\(\int \frac {\sqrt {c+d x} (A+B x+C x^2+D x^3)}{(a+b x)^4} \, dx\) [62]

Optimal result

Integrand size = 32, antiderivative size = 380 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\frac {2 D \sqrt {c+d x}}{b^4}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{3 b^4 (a+b x)^3}-\frac {\left (b^3 (6 B c+A d)-a b^2 (12 c C+7 B d)-19 a^3 d D+a^2 b (13 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^4 (b c-a d) (a+b x)^2}-\frac {\left (b^3 \left (8 c^2 C+2 B c d-A d^2\right )-29 a^3 d^2 D+a^2 b d (11 C d+54 c D)-a b^2 \left (20 c C d+B d^2+24 c^2 D\right )\right ) \sqrt {c+d x}}{8 b^4 (b c-a d)^2 (a+b x)}+\frac {\left (35 a^3 d^3 D-5 a^2 b d^2 (C d+18 c D)+a b^2 d \left (12 c C d-B d^2+72 c^2 D\right )-b^3 \left (8 c^2 C d-2 B c d^2+A d^3+16 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{9/2} (b c-a d)^{5/2}} \] Output:

2*D*(d*x+c)^(1/2)/b^4-1/3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(1/2)/b^4/ 
(b*x+a)^3-1/12*(b^3*(A*d+6*B*c)-a*b^2*(7*B*d+12*C*c)-19*a^3*d*D+a^2*b*(13* 
C*d+18*D*c))*(d*x+c)^(1/2)/b^4/(-a*d+b*c)/(b*x+a)^2-1/8*(b^3*(-A*d^2+2*B*c 
*d+8*C*c^2)-29*a^3*d^2*D+a^2*b*d*(11*C*d+54*D*c)-a*b^2*(B*d^2+20*C*c*d+24* 
D*c^2))*(d*x+c)^(1/2)/b^4/(-a*d+b*c)^2/(b*x+a)+1/8*(35*a^3*d^3*D-5*a^2*b*d 
^2*(C*d+18*D*c)+a*b^2*d*(-B*d^2+12*C*c*d+72*D*c^2)-b^3*(A*d^3-2*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^(9/2) 
/(-a*d+b*c)^(5/2)
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\frac {\sqrt {c+d x} \left (105 a^5 d^2 D-5 a^4 b d (3 C d+8 D (5 c-7 d x))-6 b^5 c x (B (2 c+d x)+4 c x (C-2 D x))+A b^3 \left (-3 a^2 d^2+2 a b d (7 c+4 d x)+b^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )+a^3 b^2 \left (92 c^2 D+c (26 C d-538 d D x)+d^2 \left (-3 B-40 C x+231 D x^2\right )\right )+a^2 b^3 \left (c^2 (-8 C+252 D x)+d^2 x \left (-8 B-33 C x+48 D x^2\right )+2 c d (2 B+5 x (7 C-45 D x))\right )+a b^4 \left (B \left (-4 c^2+14 c d x+3 d^2 x^2\right )+12 c x (-2 c (C-9 D x)+d x (5 C-8 D x))\right )\right )}{24 b^4 (b c-a d)^2 (a+b x)^3}+\frac {\left (-35 a^3 d^3 D+5 a^2 b d^2 (C d+18 c D)+a b^2 d \left (-12 c C d+B d^2-72 c^2 D\right )+b^3 \left (8 c^2 C d-2 B c d^2+A d^3+16 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{9/2} (-b c+a d)^{5/2}} \] Input:

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

Output:

