b^3 \left (8 c^2 C d-6 B c d^2+5 A d^3-16 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{7/2} (b c-a d)^{7/2}} \] Output:

-1/3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)/(b*x+a)^3- 
1/12*(b^3*(-5*A*d+6*B*c)-a*b^2*(B*d+12*C*c)-13*a^3*d*D+a^2*b*(7*C*d+18*D*c 
))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)^2/(b*x+a)^2-1/8*(b^3*(5*A*d^2-6*B*c*d+8*C* 
c^2)-11*a^3*d^2*D+a^2*b*d*(C*d+30*D*c)-a*b^2*(-B*d^2+4*C*c*d+24*D*c^2))*(d 
*x+c)^(1/2)/b^3/(-a*d+b*c)^3/(b*x+a)+1/8*(5*a^3*d^3*D+a^2*b*d^2*(C*d-18*D* 
c)-a*b^2*d*(-B*d^2+4*C*c*d-24*D*c^2)+b^3*(5*A*d^3-6*B*c*d^2+8*C*c^2*d-16*D 
*c^3))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(7/2)/(-a*d+b*c)^ 
(7/2)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x} \left (-15 a^5 d^2 D+6 b^5 c x (2 B c+4 c C x-3 B d x)+a^4 b d (-3 C d+44 c D-40 d D x)+a b^4 \left (-12 c x (-2 c C+C d x+6 c D x)+B \left (4 c^2-50 c d x+3 d^2 x^2\right )\right )+A b^3 \left (33 a^2 d^2+2 a b d (-13 c+20 d x)+b^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )-a^3 b^2 \left (44 c^2 D-2 c d (5 C+59 D x)+d^2 \left (3 B+8 C x+33 D x^2\right )\right )+a^2 b^3 \left (d^2 x (8 B+3 C x)+4 c^2 (2 C-27 D x)+2 c d \left (-8 B+7 C x+45 D x^2\right )\right )\right )}{24 b^3 (b c-a d)^3 (a+b x)^3}-\frac {\left (-5 a^3 d^3 D+a^2 b d^2 (-C d+18 c D)-a b^2 d \left (-4 c C d+B d^2+24 c^2 D\right )+b^3 \left (-8 c^2 C d+6 B c d^2-5 A d^3+16 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{7/2} (-b c+a d)^{7/2}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*Sqrt[c + d*x]),x]
 

Output:

-1/24*(Sqrt[c + d*x]*(-15*a^5*d^2*D + 6*b^5*c*x*(2*B*c + 4*c*C*x - 3*B*d*x 
) + a^4*b*d*(-3*C*d + 44*c*D - 40*d*D*x) + a*b^4*(-12*c*x*(-2*c*C + C*d*x 
+ 6*c*D*x) + B*(4*c^2 - 50*c*d*x + 3*d^2*x^2)) + A*b^3*(33*a^2*d^2 + 2*a*b 
*d*(-13*c + 20*d*x) + b^2*(8*c^2 - 10*c*d*x + 15*d^2*x^2)) - a^3*b^2*(44*c 
^2*D - 2*c*d*(5*C + 59*D*x) + d^2*(3*B + 8*C*x + 33*D*x^2)) + a^2*b^3*(d^2 
*x*(8*B + 3*C*x) + 4*c^2*(2*C - 27*D*x) + 2*c*d*(-8*B + 7*C*x + 45*D*x^2)) 
))/(b^3*(b*c - a*d)^3*(a + b*x)^3) - ((-5*a^3*d^3*D + a^2*b*d^2*(-(C*d) + 
18*c*D) - a*b^2*d*(-4*c*C*d + B*d^2 + 24*c^2*D) + b^3*(-8*c^2*C*d + 6*B*c* 
d^2 - 5*A*d^3 + 16*c^3*D))*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a* 
d]])/(8*b^(7/2)*(-(b*c) + a*d)^(7/2))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2124, 27, 1192, 1471, 27, 298, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {\int -\frac {6 \left (c-\frac {a d}{b}\right ) D x^2+\frac {6 (b c-a d) (b C-a D) x}{b^2}+\frac {-d D a^3+b (C d+6 c D) a^2-b^2 (6 c C+B d) a+b^3 (6 B c-5 A d)}{b^3}}{2 (a+b x)^3 \sqrt {c+d x}}dx}{3 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-\frac {d D a^3}{b^3}+\frac {(C d+6 c D) a^2}{b^2}-\frac {(6 c C+B d) a}{b}+6 \left (c-\frac {a d}{b}\right ) D x^2+6 B c-5 A d+\frac {6 (b c-a d) (b C-a D) x}{b^2}}{(a+b x)^3 \sqrt {c+d x}}dx}{6 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 1192

