\(\int \frac {\sqrt {c+d x} (A+B x+C x^2+D x^3)}{\sqrt {a+b x}} \, dx\) [132]

Optimal result

Integrand size = 34, antiderivative size = 393 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a+b x}} \, dx=-\frac {\left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)-a b^2 d \left (16 c C d-48 B d^2-9 c^2 D\right )-b^3 \left (8 c^2 C d-16 B c d^2+64 A d^3-5 c^3 D\right )\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^3}-\frac {\left (8 b c C-16 b B d+24 a C d-14 a c D-\frac {5 b c^2 D}{d}-\frac {29 a^2 d D}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{32 b^2 d^2}+\frac {(8 b C d-5 b c D-19 a d D) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^3 d^2}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}-\frac {(b c-a d) \left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)-a b^2 d \left (16 c C d-48 B d^2-9 c^2 D\right )-b^3 \left (8 c^2 C d-16 B c d^2+64 A d^3-5 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{7/2}} \] Output:

-1/64*(35*a^3*d^3*D-5*a^2*b*d^2*(8*C*d-3*D*c)-a*b^2*d*(-48*B*d^2+16*C*c*d- 
9*D*c^2)-b^3*(64*A*d^3-16*B*c*d^2+8*C*c^2*d-5*D*c^3))*(b*x+a)^(1/2)*(d*x+c 
)^(1/2)/b^4/d^3-1/32*(8*C*b*c-16*B*b*d+24*C*a*d-14*D*a*c-5*b*c^2*D/d-29*a^ 
2*d*D/b)*(b*x+a)^(1/2)*(d*x+c)^(3/2)/b^2/d^2+1/24*(8*C*b*d-19*D*a*d-5*D*b* 
c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b^3/d^2+1/4*D*(b*x+a)^(5/2)*(d*x+c)^(3/2)/b 
^3/d-1/64*(-a*d+b*c)*(35*a^3*d^3*D-5*a^2*b*d^2*(8*C*d-3*D*c)-a*b^2*d*(-48* 
B*d^2+16*C*c*d-9*D*c^2)-b^3*(64*A*d^3-16*B*c*d^2+8*C*c^2*d-5*D*c^3))*arcta 
nh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 11.05 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a+b x}} \, dx=\frac {\sqrt {c+d x} \left (\sqrt {d} \sqrt {a+b x} \left (-105 a^3 d^3 D+5 a^2 b d^2 (24 C d+5 c D+14 d D x)-a b^2 d \left (-17 c^2 D+4 c d (8 C+3 D x)+8 d^2 \left (18 B+10 C x+7 D x^2\right )\right )+b^3 \left (15 c^3 D-2 c^2 d (12 C+5 D x)+8 c d^2 \left (6 B+2 C x+D x^2\right )+16 d^3 \left (12 A+6 B x+4 C x^2+3 D x^3\right )\right )\right )-\frac {3 \sqrt {b c-a d} \left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)+a b^2 d \left (-16 c C d+48 B d^2+9 c^2 D\right )+b^3 \left (-8 c^2 C d+16 B c d^2-64 A d^3+5 c^3 D\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{192 b^4 d^{7/2}} \] Input:

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

Output:

(Sqrt[c + d*x]*(Sqrt[d]*Sqrt[a + b*x]*(-105*a^3*d^3*D + 5*a^2*b*d^2*(24*C* 
d + 5*c*D + 14*d*D*x) - a*b^2*d*(-17*c^2*D + 4*c*d*(8*C + 3*D*x) + 8*d^2*( 
18*B + 10*C*x + 7*D*x^2)) + b^3*(15*c^3*D - 2*c^2*d*(12*C + 5*D*x) + 8*c*d 
^2*(6*B + 2*C*x + D*x^2) + 16*d^3*(12*A + 6*B*x + 4*C*x^2 + 3*D*x^3))) - ( 
3*Sqrt[b*c - a*d]*(35*a^3*d^3*D - 5*a^2*b*d^2*(8*C*d - 3*c*D) + a*b^2*d*(- 
16*c*C*d + 48*B*d^2 + 9*c^2*D) + b^3*(-8*c^2*C*d + 16*B*c*d^2 - 64*A*d^3 + 
 5*c^3*D))*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + 
d*x))/(b*c - a*d)]))/(192*b^4*d^(7/2))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2125, 27, 1194, 27, 90, 60, 66, 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.