(Sqrt[c + d*x]*(105*a^5*d^2*D - 5*a^4*b*d*(3*C*d + 8*D*(5*c - 7*d*x)) - 6* 
b^5*c*x*(B*(2*c + d*x) + 4*c*x*(C - 2*D*x)) + A*b^3*(-3*a^2*d^2 + 2*a*b*d* 
(7*c + 4*d*x) + b^2*(-8*c^2 - 2*c*d*x + 3*d^2*x^2)) + a^3*b^2*(92*c^2*D + 
c*(26*C*d - 538*d*D*x) + d^2*(-3*B - 40*C*x + 231*D*x^2)) + a^2*b^3*(c^2*( 
-8*C + 252*D*x) + d^2*x*(-8*B - 33*C*x + 48*D*x^2) + 2*c*d*(2*B + 5*x*(7*C 
 - 45*D*x))) + a*b^4*(B*(-4*c^2 + 14*c*d*x + 3*d^2*x^2) + 12*c*x*(-2*c*(C 
- 9*D*x) + d*x*(5*C - 8*D*x)))))/(24*b^4*(b*c - a*d)^2*(a + b*x)^3) + ((-3 
5*a^3*d^3*D + 5*a^2*b*d^2*(C*d + 18*c*D) + a*b^2*d*(-12*c*C*d + B*d^2 - 72 
*c^2*D) + b^3*(8*c^2*C*d - 2*B*c*d^2 + A*d^3 + 16*c^3*D))*ArcTan[(Sqrt[b]* 
Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(8*b^(9/2)*(-(b*c) + a*d)^(5/2))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.18, 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, 1580, 1471, 299, 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.

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1192

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

\(\Big \downarrow \) 1580

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

\(\Big \downarrow \) 1471

\(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-29 a^3 d^2 D+a^2 b d (54 c D+11 C d)-a b^2 \left (B d^2+24 c^2 D+20 c C d\right )+b^3 \left (-A d^2+2 B c d+8 c^2 C\right )\right )}{2 b (b c-a d) (-a d-b (c+d x)+b c)}-\frac {\int \frac {16 D (c+d x) (b c-a d)^2+d \left (-\frac {19 d^2 D a^3}{b}+d (5 C d+42 c D) a^2-b \left (24 D c^2+12 C d c-B d^2\right ) a+b^2 \left (8 C c^2-2 B d c+A d^2\right )\right )}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{2 (b c-a d)}}{4 b^3}-\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{4 b^4 (-a d-b (c+d x)+b c)^2}}{b c-a d}-\frac {(c+d x)^{3/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-29 a^3 d^2 D+a^2 b d (54 c D+11 C d)-a b^2 \left (B d^2+24 c^2 D+20 c C d\right )+b^3 \left (-A d^2+2 B c d+8 c^2 C\right )\right )}{2 b (b c-a d) (-a d-b (c+d x)+b c)}-\frac {-\frac {\left (35 a^3 d^3 D-5 a^2 b d^2 (18 c D+C d)+a b^2 d \left (-B d^2+72 c^2 D+12 c C d\right )-\left (b^3 \left (A d^3-2 B c d^2+16 c^3 D+8 c^2 C d\right )\right )\right ) \int \frac {1}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{b}-\frac {16 D \sqrt {c+d x} (b c-a d)^2}{b}}{2 (b c-a d)}}{4 b^3}-\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{4 b^4 (-a d-b (c+d x)+b c)^2}}{b c-a d}-\frac {(c+d x)^{3/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-29 a^3 d^2 D+a^2 b d (54 c D+11 C d)-a b^2 \left (B d^2+24 c^2 D+20 c C d\right )+b^3 \left (-A d^2+2 B c d+8 c^2 C\right )\right )}{2 b (b c-a d) (-a d-b (c+d x)+b c)}-\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (35 a^3 d^3 D-5 a^2 b d^2 (18 c D+C d)+a b^2 d \left (-B d^2+72 c^2 D+12 c C d\right )-\left (b^3 \left (A d^3-2 B c d^2+16 c^3 D+8 c^2 C d\right )\right )\right )}{b^{3/2} \sqrt {b c-a d}}-\frac {16 D \sqrt {c+d x} (b c-a d)^2}{b}}{2 (b c-a d)}}{4 b^3}-\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+3 a^2 b (2 c D+C d)-a b^2 (B d+4 c C)+b^3 (2 B c-A d)\right )}{4 b^4 (-a d-b (c+d x)+b c)^2}}{b c-a d}-\frac {(c+d x)^{3/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\)