\(\displaystyle \frac {\int \frac {-6 D c^3+6 C d c^2-6 B d^2 c+5 A d^3-6 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+\frac {a d^3 \left (D a^2-b C a+b^2 B\right )}{b^3}-\frac {6 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{(b c-a d-b (c+d x))^3}d\sqrt {c+d x}}{3 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {-\frac {\int \frac {3 \left (-\left (\left (-8 D c^3+8 C d c^2-6 B d^2 c+5 A d^3\right ) b^3\right )+a d^2 (4 c C-B d) b^2-a^2 d^2 (C d+6 c D) b-8 (b c-a d)^2 D (c+d x) b+3 a^3 d^3 D\right )}{b^3 (b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 (b c-a d)}-\frac {d^2 \sqrt {c+d x} \left (-13 a^3 d D+a^2 b (18 c D+7 C d)-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right )}{4 b^3 (b c-a d) (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {-\left (\left (-8 D c^3+8 C d c^2-6 B d^2 c+5 A d^3\right ) b^3\right )+a d^2 (4 c C-B d) b^2-a^2 d^2 (C d+6 c D) b-8 (b c-a d)^2 D (c+d x) b+3 a^3 d^3 D}{(b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 b^3 (b c-a d)}-\frac {d^2 \sqrt {c+d x} \left (-13 a^3 d D+a^2 b (18 c D+7 C d)-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right )}{4 b^3 (b c-a d) (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {-\frac {3 \left (-\frac {\left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (-B d^2-24 c^2 D+4 c C d\right )+b^3 \left (5 A d^3-6 B c d^2-16 c^3 D+8 c^2 C d\right )\right ) \int \frac {1}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{2 (b c-a d)}-\frac {d \sqrt {c+d x} \left (-11 a^3 d^2 D+a^2 b d (30 c D+C d)-a b^2 \left (-B d^2+24 c^2 D+4 c C d\right )+b^3 \left (5 A d^2-6 B c d+8 c^2 C\right )\right )}{2 (b c-a d) (-a d-b (c+d x)+b c)}\right )}{4 b^3 (b c-a d)}-\frac {d^2 \sqrt {c+d x} \left (-13 a^3 d D+a^2 b (18 c D+7 C d)-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right )}{4 b^3 (b c-a d) (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {3 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (5 a^3 d^3 D+a^2 b d^2 (C d-18 c D)-a b^2 d \left (-B d^2-24 c^2 D+4 c C d\right )+b^3 \left (5 A d^3-6 B c d^2-16 c^3 D+8 c^2 C d\right )\right )}{2 \sqrt {b} (b c-a d)^{3/2}}-\frac {d \sqrt {c+d x} \left (-11 a^3 d^2 D+a^2 b d (30 c D+C d)-a b^2 \left (-B d^2+24 c^2 D+4 c C d\right )+b^3 \left (5 A d^2-6 B c d+8 c^2 C\right )\right )}{2 (b c-a d) (-a d-b (c+d x)+b c)}\right )}{4 b^3 (b c-a d)}-\frac {d^2 \sqrt {c+d x} \left (-13 a^3 d D+a^2 b (18 c D+7 C d)-a b^2 (B d+12 c C)+b^3 (6 B c-5 A d)\right )}{4 b^3 (b c-a d) (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)^4*Sqrt[c + d*x]),x]
 

Output:

