\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (c^2-d^2 x^2)^{3/2}} \, dx\) [224]

Optimal result

Integrand size = 41, antiderivative size = 233 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{2 c d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}+\frac {\left (3 c^2 C d+B c d^2+3 A d^3+c^3 D\right ) \sqrt {c+d x}}{4 c^2 d^4 \sqrt {c^2-d^2 x^2}}+\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}+\frac {\left (5 c^2 C d-B c d^2-3 A d^3-9 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{4 \sqrt {2} c^{5/2} d^4} \] Output:

-1/2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/c/d^4/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(1/2 
)+1/4*(3*A*d^3+B*c*d^2+3*C*c^2*d+D*c^3)*(d*x+c)^(1/2)/c^2/d^4/(-d^2*x^2+c^ 
2)^(1/2)+2*D*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(1/2)+1/8*(-3*A*d^3-B*c*d^2+ 
5*C*c^2*d-9*D*c^3)*arctanh(2^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(1 
/2))*2^(1/2)/c^(5/2)/d^4
 

Mathematica [A] (verified)

Time = 3.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (11 c^4 D+3 A d^4 x+c d^3 (A+B x)+c^3 d (C+D x)+c^2 d^2 (3 B+x (3 C-8 D x))\right )}{(c-d x) (c+d x)^{3/2}}-\sqrt {2} \left (-5 c^2 C d+B c d^2+3 A d^3+9 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{8 c^{5/2} d^4} \] Input:

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

Output:

((2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(11*c^4*D + 3*A*d^4*x + c*d^3*(A + B*x) + 
c^3*d*(C + D*x) + c^2*d^2*(3*B + x*(3*C - 8*D*x))))/((c - d*x)*(c + d*x)^( 
3/2)) - Sqrt[2]*(-5*c^2*C*d + B*c*d^2 + 3*A*d^3 + 9*c^3*D)*ArcTanh[(Sqrt[2 
]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(8*c^(5/2)*d^4)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2170, 27, 2170, 27, 671, 467, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {2 \int -\frac {d^6 \left (-2 D c^3+C d c^2-A d^3+d \left (-3 D c^2+2 C d c-B d^2\right ) x\right )}{2 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}dx}{d^4}+\frac {2 d \sqrt {c+d x} (c D+C d)}{\sqrt {c^2-d^2 x^2}}}{d^5}-\frac {2 D (c+d x)^{3/2}}{d^4 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (c D+C d)}{\sqrt {c^2-d^2 x^2}}-d^2 \left (\frac {\left (-3 A d^3-B c d^2-9 c^3 D+5 c^2 C d\right ) \int \frac {\sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{4 c}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{d^5}-\frac {2 D (c+d x)^{3/2}}{d^4 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 467

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (c D+C d)}{\sqrt {c^2-d^2 x^2}}-d^2 \left (\frac {\left (-3 A d^3-B c d^2-9 c^3 D+5 c^2 C d\right ) \left (\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{2 c}+\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}\right )}{4 c}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{d^5}-\frac {2 D (c+d x)^{3/2}}{d^4 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (c D+C d)}{\sqrt {c^2-d^2 x^2}}-d^2 \left (\frac {\left (-3 A d^3-B c d^2-9 c^3 D+5 c^2 C d\right ) \left (\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{c}+\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}\right )}{4 c}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{d^5}-\frac {2 D (c+d x)^{3/2}}{d^4 \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (c D+C d)}{\sqrt {c^2-d^2 x^2}}-d^2 \left (\frac {\left (\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {2} c^{3/2} d}\right ) \left (-3 A d^3-B c d^2-9 c^3 D+5 c^2 C d\right )}{4 c}+\frac {A d^3-B c d^2+c^3 (-D)+c^2 C d}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{d^5}-\frac {2 D (c+d x)^{3/2}}{d^4 \sqrt {c^2-d^2 x^2}}\)

Input:

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

Output:

(-2*D*(c + d*x)^(3/2))/(d^4*Sqrt[c^2 - d^2*x^2]) + ((2*d*(C*d + c*D)*Sqrt[ 
c + d*x])/Sqrt[c^2 - d^2*x^2] - d^2*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/( 
2*c*d*Sqrt[c + d*x]*Sqrt[c^2 - d^2*x^2]) + ((5*c^2*C*d - B*c*d^2 - 3*A*d^3 
 - 9*c^3*D)*(Sqrt[c + d*x]/(c*d*Sqrt[c^2 - d^2*x^2]) - ArcTanh[Sqrt[c^2 - 
d^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])]/(Sqrt[2]*c^(3/2)*d)))/(4*c)))/d^ 
5
 