Input:

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

Output:

-1/3*((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(3/2))/(b^3*(b*c - a*d 
)*(a + b*x)^3) + (-1/4*(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))*Sqrt[c + d*x])/(b^4*(b*c - a*d - b*(c + d* 
x))^2) + ((d*(b^3*(8*c^2*C + 2*B*c*d - A*d^2) - 29*a^3*d^2*D + a^2*b*d*(11 
*C*d + 54*c*D) - a*b^2*(20*c*C*d + B*d^2 + 24*c^2*D))*Sqrt[c + d*x])/(2*b* 
(b*c - a*d)*(b*c - a*d - b*(c + d*x))) - ((-16*(b*c - a*d)^2*D*Sqrt[c + d* 
x])/b - ((35*a^3*d^3*D - 5*a^2*b*d^2*(C*d + 18*c*D) + a*b^2*d*(12*c*C*d - 
B*d^2 + 72*c^2*D) - b^3*(8*c^2*C*d - 2*B*c*d^2 + A*d^3 + 16*c^3*D))*ArcTan 
h[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*Sqrt[b*c - a*d]))/(2* 
(b*c - a*d)))/(4*b^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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 1580

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) 
^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d 
 + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* 
(q + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* 
e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b 
*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e 
}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
 

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 = 398, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(-\frac {-\left (b^{3} \left (A \,d^{3}-2 B c \,d^{2}+8 C \,c^{2} d +16 D c^{3}\right )+a d \left (B \,d^{2}-12 C c d -72 D c^{2}\right ) b^{2}+5 a^{2} b \,d^{2} \left (C d +18 D c \right )-35 a^{3} d^{3} D\right ) \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (\left (-A \,d^{2} x^{2}+\frac {2 c x \left (3 B x +A \right ) d}{3}+\frac {8 c^{2} \left (-6 D x^{3}+3 C \,x^{2}+\frac {3}{2} B x +A \right )}{3}\right ) b^{5}-\frac {14 \left (\frac {4 x \left (\frac {3 B x}{8}+A \right ) d^{2}}{7}+c \left (-\frac {48}{7} D x^{3}+\frac {30}{7} C \,x^{2}+B x +A \right ) d -\frac {2 c^{2} \left (-54 D x^{2}+6 C x +B \right )}{7}\right ) a \,b^{4}}{3}+\left (\left (11 C \,x^{2}+\frac {8}{3} B x -16 D x^{3}+A \right ) d^{2}-\frac {4 c \left (-\frac {225}{2} D x^{2}+\frac {35}{2} C x +B \right ) d}{3}+\frac {8 \left (-\frac {63 D x}{2}+C \right ) c^{2}}{3}\right ) a^{2} b^{3}+\left (\left (-77 D x^{2}+\frac {40}{3} C x +B \right ) d^{2}-\frac {26 c \left (-\frac {269 D x}{13}+C \right ) d}{3}-\frac {92 D c^{2}}{3}\right ) a^{3} b^{2}+5 \left (\left (-\frac {56 D x}{3}+C \right ) d +\frac {40 D c}{3}\right ) d \,a^{4} b -35 D a^{5} d^{2}\right ) \sqrt {x d +c}}{8 \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right )^{2} b^{4} \left (b x +a \right )^{3}}\)	\(398\)