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

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (8 A d b^3+(8 b C d-19 a D d-5 b c D) x^2 b^2+2 \left (-7 d D a^2-5 b c D a+4 b^2 B d\right ) x b-a^2 (5 b c+3 a d) D\right )}{2 \sqrt {a+b x}}dx}{4 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (8 A d b^3+(8 b C d-19 a D d-5 b c D) x^2 b^2+2 \left (-7 d D a^2-5 b c D a+4 b^2 B d\right ) x b-a^2 (5 b c+3 a d) D\right )}{\sqrt {a+b x}}dx}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\frac {\int \frac {3 b^2 \sqrt {c+d x} \left (13 d^2 D a^3-2 b d (4 C d-7 c D) a^2-b^2 c (8 C d-5 c D) a+16 A b^3 d^2+b \left (-\left (\left (-5 D c^2+8 C d c-16 B d^2\right ) b^2\right )-2 a d (12 C d-7 c D) b+29 a^2 d^2 D\right ) x\right )}{2 \sqrt {a+b x}}dx}{3 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (-19 a d D-5 b c D+8 b C d)}{3 d}}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d x} \left (13 d^2 D a^3-2 b d (4 C d-7 c D) a^2-b^2 c (8 C d-5 c D) a+16 A b^3 d^2+b \left (-\left (\left (-5 D c^2+8 C d c-16 B d^2\right ) b^2\right )-2 a d (12 C d-7 c D) b+29 a^2 d^2 D\right ) x\right )}{\sqrt {a+b x}}dx}{2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (-19 a d D-5 b c D+8 b C d)}{3 d}}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (29 a^2 d^2 D-2 a b d (12 C d-7 c D)-\left (b^2 \left (-16 B d^2-5 c^2 D+8 c C d\right )\right )\right )}{2 d}-\frac {\left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)-a b^2 d \left (-48 B d^2-9 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3-16 B c d^2-5 c^3 D+8 c^2 C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 d}}{2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (-19 a d D-5 b c D+8 b C d)}{3 d}}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (29 a^2 d^2 D-2 a b d (12 C d-7 c D)-\left (b^2 \left (-16 B d^2-5 c^2 D+8 c C d\right )\right )\right )}{2 d}-\frac {\left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)-a b^2 d \left (-48 B d^2-9 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3-16 B c d^2-5 c^3 D+8 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 d}}{2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (-19 a d D-5 b c D+8 b C d)}{3 d}}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (29 a^2 d^2 D-2 a b d (12 C d-7 c D)-\left (b^2 \left (-16 B d^2-5 c^2 D+8 c C d\right )\right )\right )}{2 d}-\frac {\left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)-a b^2 d \left (-48 B d^2-9 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3-16 B c d^2-5 c^3 D+8 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 d}}{2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (-19 a d D-5 b c D+8 b C d)}{3 d}}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (29 a^2 d^2 D-2 a b d (12 C d-7 c D)-\left (b^2 \left (-16 B d^2-5 c^2 D+8 c C d\right )\right )\right )}{2 d}-\frac {\left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right ) \left (35 a^3 d^3 D-5 a^2 b d^2 (8 C d-3 c D)-a b^2 d \left (-48 B d^2-9 c^2 D+16 c C d\right )-\left (b^3 \left (64 A d^3-16 B c d^2-5 c^3 D+8 c^2 C d\right )\right )\right )}{4 d}}{2 d}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (-19 a d D-5 b c D+8 b C d)}{3 d}}{8 b^3 d}+\frac {D (a+b x)^{5/2} (c+d x)^{3/2}}{4 b^3 d}\)

Input:

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

Output:

(D*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*b^3*d) + (((8*b*C*d - 5*b*c*D - 19* 
a*d*D)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(3*d) + (((29*a^2*d^2*D - 2*a*b*d* 
(12*C*d - 7*c*D) - b^2*(8*c*C*d - 16*B*d^2 - 5*c^2*D))*Sqrt[a + b*x]*(c + 
d*x)^(3/2))/(2*d) - ((35*a^3*d^3*D - 5*a^2*b*d^2*(8*C*d - 3*c*D) - a*b^2*d 
*(16*c*C*d - 48*B*d^2 - 9*c^2*D) - b^3*(8*c^2*C*d - 16*B*c*d^2 + 64*A*d^3 
- 5*c^3*D))*((Sqrt[a + b*x]*Sqrt[c + d*x])/b + ((b*c - a*d)*ArcTanh[(Sqrt[ 
d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*Sqrt[d])))/(4*d))/(2* 
d))/(8*b^3*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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

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

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 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

rule 2125

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x 
)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( 
m + n + q + 1))   Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q 
+ 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) 
^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, 
 x] && PolyQ[Px, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1333\) vs. \(2(355)=710\).

Time = 0.53 (sec) , antiderivative size = 1334, normalized size of antiderivative = 3.39

method	result	size
default	\(\text {Expression too large to display}\)	\(1334\)

Input:

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

Output:

-1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(48*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c 
))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^4*c^2*d^2-24*C*ln(1/2*(2*b*d* 
x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^4*c^3*d+48 
*C*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^2*d-384*A*(d*b)^(1/2)*((b*x+a 
)*(d*x+c))^(1/2)*b^3*d^3+24*D*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^2*c* 
d^2*x+120*C*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c) 
/(d*b)^(1/2))*a^3*b*d^4+12*D*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d* 
b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^3*c^3*d+210*D*((b*x+a)*(d*x+c))^(1/2)*( 
d*b)^(1/2)*a^3*d^3-144*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^( 
1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*b^2*d^4-30*D*(d*b)^(1/2)*((b*x+a)*(d*x+c))^ 
(1/2)*b^3*c^3-240*C*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b*d^3+60*D*ln( 
1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a 
^3*b*c*d^3+18*D*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+ 
b*c)/(d*b)^(1/2))*a^2*b^2*c^2*d^2+288*B*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2 
)*a*b^2*d^3-72*C*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d 
+b*c)/(d*b)^(1/2))*a^2*b^2*c*d^3-192*B*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2) 
*b^3*d^3*x+192*A*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d 
+b*c)/(d*b)^(1/2))*a*b^3*d^4-192*A*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/ 
2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^4*c*d^3+112*D*a*b^2*d^3*x^2*((b*x+a 
)*(d*x+c))^(1/2)*(d*b)^(1/2)-16*D*b^3*c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)...
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.26 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/768*(3*(5*D*b^4*c^4 + 4*(D*a*b^3 - 2*C*b^4)*c^3*d + 2*(3*D*a^2*b^2 - 4* 
C*a*b^3 + 8*B*b^4)*c^2*d^2 + 4*(5*D*a^3*b - 6*C*a^2*b^2 + 8*B*a*b^3 - 16*A 
*b^4)*c*d^3 - (35*D*a^4 - 40*C*a^3*b + 48*B*a^2*b^2 - 64*A*a*b^3)*d^4)*sqr 
t(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b* 
c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) 
+ 4*(48*D*b^4*d^4*x^3 + 15*D*b^4*c^3*d + (17*D*a*b^3 - 24*C*b^4)*c^2*d^2 + 
 (25*D*a^2*b^2 - 32*C*a*b^3 + 48*B*b^4)*c*d^3 - 3*(35*D*a^3*b - 40*C*a^2*b 
^2 + 48*B*a*b^3 - 64*A*b^4)*d^4 + 8*(D*b^4*c*d^3 - (7*D*a*b^3 - 8*C*b^4)*d 
^4)*x^2 - 2*(5*D*b^4*c^2*d^2 + 2*(3*D*a*b^3 - 4*C*b^4)*c*d^3 - (35*D*a^2*b 
^2 - 40*C*a*b^3 + 48*B*b^4)*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^4) 
, 1/384*(3*(5*D*b^4*c^4 + 4*(D*a*b^3 - 2*C*b^4)*c^3*d + 2*(3*D*a^2*b^2 - 4 
*C*a*b^3 + 8*B*b^4)*c^2*d^2 + 4*(5*D*a^3*b - 6*C*a^2*b^2 + 8*B*a*b^3 - 16* 
A*b^4)*c*d^3 - (35*D*a^4 - 40*C*a^3*b + 48*B*a^2*b^2 - 64*A*a*b^3)*d^4)*sq 
rt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d* 
x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(48*D*b^4*d^4* 
x^3 + 15*D*b^4*c^3*d + (17*D*a*b^3 - 24*C*b^4)*c^2*d^2 + (25*D*a^2*b^2 - 3 
2*C*a*b^3 + 48*B*b^4)*c*d^3 - 3*(35*D*a^3*b - 40*C*a^2*b^2 + 48*B*a*b^3 - 
64*A*b^4)*d^4 + 8*(D*b^4*c*d^3 - (7*D*a*b^3 - 8*C*b^4)*d^4)*x^2 - 2*(5*D*b 
^4*c^2*d^2 + 2*(3*D*a*b^3 - 4*C*b^4)*c*d^3 - (35*D*a^2*b^2 - 40*C*a*b^3 + 
48*B*b^4)*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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. 710 vs. \(2 (351) = 702\).