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 467

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

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 671

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.73

method	result	size
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (-22 D c^{\frac {9}{2}}-2 D c^{\frac {7}{2}} d x +16 D c^{\frac {5}{2}} d^{2} x^{2}+3 A \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{4} x +B \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{3} x -5 C \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{2} x -2 C \,c^{\frac {7}{2}} d -6 C \,c^{\frac {5}{2}} d^{2} x +9 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d x +3 A \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{3}+B \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{2}-6 B \,c^{\frac {5}{2}} d^{2}-2 B \,c^{\frac {3}{2}} d^{3} x -5 C \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d +9 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}-2 A \,c^{\frac {3}{2}} d^{3}-6 A \sqrt {c}\, d^{4} x \right )}{8 \left (d x +c \right )^{\frac {3}{2}} \left (-d x +c \right ) d^{4} c^{\frac {5}{2}}}\)	\(404\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-1/8/(d*x+c)^(3/2)*(-d^2*x^2+c^2)^(1/2)*(-22*D*c^(9/2)-2*D*c^(7/2)*d*x+16* 
D*c^(5/2)*d^2*x^2+3*A*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^ 
(1/2)/c^(1/2))*d^4*x+B*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2 
^(1/2)/c^(1/2))*c*d^3*x-5*C*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c)^(1 
/2)*2^(1/2)/c^(1/2))*c^2*d^2*x-2*C*c^(7/2)*d-6*C*c^(5/2)*d^2*x+9*D*(-d*x+c 
)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d*x+3*A*(- 
d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^3+B*( 
-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^2- 
6*B*c^(5/2)*d^2-2*B*c^(3/2)*d^3*x-5*C*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*( 
-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d+9*D*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/ 
2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4-2*A*c^(3/2)*d^3-6*A*c^(1/2)*d^4*x)/( 
-d*x+c)/d^4/c^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.00 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\left [-\frac {\sqrt {2} {\left (9 \, D c^{6} - 5 \, C c^{5} d + B c^{4} d^{2} + 3 \, A c^{3} d^{3} - {\left (9 \, D c^{3} d^{3} - 5 \, C c^{2} d^{4} + B c d^{5} + 3 \, A d^{6}\right )} x^{3} - {\left (9 \, D c^{4} d^{2} - 5 \, C c^{3} d^{3} + B c^{2} d^{4} + 3 \, A c d^{5}\right )} x^{2} + {\left (9 \, D c^{5} d - 5 \, C c^{4} d^{2} + B c^{3} d^{3} + 3 \, A c^{2} d^{4}\right )} x\right )} \sqrt {c} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x + 2 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {c} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 4 \, {\left (8 \, D c^{3} d^{2} x^{2} - 11 \, D c^{5} - C c^{4} d - 3 \, B c^{3} d^{2} - A c^{2} d^{3} - {\left (D c^{4} d + 3 \, C c^{3} d^{2} + B c^{2} d^{3} + 3 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{16 \, {\left (c^{3} d^{7} x^{3} + c^{4} d^{6} x^{2} - c^{5} d^{5} x - c^{6} d^{4}\right )}}, -\frac {\sqrt {2} {\left (9 \, D c^{6} - 5 \, C c^{5} d + B c^{4} d^{2} + 3 \, A c^{3} d^{3} - {\left (9 \, D c^{3} d^{3} - 5 \, C c^{2} d^{4} + B c d^{5} + 3 \, A d^{6}\right )} x^{3} - {\left (9 \, D c^{4} d^{2} - 5 \, C c^{3} d^{3} + B c^{2} d^{4} + 3 \, A c d^{5}\right )} x^{2} + {\left (9 \, D c^{5} d - 5 \, C c^{4} d^{2} + B c^{3} d^{3} + 3 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) - 2 \, {\left (8 \, D c^{3} d^{2} x^{2} - 11 \, D c^{5} - C c^{4} d - 3 \, B c^{3} d^{2} - A c^{2} d^{3} - {\left (D c^{4} d + 3 \, C c^{3} d^{2} + B c^{2} d^{3} + 3 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{8 \, {\left (c^{3} d^{7} x^{3} + c^{4} d^{6} x^{2} - c^{5} d^{5} x - c^{6} d^{4}\right )}}\right ] \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algori 
thm="fricas")
                                                                                    
                                                                                    
 

Output:

[-1/16*(sqrt(2)*(9*D*c^6 - 5*C*c^5*d + B*c^4*d^2 + 3*A*c^3*d^3 - (9*D*c^3* 
d^3 - 5*C*c^2*d^4 + B*c*d^5 + 3*A*d^6)*x^3 - (9*D*c^4*d^2 - 5*C*c^3*d^3 + 
B*c^2*d^4 + 3*A*c*d^5)*x^2 + (9*D*c^5*d - 5*C*c^4*d^2 + B*c^3*d^3 + 3*A*c^ 
2*d^4)*x)*sqrt(c)*log(-(d^2*x^2 - 2*c*d*x + 2*sqrt(2)*sqrt(-d^2*x^2 + c^2) 
*sqrt(d*x + c)*sqrt(c) - 3*c^2)/(d^2*x^2 + 2*c*d*x + c^2)) - 4*(8*D*c^3*d^ 
2*x^2 - 11*D*c^5 - C*c^4*d - 3*B*c^3*d^2 - A*c^2*d^3 - (D*c^4*d + 3*C*c^3* 
d^2 + B*c^2*d^3 + 3*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^3*d 
^7*x^3 + c^4*d^6*x^2 - c^5*d^5*x - c^6*d^4), -1/8*(sqrt(2)*(9*D*c^6 - 5*C* 
c^5*d + B*c^4*d^2 + 3*A*c^3*d^3 - (9*D*c^3*d^3 - 5*C*c^2*d^4 + B*c*d^5 + 3 
*A*d^6)*x^3 - (9*D*c^4*d^2 - 5*C*c^3*d^3 + B*c^2*d^4 + 3*A*c*d^5)*x^2 + (9 
*D*c^5*d - 5*C*c^4*d^2 + B*c^3*d^3 + 3*A*c^2*d^4)*x)*sqrt(-c)*arctan(1/2*s 
qrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(-c)/(c*d*x + c^2)) - 2*(8*D 
*c^3*d^2*x^2 - 11*D*c^5 - C*c^4*d - 3*B*c^3*d^2 - A*c^2*d^3 - (D*c^4*d + 3 
*C*c^3*d^2 + B*c^2*d^3 + 3*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)) 
/(c^3*d^7*x^3 + c^4*d^6*x^2 - c^5*d^5*x - c^6*d^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algori 
thm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\frac {16 \, \sqrt {-d x + c} D}{d^{3}} + \frac {\sqrt {2} {\left (9 \, D c^{3} - 5 \, C c^{2} d + B c d^{2} + 3 \, A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{2} d^{3}} - \frac {2 \, {\left ({\left (d x - c\right )} D c^{3} + 4 \, D c^{4} + 3 \, {\left (d x - c\right )} C c^{2} d + 4 \, C c^{3} d + {\left (d x - c\right )} B c d^{2} + 4 \, B c^{2} d^{2} + 3 \, {\left (d x - c\right )} A d^{3} + 4 \, A c d^{3}\right )}}{{\left ({\left (-d x + c\right )}^{\frac {3}{2}} - 2 \, \sqrt {-d x + c} c\right )} c^{2} d^{3}}}{8 \, d} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(3/2),x, algori 
thm="giac")
 

Output:

1/8*(16*sqrt(-d*x + c)*D/d^3 + sqrt(2)*(9*D*c^3 - 5*C*c^2*d + B*c*d^2 + 3* 
A*d^3)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c^2*d^3) - 2* 
((d*x - c)*D*c^3 + 4*D*c^4 + 3*(d*x - c)*C*c^2*d + 4*C*c^3*d + (d*x - c)*B 
*c*d^2 + 4*B*c^2*d^2 + 3*(d*x - c)*A*d^3 + 4*A*c*d^3)/(((-d*x + c)^(3/2) - 
 2*sqrt(-d*x + c)*c)*c^2*d^3))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {3 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a c \,d^{2}+3 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a \,d^{3} x +\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d +\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b c \,d^{2} x +4 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{4}+4 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{3} d x -3 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a c \,d^{2}-3 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a \,d^{3} x -\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d -\sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b c \,d^{2} x -4 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{4}-4 \sqrt {c}\, \sqrt {-d x +c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{3} d x +4 a \,c^{2} d^{2}+12 a c \,d^{3} x +12 b \,c^{3} d +4 b \,c^{2} d^{2} x +48 c^{5}+16 c^{4} d x -32 c^{3} d^{2} x^{2}}{16 \sqrt {-d x +c}\, c^{3} d^{3} \left (d x +c \right )} \] Input:

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

Output:

(3*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*c* 
d**2 + 3*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2) 
)*a*d**3*x + sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqr 
t(2))*b*c**2*d + sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c) 
*sqrt(2))*b*c*d**2*x + 4*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - 
 sqrt(c)*sqrt(2))*c**4 + 4*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) 
 - sqrt(c)*sqrt(2))*c**3*d*x - 3*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c 
- d*x) + sqrt(c)*sqrt(2))*a*c*d**2 - 3*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(s 
qrt(c - d*x) + sqrt(c)*sqrt(2))*a*d**3*x - sqrt(c)*sqrt(c - d*x)*sqrt(2)*l 
og(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**2*d - sqrt(c)*sqrt(c - d*x)*sqrt( 
2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c*d**2*x - 4*sqrt(c)*sqrt(c - d* 
x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**4 - 4*sqrt(c)*sqrt(c - 
d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**3*d*x + 4*a*c**2*d**2 
 + 12*a*c*d**3*x + 12*b*c**3*d + 4*b*c**2*d**2*x + 48*c**5 + 16*c**4*d*x - 
 32*c**3*d**2*x**2)/(16*sqrt(c - d*x)*c**3*d**3*(c + d*x))