derivativedivides	\(\frac {2 D \sqrt {x d +c}}{b^{4}}+\frac {\frac {2 \left (\frac {b^{2} d \left (b^{3} d^{2} A +B a \,b^{2} d^{2}-2 B \,b^{3} c d -11 C \,a^{2} b \,d^{2}+20 C a \,b^{2} c d -8 C \,b^{3} c^{2}+29 a^{3} d^{2} D-54 a^{2} b c d D+24 a \,b^{2} c^{2} D\right ) \left (x d +c \right )^{\frac {5}{2}}}{16 a^{2} d^{2}-32 a b c d +16 b^{2} c^{2}}+\frac {\left (b^{3} d^{2} A -B a \,b^{2} d^{2}-5 C \,a^{2} b \,d^{2}+12 C a \,b^{2} c d -6 C \,b^{3} c^{2}+17 a^{3} d^{2} D-36 a^{2} b c d D+18 a \,b^{2} c^{2} D\right ) b d \left (x d +c \right )^{\frac {3}{2}}}{6 a d -6 b c}+\left (-\frac {1}{16} A \,b^{3} d^{3}-\frac {1}{16} B a \,b^{2} d^{3}+\frac {1}{8} B \,b^{3} c \,d^{2}-\frac {5}{16} a^{2} b C \,d^{3}+\frac {3}{4} C a \,b^{2} c \,d^{2}-\frac {1}{2} C \,b^{3} c^{2} d +\frac {19}{16} a^{3} d^{3} D-\frac {21}{8} D a^{2} b c \,d^{2}+\frac {3}{2} D a \,b^{2} c^{2} d \right ) \sqrt {x d +c}\right )}{\left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\left (A \,b^{3} d^{3}+B a \,b^{2} d^{3}-2 B \,b^{3} c \,d^{2}+5 a^{2} b C \,d^{3}-12 C a \,b^{2} c \,d^{2}+8 C \,b^{3} c^{2} d -35 a^{3} d^{3} D+90 D a^{2} b c \,d^{2}-72 D a \,b^{2} c^{2} d +16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}}{b^{4}}\)	\(512\)
default	\(\frac {2 D \sqrt {x d +c}}{b^{4}}+\frac {\frac {2 \left (\frac {b^{2} d \left (b^{3} d^{2} A +B a \,b^{2} d^{2}-2 B \,b^{3} c d -11 C \,a^{2} b \,d^{2}+20 C a \,b^{2} c d -8 C \,b^{3} c^{2}+29 a^{3} d^{2} D-54 a^{2} b c d D+24 a \,b^{2} c^{2} D\right ) \left (x d +c \right )^{\frac {5}{2}}}{16 a^{2} d^{2}-32 a b c d +16 b^{2} c^{2}}+\frac {\left (b^{3} d^{2} A -B a \,b^{2} d^{2}-5 C \,a^{2} b \,d^{2}+12 C a \,b^{2} c d -6 C \,b^{3} c^{2}+17 a^{3} d^{2} D-36 a^{2} b c d D+18 a \,b^{2} c^{2} D\right ) b d \left (x d +c \right )^{\frac {3}{2}}}{6 a d -6 b c}+\left (-\frac {1}{16} A \,b^{3} d^{3}-\frac {1}{16} B a \,b^{2} d^{3}+\frac {1}{8} B \,b^{3} c \,d^{2}-\frac {5}{16} a^{2} b C \,d^{3}+\frac {3}{4} C a \,b^{2} c \,d^{2}-\frac {1}{2} C \,b^{3} c^{2} d +\frac {19}{16} a^{3} d^{3} D-\frac {21}{8} D a^{2} b c \,d^{2}+\frac {3}{2} D a \,b^{2} c^{2} d \right ) \sqrt {x d +c}\right )}{\left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\left (A \,b^{3} d^{3}+B a \,b^{2} d^{3}-2 B \,b^{3} c \,d^{2}+5 a^{2} b C \,d^{3}-12 C a \,b^{2} c \,d^{2}+8 C \,b^{3} c^{2} d -35 a^{3} d^{3} D+90 D a^{2} b c \,d^{2}-72 D a \,b^{2} c^{2} d +16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}}{b^{4}}\)	\(512\)

Input:

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

Output:

-1/8*(-(b^3*(A*d^3-2*B*c*d^2+8*C*c^2*d+16*D*c^3)+a*d*(B*d^2-12*C*c*d-72*D* 
c^2)*b^2+5*a^2*b*d^2*(C*d+18*D*c)-35*a^3*d^3*D)*(b*x+a)^3*arctan(b*(d*x+c) 
^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*((-A*d^2*x^2+2/3*c*x*(3*B* 
x+A)*d+8/3*c^2*(-6*D*x^3+3*C*x^2+3/2*B*x+A))*b^5-14/3*(4/7*x*(3/8*B*x+A)*d 
^2+c*(-48/7*D*x^3+30/7*C*x^2+B*x+A)*d-2/7*c^2*(-54*D*x^2+6*C*x+B))*a*b^4+( 
(11*C*x^2+8/3*B*x-16*D*x^3+A)*d^2-4/3*c*(-225/2*D*x^2+35/2*C*x+B)*d+8/3*(- 
63/2*D*x+C)*c^2)*a^2*b^3+((-77*D*x^2+40/3*C*x+B)*d^2-26/3*c*(-269/13*D*x+C 
)*d-92/3*D*c^2)*a^3*b^2+5*((-56/3*D*x+C)*d+40/3*D*c)*d*a^4*b-35*D*a^5*d^2) 
*(d*x+c)^(1/2))/((a*d-b*c)*b)^(1/2)/(a*d-b*c)^2/b^4/(b*x+a)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (356) = 712\).

Time = 0.17 (sec) , antiderivative size = 2354, normalized size of antiderivative = 6.19 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(3*(16*D*a^3*b^3*c^3 - 8*(9*D*a^4*b^2 - C*a^3*b^3)*c^2*d + 2*(45*D*a 
^5*b - 6*C*a^4*b^2 - B*a^3*b^3)*c*d^2 - (35*D*a^6 - 5*C*a^5*b - B*a^4*b^2 
- A*a^3*b^3)*d^3 + (16*D*b^6*c^3 - 8*(9*D*a*b^5 - C*b^6)*c^2*d + 2*(45*D*a 
^2*b^4 - 6*C*a*b^5 - B*b^6)*c*d^2 - (35*D*a^3*b^3 - 5*C*a^2*b^4 - B*a*b^5 
- A*b^6)*d^3)*x^3 + 3*(16*D*a*b^5*c^3 - 8*(9*D*a^2*b^4 - C*a*b^5)*c^2*d + 
2*(45*D*a^3*b^3 - 6*C*a^2*b^4 - B*a*b^5)*c*d^2 - (35*D*a^4*b^2 - 5*C*a^3*b 
^3 - B*a^2*b^4 - A*a*b^5)*d^3)*x^2 + 3*(16*D*a^2*b^4*c^3 - 8*(9*D*a^3*b^3 
- C*a^2*b^4)*c^2*d + 2*(45*D*a^4*b^2 - 6*C*a^3*b^3 - B*a^2*b^4)*c*d^2 - (3 
5*D*a^5*b - 5*C*a^4*b^2 - B*a^3*b^3 - 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*(23*D*a^3*b^4 - 2*C*a^2*b^5 - B*a*b^6 - 2*A*b^7)*c^3 - 2*(146*D 
*a^4*b^3 - 17*C*a^3*b^4 - 4*B*a^2*b^5 - 11*A*a*b^6)*c^2*d + (305*D*a^5*b^2 
 - 41*C*a^4*b^3 - 7*B*a^3*b^4 - 17*A*a^2*b^5)*c*d^2 - 3*(35*D*a^6*b - 5*C* 
a^5*b^2 - B*a^4*b^3 - A*a^3*b^4)*d^3 + 48*(D*b^7*c^3 - 3*D*a*b^6*c^2*d + 3 
*D*a^2*b^5*c*d^2 - D*a^3*b^4*d^3)*x^3 + 3*(8*(9*D*a*b^6 - C*b^7)*c^3 - 2*( 
111*D*a^2*b^5 - 14*C*a*b^6 + B*b^7)*c^2*d + (227*D*a^3*b^4 - 31*C*a^2*b^5 
+ 3*B*a*b^6 + A*b^7)*c*d^2 - (77*D*a^4*b^3 - 11*C*a^3*b^4 + B*a^2*b^5 + A* 
a*b^6)*d^3)*x^2 + 2*(6*(21*D*a^2*b^5 - 2*C*a*b^6 - B*b^7)*c^3 - (395*D*a^3 
*b^4 - 47*C*a^2*b^5 - 13*B*a*b^6 + A*b^7)*c^2*d + (409*D*a^4*b^3 - 55*C*a^ 
3*b^4 - 11*B*a^2*b^5 + 5*A*a*b^6)*c*d^2 - 4*(35*D*a^5*b^2 - 5*C*a^4*b^3...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^4,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. 938 vs. \(2 (356) = 712\).

