\(\int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} (c^2-d^2 x^2)^{5/2}} \, dx\) [232]

Optimal result

Integrand size = 41, antiderivative size = 326 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx=-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{4 c d^4 \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}+\frac {\left (7 c^2 C d+B c d^2+7 A d^3+c^3 D\right ) \sqrt {c+d x}}{24 c^2 d^4 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {\left (13 c^2 C d-5 B c d^2-35 A d^3+43 c^3 D\right ) \sqrt {c+d x}}{96 c^4 d^4 \sqrt {c^2-d^2 x^2}}+\frac {\left (13 c^2 C d-5 B c d^2-35 A d^3-53 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{192 c^4 d^4 (c+d x)^{3/2}}+\frac {\left (13 c^2 C d-5 B c d^2-35 A d^3+11 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{64 \sqrt {2} c^{9/2} d^4} \] Output:

-1/4*(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)^(3/2 
)+1/24*(7*A*d^3+B*c*d^2+7*C*c^2*d+D*c^3)*(d*x+c)^(1/2)/c^2/d^4/(-d^2*x^2+c 
^2)^(3/2)-1/96*(-35*A*d^3-5*B*c*d^2+13*C*c^2*d+43*D*c^3)*(d*x+c)^(1/2)/c^4 
/d^4/(-d^2*x^2+c^2)^(1/2)+1/192*(-35*A*d^3-5*B*c*d^2+13*C*c^2*d-53*D*c^3)* 
(-d^2*x^2+c^2)^(1/2)/c^4/d^4/(d*x+c)^(3/2)+1/128*(-35*A*d^3-5*B*c*d^2+13*C 
*c^2*d+11*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^(9/2)/d^4
 

Mathematica [A] (verified)

Time = 4.04 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (83 c^6 D+105 A d^6 x^3+5 c d^5 x^2 (7 A+3 B x)+5 c^5 d (C+5 D x)+c^2 d^4 x (-161 A+x (5 B-39 C x))-c^4 d^2 (61 B+x (17 C+139 D x))-c^3 d^3 \left (43 A+x \left (23 B+13 C x+33 D x^2\right )\right )\right )}{(c-d x)^2 (c+d x)^{5/2}}+3 \sqrt {2} \left (13 c^2 C d-5 B c d^2-35 A d^3+11 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{384 c^{9/2} d^4} \] Input:

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

Output:

((-2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(83*c^6*D + 105*A*d^6*x^3 + 5*c*d^5*x^2*( 
7*A + 3*B*x) + 5*c^5*d*(C + 5*D*x) + c^2*d^4*x*(-161*A + x*(5*B - 39*C*x)) 
 - c^4*d^2*(61*B + x*(17*C + 139*D*x)) - c^3*d^3*(43*A + x*(23*B + 13*C*x 
+ 33*D*x^2))))/((c - d*x)^2*(c + d*x)^(5/2)) + 3*Sqrt[2]*(13*c^2*C*d - 5*B 
*c*d^2 - 35*A*d^3 + 11*c^3*D)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt 
[c^2 - d^2*x^2]])/(384*c^(9/2)*d^4)
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2170, 27, 2170, 27, 671, 467, 470, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \int \frac {2 D c^3+C d c^2-5 A d^3+d \left (-3 D c^2+6 C d c-5 B d^2\right ) x}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}}dx}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \left (\frac {\left (-35 A d^3-5 B c d^2+11 c^3 D+13 c^2 C d\right ) \int \frac {\sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{8 c}+\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}\right )}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 467

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \left (\frac {\left (-35 A d^3-5 B c d^2+11 c^3 D+13 c^2 C d\right ) \left (\frac {5 \int \frac {1}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}dx}{6 c}+\frac {\sqrt {c+d x}}{3 c d \left (c^2-d^2 x^2\right )^{3/2}}\right )}{8 c}+\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}\right )}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \left (\frac {\left (-35 A d^3-5 B c d^2+11 c^3 D+13 c^2 C d\right ) \left (\frac {5 \left (\frac {3 \int \frac {\sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{6 c}+\frac {\sqrt {c+d x}}{3 c d \left (c^2-d^2 x^2\right )^{3/2}}\right )}{8 c}+\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}\right )}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 467