-1/3*((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(b^3*(b*c - a*d)* 
(a + b*x)^3) + (-1/4*(d^2*(b^3*(6*B*c - 5*A*d) - a*b^2*(12*c*C + B*d) - 13 
*a^3*d*D + a^2*b*(7*C*d + 18*c*D))*Sqrt[c + d*x])/(b^3*(b*c - a*d)*(b*c - 
a*d - b*(c + d*x))^2) - (3*(-1/2*(d*(b^3*(8*c^2*C - 6*B*c*d + 5*A*d^2) - 1 
1*a^3*d^2*D + a^2*b*d*(C*d + 30*c*D) - a*b^2*(4*c*C*d - B*d^2 + 24*c^2*D)) 
*Sqrt[c + d*x])/((b*c - a*d)*(b*c - a*d - b*(c + d*x))) - ((5*a^3*d^3*D + 
a^2*b*d^2*(C*d - 18*c*D) - a*b^2*d*(4*c*C*d - B*d^2 - 24*c^2*D) + b^3*(8*c 
^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 16*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/ 
Sqrt[b*c - a*d]])/(2*Sqrt[b]*(b*c - a*d)^(3/2))))/(4*b^3*(b*c - a*d)))/(3* 
(b*c - a*d))
 

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 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 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 1192

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]
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], 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] && LtQ[q, -1]
 

rule 2124

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])
 

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {\frac {5 \left (\left (A \,d^{3}-\frac {6}{5} B c \,d^{2}+\frac {8}{5} C \,c^{2} d -\frac {16}{5} D c^{3}\right ) b^{3}+\frac {a d \left (B \,d^{2}-4 C c d +24 D c^{2}\right ) b^{2}}{5}+\frac {a^{2} b \,d^{2} \left (C d -18 D c \right )}{5}+a^{3} d^{3} D\right ) \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8}+\frac {11 \sqrt {\left (a d -b c \right ) b}\, \left (\left (\frac {5 A \,d^{2} x^{2}}{11}-\frac {10 x \left (\frac {9 B x}{5}+A \right ) c d}{33}+\frac {8 c^{2} \left (3 C \,x^{2}+\frac {3}{2} B x +A \right )}{33}\right ) b^{5}-\frac {26 \left (-\frac {20 x \left (\frac {3 B x}{40}+A \right ) d^{2}}{13}+c \left (\frac {6}{13} C \,x^{2}+\frac {25}{13} B x +A \right ) d -\frac {2 c^{2} \left (-18 D x^{2}+6 C x +B \right )}{13}\right ) a \,b^{4}}{33}+\left (\left (\frac {1}{11} C \,x^{2}+\frac {8}{33} B x +A \right ) d^{2}-\frac {16 \left (-\frac {45}{8} D x^{2}-\frac {7}{8} C x +B \right ) c d}{33}+\frac {8 \left (-\frac {27 D x}{2}+C \right ) c^{2}}{33}\right ) a^{2} b^{3}-\frac {\left (\left (11 D x^{2}+\frac {8}{3} C x +B \right ) d^{2}-\frac {10 c \left (\frac {59 D x}{5}+C \right ) d}{3}+\frac {44 D c^{2}}{3}\right ) a^{3} b^{2}}{11}-\frac {d \left (\left (\frac {40 D x}{3}+C \right ) d -\frac {44 D c}{3}\right ) a^{4} b}{11}-\frac {5 D a^{5} d^{2}}{11}\right ) \sqrt {x d +c}}{8}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{3} \left (a d -b c \right )^{3} b^{3}}\)	\(382\)
derivativedivides	\(\frac {\frac {d \left (5 b^{3} d^{2} A +B a \,b^{2} d^{2}-6 B \,b^{3} c d +C \,a^{2} b \,d^{2}-4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-11 a^{3} d^{2} D+30 a^{2} b c d D-24 a \,b^{2} c^{2} D\right ) \left (x d +c \right )^{\frac {5}{2}}}{8 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (5 b^{3} d^{2} A +B a \,b^{2} d^{2}-6 B \,b^{3} c d -C \,a^{2} b \,d^{2}+6 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 a^{2} b c d D-18 a \,b^{2} c^{2} D\right ) d \left (x d +c \right )^{\frac {3}{2}}}{3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (11 b^{3} d^{2} A -B a \,b^{2} d^{2}-10 B \,b^{3} c d -C \,a^{2} b \,d^{2}+4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 a^{2} b c d D-24 a \,b^{2} c^{2} D\right ) d \sqrt {x d +c}}{8 b^{3} \left (a d -b c \right )}}{\left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\left (5 A \,b^{3} d^{3}+B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}+a^{2} b C \,d^{3}-4 C a \,b^{2} c \,d^{2}+8 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-18 D a^{2} b c \,d^{2}+24 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\)	\(548\)
default	\(\frac {\frac {d \left (5 b^{3} d^{2} A +B a \,b^{2} d^{2}-6 B \,b^{3} c d +C \,a^{2} b \,d^{2}-4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-11 a^{3} d^{2} D+30 a^{2} b c d D-24 a \,b^{2} c^{2} D\right ) \left (x d +c \right )^{\frac {5}{2}}}{8 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (5 b^{3} d^{2} A +B a \,b^{2} d^{2}-6 B \,b^{3} c d -C \,a^{2} b \,d^{2}+6 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 a^{2} b c d D-18 a \,b^{2} c^{2} D\right ) d \left (x d +c \right )^{\frac {3}{2}}}{3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (11 b^{3} d^{2} A -B a \,b^{2} d^{2}-10 B \,b^{3} c d -C \,a^{2} b \,d^{2}+4 C a \,b^{2} c d +8 C \,b^{3} c^{2}-5 a^{3} d^{2} D+18 a^{2} b c d D-24 a \,b^{2} c^{2} D\right ) d \sqrt {x d +c}}{8 b^{3} \left (a d -b c \right )}}{\left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\left (5 A \,b^{3} d^{3}+B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}+a^{2} b C \,d^{3}-4 C a \,b^{2} c \,d^{2}+8 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-18 D a^{2} b c \,d^{2}+24 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (a d -b c \right ) b}}\)	\(548\)

Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

11/8*(5/11*((A*d^3-6/5*B*c*d^2+8/5*C*c^2*d-16/5*D*c^3)*b^3+1/5*a*d*(B*d^2- 
4*C*c*d+24*D*c^2)*b^2+1/5*a^2*b*d^2*(C*d-18*D*c)+a^3*d^3*D)*(b*x+a)^3*arct 
an(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*((5/11*A*d^2*x 
^2-10/33*x*(9/5*B*x+A)*c*d+8/33*c^2*(3*C*x^2+3/2*B*x+A))*b^5-26/33*(-20/13 
*x*(3/40*B*x+A)*d^2+c*(6/13*C*x^2+25/13*B*x+A)*d-2/13*c^2*(-18*D*x^2+6*C*x 
+B))*a*b^4+((1/11*C*x^2+8/33*B*x+A)*d^2-16/33*(-45/8*D*x^2-7/8*C*x+B)*c*d+ 
8/33*(-27/2*D*x+C)*c^2)*a^2*b^3-1/11*((11*D*x^2+8/3*C*x+B)*d^2-10/3*c*(59/ 
5*D*x+C)*d+44/3*D*c^2)*a^3*b^2-1/11*d*((40/3*D*x+C)*d-44/3*D*c)*a^4*b-5/11 
*D*a^5*d^2)*(d*x+c)^(1/2))/((a*d-b*c)*b)^(1/2)/(b*x+a)^3/(a*d-b*c)^3/b^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (354) = 708\).

