\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} (c^2-d^2 x^2)^{3/2}} \, dx\) [226]

Optimal result

Integrand size = 41, antiderivative size = 328 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {c^2 C d-B c d^2+A d^3-c^3 D}{6 c d^4 (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}+\frac {17 c^2 C d-5 B c d^2-7 A d^3-29 c^3 D}{48 c^2 d^4 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}+\frac {\left (11 c^2 C d+25 B c d^2+35 A d^3+49 c^3 D\right ) \sqrt {c+d x}}{192 c^4 d^4 \sqrt {c^2-d^2 x^2}}-\frac {\left (11 c^2 C d+25 B c d^2+35 A d^3-143 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{384 c^4 d^4 (c+d x)^{3/2}}-\frac {\left (11 c^2 C d+25 B c d^2+35 A d^3-15 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{128 \sqrt {2} c^{9/2} d^4} \] Output:

-1/6*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/c/d^4/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2 
)+1/48*(-7*A*d^3-5*B*c*d^2+17*C*c^2*d-29*D*c^3)/c^2/d^4/(d*x+c)^(3/2)/(-d^ 
2*x^2+c^2)^(1/2)+1/192*(35*A*d^3+25*B*c*d^2+11*C*c^2*d+49*D*c^3)*(d*x+c)^( 
1/2)/c^4/d^4/(-d^2*x^2+c^2)^(1/2)-1/384*(35*A*d^3+25*B*c*d^2+11*C*c^2*d-14 
3*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c^4/d^4/(d*x+c)^(3/2)-1/256*(35*A*d^3+25*B*c 
*d^2+11*C*c^2*d-15*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 = 3.96 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (73 c^6 D+105 A d^6 x^3+5 c d^5 x^2 (49 A+15 B x)+c^5 d (83 C+205 D x)+c^2 d^4 x (119 A+x (175 B+33 C x))+c^4 d^2 (49 B+x (191 C+151 D x))+c^3 d^3 \left (-85 A+x \left (85 B+77 C x-45 D x^2\right )\right )\right )}{(c-d x) (c+d x)^{7/2}}+3 \sqrt {2} \left (-11 c^2 C d-25 B c d^2-35 A d^3+15 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{768 c^{9/2} d^4} \] Input:

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

Output:

((2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(73*c^6*D + 105*A*d^6*x^3 + 5*c*d^5*x^2*(4 
9*A + 15*B*x) + c^5*d*(83*C + 205*D*x) + c^2*d^4*x*(119*A + x*(175*B + 33* 
C*x)) + c^4*d^2*(49*B + x*(191*C + 151*D*x)) + c^3*d^3*(-85*A + x*(85*B + 
77*C*x - 45*D*x^2))))/((c - d*x)*(c + d*x)^(7/2)) + 3*Sqrt[2]*(-11*c^2*C*d 
 - 25*B*c*d^2 - 35*A*d^3 + 15*c^3*D)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x 
])/Sqrt[c^2 - d^2*x^2]])/(768*c^(9/2)*d^4)
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.08, 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, 470, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {2 \int \frac {d^6 \left (-10 D c^3+9 C d c^2+15 A d^3-d \left (-5 D c^2+6 C d c-15 B d^2\right ) x\right )}{2 (c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}dx}{5 d^4}+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} d^2 \int \frac {-10 D c^3+9 C d c^2+15 A d^3-d \left (-5 D c^2+6 C d c-15 B d^2\right ) x}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}}dx+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (35 A d^3+25 B c d^2-15 c^3 D+11 c^2 C d\right ) \int \frac {1}{(c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{3/2}}dx}{4 c}-\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}\right )+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (35 A d^3+25 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (\frac {5 \int \frac {1}{\sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}dx}{8 c}-\frac {1}{4 c d (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}\right )+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (35 A d^3+25 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (\frac {5 \left (\frac {3 \int \frac {\sqrt {c+d x}}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{8 c}-\frac {1}{4 c d (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}\right )+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 467