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \left (\frac {\left (-35 A d^3-5 B c d^2+11 c^3 D+13 c^2 C d\right ) \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{2 c}+\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{6 c}+\frac {\sqrt {c+d x}}{3 c d \left (c^2-d^2 x^2\right )^{3/2}}\right )}{8 c}+\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}\right )}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \left (\frac {\left (-35 A d^3-5 B c d^2+11 c^3 D+13 c^2 C d\right ) \left (\frac {5 \left (\frac {3 \left (\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{c}+\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{6 c}+\frac {\sqrt {c+d x}}{3 c d \left (c^2-d^2 x^2\right )^{3/2}}\right )}{8 c}+\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}\right )}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 d \sqrt {c+d x} (C d-3 c D)}{5 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {1}{5} d^2 \left (\frac {\left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {2} c^{3/2} d}\right )}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{6 c}+\frac {\sqrt {c+d x}}{3 c d \left (c^2-d^2 x^2\right )^{3/2}}\right ) \left (-35 A d^3-5 B c d^2+11 c^3 D+13 c^2 C d\right )}{8 c}+\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}\right )}{d^5}+\frac {2 D (c+d x)^{3/2}}{3 d^4 \left (c^2-d^2 x^2\right )^{3/2}}\)

Input:

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

Output:

(2*D*(c + d*x)^(3/2))/(3*d^4*(c^2 - d^2*x^2)^(3/2)) + ((2*d*(C*d - 3*c*D)* 
Sqrt[c + d*x])/(5*(c^2 - d^2*x^2)^(3/2)) - (d^2*((5*(c^2*C*d - B*c*d^2 + A 
*d^3 - c^3*D))/(4*c*d*Sqrt[c + d*x]*(c^2 - d^2*x^2)^(3/2)) + ((13*c^2*C*d 
- 5*B*c*d^2 - 35*A*d^3 + 11*c^3*D)*(Sqrt[c + d*x]/(3*c*d*(c^2 - d^2*x^2)^( 
3/2)) + (5*(-1/2*1/(c*d*Sqrt[c + d*x]*Sqrt[c^2 - d^2*x^2]) + (3*(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)))/(6*c)))/(8*c)))/5)/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 470

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

Leaf count of result is larger than twice the leaf count of optimal. \(809\) vs. \(2(288)=576\).

Time = 0.37 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.48

method	result	size
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (105 A \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3}+15 B \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2}-39 C \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5} d -105 A \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{6} x^{3}-105 A \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{5} x^{2}-15 B \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x^{2}+39 C \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x^{2}+33 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2} x^{2}+105 A \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x +15 B \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x +10 C \,c^{\frac {11}{2}} d -15 B \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{5} x^{3}+39 C \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x^{3}+33 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x^{3}-78 C \,c^{\frac {5}{2}} d^{4} x^{3}-39 C \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2} x -33 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5} d x +70 A \,c^{\frac {3}{2}} d^{5} x^{2}+10 B \,c^{\frac {5}{2}} d^{4} x^{2}-26 C \,c^{\frac {7}{2}} d^{3} x^{2}-33 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}-278 D c^{\frac {9}{2}} d^{2} x^{2}-322 A \,c^{\frac {5}{2}} d^{4} x -46 B \,c^{\frac {7}{2}} d^{3} x -34 C \,c^{\frac {9}{2}} d^{2} x +50 D c^{\frac {11}{2}} d x +30 B \,c^{\frac {3}{2}} d^{5} x^{3}-122 B \,c^{\frac {9}{2}} d^{2}-66 D c^{\frac {7}{2}} d^{3} x^{3}+210 A \sqrt {c}\, d^{6} x^{3}-86 A \,c^{\frac {7}{2}} d^{3}+166 D c^{\frac {13}{2}}\right )}{384 \left (d x +c \right )^{\frac {5}{2}} \left (-d x +c \right )^{2} d^{4} c^{\frac {9}{2}}}\)	\(810\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(5/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-1/384*(-d^2*x^2+c^2)^(1/2)*(105*A*(-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^3+15*B*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/ 
2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^2-39*C*(-d*x+c)^(1/2)*2^(1/2)*arct 
anh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d-105*A*(-d*x+c)^(1/2)*2^(1/2) 
*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*d^6*x^3-105*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^5*x^2-15*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^4*x^2+39 
*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^3*x^2+33*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*d^2*x^2+105*A*(-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^4*x+15*B*(-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^3*x+10*C*c^(11/2)*d-15*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^5*x^3+39*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^4*x^3+33*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^3 
*x^3-78*C*c^(5/2)*d^4*x^3-39*C*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x+c) 
^(1/2)*2^(1/2)/c^(1/2))*c^4*d^2*x-33*D*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2* 
(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d*x+70*A*c^(3/2)*d^5*x^2+10*B*c^(5/2)* 
d^4*x^2-26*C*c^(7/2)*d^3*x^2-33*D*(-d*x+c)^(1/2)*2^(1/2)*arctanh(1/2*(-d*x 
+c)^(1/2)*2^(1/2)/c^(1/2))*c^6-278*D*c^(9/2)*d^2*x^2-322*A*c^(5/2)*d^4*...
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 1059, normalized size of antiderivative = 3.25 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(5/2),x, algori 
thm="fricas")
 