Time = 0.17 (sec) , antiderivative size = 2338, normalized size of antiderivative = 6.23 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2),x, algorithm="fricas 
")
 

Output:

[1/48*(3*(16*D*a^3*b^3*c^3 - 8*(3*D*a^4*b^2 + C*a^3*b^3)*c^2*d + 2*(9*D*a^ 
5*b + 2*C*a^4*b^2 + 3*B*a^3*b^3)*c*d^2 - (5*D*a^6 + C*a^5*b + B*a^4*b^2 + 
5*A*a^3*b^3)*d^3 + (16*D*b^6*c^3 - 8*(3*D*a*b^5 + C*b^6)*c^2*d + 2*(9*D*a^ 
2*b^4 + 2*C*a*b^5 + 3*B*b^6)*c*d^2 - (5*D*a^3*b^3 + C*a^2*b^4 + B*a*b^5 + 
5*A*b^6)*d^3)*x^3 + 3*(16*D*a*b^5*c^3 - 8*(3*D*a^2*b^4 + C*a*b^5)*c^2*d + 
2*(9*D*a^3*b^3 + 2*C*a^2*b^4 + 3*B*a*b^5)*c*d^2 - (5*D*a^4*b^2 + C*a^3*b^3 
 + B*a^2*b^4 + 5*A*a*b^5)*d^3)*x^2 + 3*(16*D*a^2*b^4*c^3 - 8*(3*D*a^3*b^3 
+ C*a^2*b^4)*c^2*d + 2*(9*D*a^4*b^2 + 2*C*a^3*b^3 + 3*B*a^2*b^4)*c*d^2 - ( 
5*D*a^5*b + C*a^4*b^2 + B*a^3*b^3 + 5*A*a^2*b^4)*d^3)*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*(11*D*a^3*b^4 - 2*C*a^2*b^5 - B*a*b^6 - 2*A*b^7)*c^3 - 2*(44*D* 
a^4*b^3 + C*a^3*b^4 - 10*B*a^2*b^5 - 17*A*a*b^6)*c^2*d + (59*D*a^5*b^2 + 1 
3*C*a^4*b^3 - 13*B*a^3*b^4 - 59*A*a^2*b^5)*c*d^2 - 3*(5*D*a^6*b + C*a^5*b^ 
2 + B*a^4*b^3 - 11*A*a^3*b^4)*d^3 + 3*(8*(3*D*a*b^6 - C*b^7)*c^3 - 6*(9*D* 
a^2*b^5 - 2*C*a*b^6 - B*b^7)*c^2*d + (41*D*a^3*b^4 - 5*C*a^2*b^5 - 7*B*a*b 
^6 - 5*A*b^7)*c*d^2 - (11*D*a^4*b^3 - C*a^3*b^4 - B*a^2*b^5 - 5*A*a*b^6)*d 
^3)*x^2 + 2*(6*(9*D*a^2*b^5 - 2*C*a*b^6 - B*b^7)*c^3 - (113*D*a^3*b^4 - 5* 
C*a^2*b^5 - 31*B*a*b^6 - 5*A*b^7)*c^2*d + (79*D*a^4*b^3 + 11*C*a^3*b^4 - 2 
9*B*a^2*b^5 - 25*A*a*b^6)*c*d^2 - 4*(5*D*a^5*b^2 + C*a^4*b^3 - B*a^3*b^4 - 
 5*A*a^2*b^5)*d^3)*x)*sqrt(d*x + c))/(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**4/(d*x+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2),x, algorithm="maxima 
")
 

Output:

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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 976 vs. \(2 (354) = 708\).

Time = 0.14 (sec) , antiderivative size = 976, normalized size of antiderivative = 2.60 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