Time = 0.22 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/192*(192*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c 
 + (b*x + a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d 
)*sqrt(b*x + a))*A*abs(b)/b^2 - (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b 
*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14 
*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) 
 + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11* 
d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2* 
c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + 
sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*D*abs(b)/b^2 - 
48*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d 
^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b* 
d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*B* 
abs(b)/b^3 - 8*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*b*x + 4*a + (b*c 
*d^3 - 13*a*d^4)/d^4)*(b*x + a) - 3*(b^2*c^2*d^2 + 2*a*b*c*d^3 - 11*a^2*d^ 
4)/d^4)*sqrt(b*x + a) - 3*(b^4*c^3 + a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - 5*a^3 
*b*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a* 
b*d)))/(sqrt(b*d)*d^2))*C*abs(b)/b^4)/b
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a+b x}} \, dx=\frac {-105 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b \,d^{4}+145 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} c \,d^{3}+70 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x +48 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} d^{3}-15 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c^{2} d^{2}-92 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c \,d^{3} x -56 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c \,d^{2}+96 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} d^{3} x -9 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{3} d +6 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{2} d^{2} x +72 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c \,d^{3} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} d^{4} x^{3}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} d^{4}-180 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b c \,d^{3}-48 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{3} d^{3}+54 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{2} c^{2} d^{2}+96 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{4} c \,d^{2}+12 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{3} c^{3} d -48 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{5} c^{2} d +9 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{4} c^{4}}{192 b^{5} d^{3}} \] Input:

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

Output:

( - 105*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*d**4 + 145*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**2*b**2*c*d**3 + 70*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*d**4*x 
 + 48*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*d**3 - 15*sqrt(c + d*x)*sqrt(a + 
b*x)*a*b**3*c**2*d**2 - 92*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c*d**3*x - 5 
6*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*d**4*x**2 + 48*sqrt(c + d*x)*sqrt(a + 
 b*x)*b**5*c*d**2 + 96*sqrt(c + d*x)*sqrt(a + b*x)*b**5*d**3*x - 9*sqrt(c 
+ d*x)*sqrt(a + b*x)*b**4*c**3*d + 6*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c**2 
*d**2*x + 72*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c*d**3*x**2 + 48*sqrt(c + d* 
x)*sqrt(a + b*x)*b**4*d**4*x**3 + 105*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a 
+ b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d**4 - 180*sqrt(d)*s 
qrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c) 
)*a**3*b*c*d**3 - 48*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)* 
sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**3*d**3 + 54*sqrt(d)*sqrt(b)*log((s 
qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*c 
**2*d**2 + 96*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
+ d*x))/sqrt(a*d - b*c))*a*b**4*c*d**2 + 12*sqrt(d)*sqrt(b)*log((sqrt(d)*s 
qrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**3*c**3*d - 48* 
sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a 
*d - b*c))*b**5*c**2*d + 9*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sq 
rt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**4*c**4)/(192*b**5*d**3)