Output:

[-1/768*(3*sqrt(2)*(11*D*c^8 + 13*C*c^7*d - 5*B*c^6*d^2 - 35*A*c^5*d^3 + ( 
11*D*c^3*d^5 + 13*C*c^2*d^6 - 5*B*c*d^7 - 35*A*d^8)*x^5 + (11*D*c^4*d^4 + 
13*C*c^3*d^5 - 5*B*c^2*d^6 - 35*A*c*d^7)*x^4 - 2*(11*D*c^5*d^3 + 13*C*c^4* 
d^4 - 5*B*c^3*d^5 - 35*A*c^2*d^6)*x^3 - 2*(11*D*c^6*d^2 + 13*C*c^5*d^3 - 5 
*B*c^4*d^4 - 35*A*c^3*d^5)*x^2 + (11*D*c^7*d + 13*C*c^6*d^2 - 5*B*c^5*d^3 
- 35*A*c^4*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 
3*D*c^7 + 5*C*c^6*d - 61*B*c^5*d^2 - 43*A*c^4*d^3 - 3*(11*D*c^4*d^3 + 13*C 
*c^3*d^4 - 5*B*c^2*d^5 - 35*A*c*d^6)*x^3 - (139*D*c^5*d^2 + 13*C*c^4*d^3 - 
 5*B*c^3*d^4 - 35*A*c^2*d^5)*x^2 + (25*D*c^6*d - 17*C*c^5*d^2 - 23*B*c^4*d 
^3 - 161*A*c^3*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^5*d^9*x^5 + 
c^6*d^8*x^4 - 2*c^7*d^7*x^3 - 2*c^8*d^6*x^2 + c^9*d^5*x + c^10*d^4), -1/38 
4*(3*sqrt(2)*(11*D*c^8 + 13*C*c^7*d - 5*B*c^6*d^2 - 35*A*c^5*d^3 + (11*D*c 
^3*d^5 + 13*C*c^2*d^6 - 5*B*c*d^7 - 35*A*d^8)*x^5 + (11*D*c^4*d^4 + 13*C*c 
^3*d^5 - 5*B*c^2*d^6 - 35*A*c*d^7)*x^4 - 2*(11*D*c^5*d^3 + 13*C*c^4*d^4 - 
5*B*c^3*d^5 - 35*A*c^2*d^6)*x^3 - 2*(11*D*c^6*d^2 + 13*C*c^5*d^3 - 5*B*c^4 
*d^4 - 35*A*c^3*d^5)*x^2 + (11*D*c^7*d + 13*C*c^6*d^2 - 5*B*c^5*d^3 - 35*A 
*c^4*d^4)*x)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c 
)*sqrt(-c)/(c*d*x + c^2)) + 2*(83*D*c^7 + 5*C*c^6*d - 61*B*c^5*d^2 - 43*A* 
c^4*d^3 - 3*(11*D*c^4*d^3 + 13*C*c^3*d^4 - 5*B*c^2*d^5 - 35*A*c*d^6)*x^...
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2)/(-d**2*x**2+c**2)**(5/2),x)
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/((-(-c + d*x)*(c + d*x))**(5/2)*sqrt( 
c + d*x)), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(5/2),x, algori 
thm="maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)/((-d^2*x^2 + c^2)^(5/2)*sqrt(d*x + c)) 
, x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} {\left (11 \, D c^{3} + 13 \, C c^{2} d - 5 \, B c d^{2} - 35 \, A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{4} d^{3}} + \frac {16 \, {\left (9 \, {\left (d x - c\right )} D c^{3} + 2 \, D c^{4} + 3 \, {\left (d x - c\right )} C c^{2} d + 2 \, C c^{3} d - 3 \, {\left (d x - c\right )} B c d^{2} + 2 \, B c^{2} d^{2} - 9 \, {\left (d x - c\right )} A d^{3} + 2 \, A c d^{3}\right )}}{{\left (d x - c\right )} \sqrt {-d x + c} c^{4} d^{3}} - \frac {6 \, {\left (13 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{3} - 22 \, \sqrt {-d x + c} D c^{4} - 5 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{2} d + 6 \, \sqrt {-d x + c} C c^{3} d - 3 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c d^{2} + 10 \, \sqrt {-d x + c} B c^{2} d^{2} + 11 \, {\left (-d x + c\right )}^{\frac {3}{2}} A d^{3} - 26 \, \sqrt {-d x + c} A c d^{3}\right )}}{{\left (d x + c\right )}^{2} c^{4} d^{3}}}{384 \, d} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(5/2),x, algori 
thm="giac")
 