1/8*(16*D*b^3*c^3 - 24*D*a*b^2*c^2*d - 8*C*b^3*c^2*d + 18*D*a^2*b*c*d^2 + 
4*C*a*b^2*c*d^2 + 6*B*b^3*c*d^2 - 5*D*a^3*d^3 - C*a^2*b*d^3 - B*a*b^2*d^3 
- 5*A*b^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^6*c^3 - 3* 
a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*sqrt(-b^2*c + a*b*d)) + 1/24* 
(72*(d*x + c)^(5/2)*D*a*b^4*c^2*d - 24*(d*x + c)^(5/2)*C*b^5*c^2*d - 144*( 
d*x + c)^(3/2)*D*a*b^4*c^3*d + 48*(d*x + c)^(3/2)*C*b^5*c^3*d + 72*sqrt(d* 
x + c)*D*a*b^4*c^4*d - 24*sqrt(d*x + c)*C*b^5*c^4*d - 90*(d*x + c)^(5/2)*D 
*a^2*b^3*c*d^2 + 12*(d*x + c)^(5/2)*C*a*b^4*c*d^2 + 18*(d*x + c)^(5/2)*B*b 
^5*c*d^2 + 288*(d*x + c)^(3/2)*D*a^2*b^3*c^2*d^2 - 48*(d*x + c)^(3/2)*C*a* 
b^4*c^2*d^2 - 48*(d*x + c)^(3/2)*B*b^5*c^2*d^2 - 198*sqrt(d*x + c)*D*a^2*b 
^3*c^3*d^2 + 36*sqrt(d*x + c)*C*a*b^4*c^3*d^2 + 30*sqrt(d*x + c)*B*b^5*c^3 
*d^2 + 33*(d*x + c)^(5/2)*D*a^3*b^2*d^3 - 3*(d*x + c)^(5/2)*C*a^2*b^3*d^3 
- 3*(d*x + c)^(5/2)*B*a*b^4*d^3 - 15*(d*x + c)^(5/2)*A*b^5*d^3 - 184*(d*x 
+ c)^(3/2)*D*a^3*b^2*c*d^3 - 8*(d*x + c)^(3/2)*C*a^2*b^3*c*d^3 + 56*(d*x + 
 c)^(3/2)*B*a*b^4*c*d^3 + 40*(d*x + c)^(3/2)*A*b^5*c*d^3 + 195*sqrt(d*x + 
c)*D*a^3*b^2*c^2*d^3 + 3*sqrt(d*x + c)*C*a^2*b^3*c^2*d^3 - 57*sqrt(d*x + c 
)*B*a*b^4*c^2*d^3 - 33*sqrt(d*x + c)*A*b^5*c^2*d^3 + 40*(d*x + c)^(3/2)*D* 
a^4*b*d^4 + 8*(d*x + c)^(3/2)*C*a^3*b^2*d^4 - 8*(d*x + c)^(3/2)*B*a^2*b^3* 
d^4 - 40*(d*x + c)^(3/2)*A*a*b^4*d^4 - 84*sqrt(d*x + c)*D*a^4*b*c*d^4 - 18 
*sqrt(d*x + c)*C*a^3*b^2*c*d^4 + 24*sqrt(d*x + c)*B*a^2*b^3*c*d^4 + 66*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^4\,\sqrt {c+d\,x}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^4*(c + d*x)^(1/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)^4*(c + d*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1305, normalized size of antiderivative = 3.48 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2),x)
 

Output:

(15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c 
)))*a**5*d**3 - 36*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b) 
*sqrt(a*d - b*c)))*a**4*b*c*d**2 + 45*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c 
 + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**4*b*d**3*x + 18*sqrt(b)*sqrt(a*d 
- b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**3*d**2 + 
24*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c) 
))*a**3*b**2*c**2*d - 108*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/( 
sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*c*d**2*x + 45*sqrt(b)*sqrt(a*d - b*c)* 
atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*d**3*x**2 + 54 
*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c))) 
*a**2*b**4*d**2*x + 72*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(a*d - b*c)))*a**2*b**3*c**2*d*x - 108*sqrt(b)*sqrt(a*d - b*c)*at 
an((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*c*d**2*x**2 + 15 
*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c))) 
*a**2*b**3*d**3*x**3 + 54*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/( 
sqrt(b)*sqrt(a*d - b*c)))*a*b**5*d**2*x**2 + 72*sqrt(b)*sqrt(a*d - b*c)*at 
an((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**4*c**2*d*x**2 - 36*sq 
rt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a* 
b**4*c*d**2*x**3 + 18*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt 
(b)*sqrt(a*d - b*c)))*b**6*d**2*x**3 + 24*sqrt(b)*sqrt(a*d - b*c)*atan(...