Time = 0.15 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.47 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(16*D*b^3*c^3 - 72*D*a*b^2*c^2*d + 8*C*b^3*c^2*d + 90*D*a^2*b*c*d^2 - 
12*C*a*b^2*c*d^2 - 2*B*b^3*c*d^2 - 35*D*a^3*d^3 + 5*C*a^2*b*d^3 + B*a*b^2* 
d^3 + A*b^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^6*c^2 - 
2*a*b^5*c*d + a^2*b^4*d^2)*sqrt(-b^2*c + a*b*d)) + 2*sqrt(d*x + c)*D/b^4 + 
 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*s 
qrt(d*x + c)*D*a*b^4*c^4*d - 24*sqrt(d*x + c)*C*b^5*c^4*d - 162*(d*x + c)^ 
(5/2)*D*a^2*b^3*c*d^2 + 60*(d*x + c)^(5/2)*C*a*b^4*c*d^2 - 6*(d*x + c)^(5/ 
2)*B*b^5*c*d^2 + 432*(d*x + c)^(3/2)*D*a^2*b^3*c^2*d^2 - 144*(d*x + c)^(3/ 
2)*C*a*b^4*c^2*d^2 - 270*sqrt(d*x + c)*D*a^2*b^3*c^3*d^2 + 84*sqrt(d*x + c 
)*C*a*b^4*c^3*d^2 + 6*sqrt(d*x + c)*B*b^5*c^3*d^2 + 87*(d*x + c)^(5/2)*D*a 
^3*b^2*d^3 - 33*(d*x + c)^(5/2)*C*a^2*b^3*d^3 + 3*(d*x + c)^(5/2)*B*a*b^4* 
d^3 + 3*(d*x + c)^(5/2)*A*b^5*d^3 - 424*(d*x + c)^(3/2)*D*a^3*b^2*c*d^3 + 
136*(d*x + c)^(3/2)*C*a^2*b^3*c*d^3 + 8*(d*x + c)^(3/2)*B*a*b^4*c*d^3 - 8* 
(d*x + c)^(3/2)*A*b^5*c*d^3 + 381*sqrt(d*x + c)*D*a^3*b^2*c^2*d^3 - 111*sq 
rt(d*x + c)*C*a^2*b^3*c^2*d^3 - 15*sqrt(d*x + c)*B*a*b^4*c^2*d^3 - 3*sqrt( 
d*x + c)*A*b^5*c^2*d^3 + 136*(d*x + c)^(3/2)*D*a^4*b*d^4 - 40*(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 + 8*(d*x + c)^(3/2)*A* 
a*b^4*d^4 - 240*sqrt(d*x + c)*D*a^4*b*c*d^4 + 66*sqrt(d*x + c)*C*a^3*b^2*c 
*d^4 + 12*sqrt(d*x + c)*B*a^2*b^3*c*d^4 + 6*sqrt(d*x + c)*A*a*b^4*c*d^4...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1279, normalized size of antiderivative = 3.37 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 105*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - 
 b*c)))*a**5*d**3 + 180*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sq 
rt(b)*sqrt(a*d - b*c)))*a**4*b*c*d**2 - 315*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 + 6*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 - 72*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 + 540*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 - 315*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 + 18*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 - 216*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)* 
b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*c**2*d*x + 540*sqrt(b)*sqrt(a*d - 
b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*c*d**2*x* 
*2 - 105*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 + 18*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 - 216*sqrt(b)*sqrt(a*d 
- b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**4*c**2*d*x** 
2 + 180*sqrt(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 + 6*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 - 72*sqrt(b)*sqrt(a*d - b...