Output:

-1/384*(3*sqrt(2)*(11*D*c^3 + 13*C*c^2*d - 5*B*c*d^2 - 35*A*d^3)*arctan(1/ 
2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c^4*d^3) + 16*(9*(d*x - c)*D* 
c^3 + 2*D*c^4 + 3*(d*x - c)*C*c^2*d + 2*C*c^3*d - 3*(d*x - c)*B*c*d^2 + 2* 
B*c^2*d^2 - 9*(d*x - c)*A*d^3 + 2*A*c*d^3)/((d*x - c)*sqrt(-d*x + c)*c^4*d 
^3) - 6*(13*(-d*x + c)^(3/2)*D*c^3 - 22*sqrt(-d*x + c)*D*c^4 - 5*(-d*x + c 
)^(3/2)*C*c^2*d + 6*sqrt(-d*x + c)*C*c^3*d - 3*(-d*x + c)^(3/2)*B*c*d^2 + 
10*sqrt(-d*x + c)*B*c^2*d^2 + 11*(-d*x + c)^(3/2)*A*d^3 - 26*sqrt(-d*x + c 
)*A*c*d^3)/((d*x + c)^2*c^4*d^3))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (c^2-d^2\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(5/2)*(c + d*x)^(1/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(5/2)*(c + d*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(5/2),x)
 

Output:

(105*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a* 
c**3*d**2 + 105*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)* 
sqrt(2))*a*c**2*d**3*x - 105*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d* 
x) - sqrt(c)*sqrt(2))*a*c*d**4*x**2 - 105*sqrt(c)*sqrt(c - d*x)*sqrt(2)*lo 
g(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*d**5*x**3 + 15*sqrt(c)*sqrt(c - d*x)* 
sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**4*d + 15*sqrt(c)*sqrt(c 
- d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**3*d**2*x - 15*sqr 
t(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**2*d** 
3*x**2 - 15*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt 
(2))*b*c*d**4*x**3 - 72*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - 
sqrt(c)*sqrt(2))*c**6 - 72*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) 
 - sqrt(c)*sqrt(2))*c**5*d*x + 72*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c 
 - d*x) - sqrt(c)*sqrt(2))*c**4*d**2*x**2 + 72*sqrt(c)*sqrt(c - d*x)*sqrt( 
2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*c**3*d**3*x**3 - 105*sqrt(c)*sqrt( 
c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c**3*d**2 - 105*sq 
rt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c**2*d* 
*3*x + 105*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt( 
2))*a*c*d**4*x**2 + 105*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + 
sqrt(c)*sqrt(2))*a*d**5*x**3 - 15*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c 
 - d*x) + sqrt(c)*sqrt(2))*b*c**4*d - 15*sqrt(c)*sqrt(c - d*x)*sqrt(2)*...