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (35 A d^3+25 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{2 c}+\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{8 c}-\frac {1}{4 c d (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}\right )+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (35 A d^3+25 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (\frac {5 \left (\frac {3 \left (\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{c}+\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{8 c}-\frac {1}{4 c d (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}\right )}{4 c}-\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}\right )+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{5} d^2 \left (\frac {\left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {c+d x}}{c d \sqrt {c^2-d^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {2} c^{3/2} d}\right )}{4 c}-\frac {1}{2 c d \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\right )}{8 c}-\frac {1}{4 c d (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}\right ) \left (35 A d^3+25 B c d^2-15 c^3 D+11 c^2 C d\right )}{4 c}-\frac {5 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}\right )+\frac {2 d (3 C d-5 c D)}{5 (c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}}{3 d^5}+\frac {2 D}{3 d^4 \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}\)

Input:

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

Output:

(2*D)/(3*d^4*Sqrt[c + d*x]*Sqrt[c^2 - d^2*x^2]) + ((2*d*(3*C*d - 5*c*D))/( 
5*(c + d*x)^(3/2)*Sqrt[c^2 - d^2*x^2]) + (d^2*((-5*(c^2*C*d - B*c*d^2 + A* 
d^3 - c^3*D))/(2*c*d*(c + d*x)^(5/2)*Sqrt[c^2 - d^2*x^2]) + ((11*c^2*C*d + 
 25*B*c*d^2 + 35*A*d^3 - 15*c^3*D)*(-1/4*1/(c*d*(c + d*x)^(3/2)*Sqrt[c^2 - 
 d^2*x^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)))/(8*c)))/(4*c)))/5)/ 
(3*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. \(2(290)=580\).

Time = 0.37 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.47

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}+75 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}+33 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}+315 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}+225 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}+99 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}-135 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}+315 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 +225 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 -166 C \,c^{\frac {11}{2}} d +75 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}+33 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}-45 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}-66 C \,c^{\frac {5}{2}} d^{4} x^{3}+99 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 -135 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 -490 A \,c^{\frac {3}{2}} d^{5} x^{2}-350 B \,c^{\frac {5}{2}} d^{4} x^{2}-154 C \,c^{\frac {7}{2}} d^{3} x^{2}-45 D \sqrt {-d x +c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}-302 D c^{\frac {9}{2}} d^{2} x^{2}-238 A \,c^{\frac {5}{2}} d^{4} x -170 B \,c^{\frac {7}{2}} d^{3} x -382 C \,c^{\frac {9}{2}} d^{2} x -410 D c^{\frac {11}{2}} d x -150 B \,c^{\frac {3}{2}} d^{5} x^{3}-98 B \,c^{\frac {9}{2}} d^{2}+90 D c^{\frac {7}{2}} d^{3} x^{3}-210 A \sqrt {c}\, d^{6} x^{3}+170 A \,c^{\frac {7}{2}} d^{3}-146 D c^{\frac {13}{2}}\right )}{768 \left (d x +c \right )^{\frac {7}{2}} \left (-d x +c \right ) d^{4} c^{\frac {9}{2}}}\)	\(810\)

Input:

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

Output:

-1/768/(d*x+c)^(7/2)*(-d^2*x^2+c^2)^(1/2)*(105*A*(-d*x+c)^(1/2)*2^(1/2)*ar 
ctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d^3+75*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+33*C*(-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+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+315*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+225*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+99*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^2-135*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+315*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+225*B*(-d*x+c)^(1/2)*2^(1/2)*ar 
ctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d^3*x-166*C*c^(11/2)*d+75*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+33*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^3-45*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^3-66*C*c^(5/2)*d^4*x^3+99*C*(-d*x+c)^(1/2)*2^(1/2)*ar 
ctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^2*x-135*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-490*A*c^(3/2)*d 
^5*x^2-350*B*c^(5/2)*d^4*x^2-154*C*c^(7/2)*d^3*x^2-45*D*(-d*x+c)^(1/2)*2^( 
1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^6-302*D*c^(9/2)*d^2*...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1069, normalized size of antiderivative = 3.26 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/1536*(3*sqrt(2)*(15*D*c^8 - 11*C*c^7*d - 25*B*c^6*d^2 - 35*A*c^5*d^3 - 
(15*D*c^3*d^5 - 11*C*c^2*d^6 - 25*B*c*d^7 - 35*A*d^8)*x^5 - 3*(15*D*c^4*d^ 
4 - 11*C*c^3*d^5 - 25*B*c^2*d^6 - 35*A*c*d^7)*x^4 - 2*(15*D*c^5*d^3 - 11*C 
*c^4*d^4 - 25*B*c^3*d^5 - 35*A*c^2*d^6)*x^3 + 2*(15*D*c^6*d^2 - 11*C*c^5*d 
^3 - 25*B*c^4*d^4 - 35*A*c^3*d^5)*x^2 + 3*(15*D*c^7*d - 11*C*c^6*d^2 - 25* 
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)*s 
qrt(-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*(73*D*c^7 + 83*C*c^6*d + 49*B*c^5*d^2 - 85*A*c^4*d^3 - 3*(15*D*c^4 
*d^3 - 11*C*c^3*d^4 - 25*B*c^2*d^5 - 35*A*c*d^6)*x^3 + (151*D*c^5*d^2 + 77 
*C*c^4*d^3 + 175*B*c^3*d^4 + 245*A*c^2*d^5)*x^2 + (205*D*c^6*d + 191*C*c^5 
*d^2 + 85*B*c^4*d^3 + 119*A*c^3*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c) 
)/(c^5*d^9*x^5 + 3*c^6*d^8*x^4 + 2*c^7*d^7*x^3 - 2*c^8*d^6*x^2 - 3*c^9*d^5 
*x - c^10*d^4), 1/768*(3*sqrt(2)*(15*D*c^8 - 11*C*c^7*d - 25*B*c^6*d^2 - 3 
5*A*c^5*d^3 - (15*D*c^3*d^5 - 11*C*c^2*d^6 - 25*B*c*d^7 - 35*A*d^8)*x^5 - 
3*(15*D*c^4*d^4 - 11*C*c^3*d^5 - 25*B*c^2*d^6 - 35*A*c*d^7)*x^4 - 2*(15*D* 
c^5*d^3 - 11*C*c^4*d^4 - 25*B*c^3*d^5 - 35*A*c^2*d^6)*x^3 + 2*(15*D*c^6*d^ 
2 - 11*C*c^5*d^3 - 25*B*c^4*d^4 - 35*A*c^3*d^5)*x^2 + 3*(15*D*c^7*d - 11*C 
*c^6*d^2 - 25*B*c^5*d^3 - 35*A*c^4*d^4)*x)*sqrt(-c)*arctan(1/2*sqrt(2)*sqr 
t(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(-c)/(c*d*x + c^2)) - 2*(73*D*c^7 + 83 
*C*c^6*d + 49*B*c^5*d^2 - 85*A*c^4*d^3 - 3*(15*D*c^4*d^3 - 11*C*c^3*d^4...
 

Sympy [F]

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

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

Maxima [F]

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

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} {\left (15 \, D c^{3} - 11 \, C c^{2} d - 25 \, B c d^{2} - 35 \, A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{4} d^{3}} - \frac {96 \, {\left (D c^{3} + C c^{2} d + B c d^{2} + A d^{3}\right )}}{\sqrt {-d x + c} c^{4} d^{3}} - \frac {2 \, {\left (93 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} D c^{3} - 272 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{4} + 204 \, \sqrt {-d x + c} D c^{5} + 15 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} C c^{2} d - 112 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{3} d + 132 \, \sqrt {-d x + c} C c^{4} d - 27 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} B c d^{2} + 112 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c^{2} d^{2} - 84 \, \sqrt {-d x + c} B c^{3} d^{2} - 57 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} A d^{3} + 272 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c d^{3} - 348 \, \sqrt {-d x + c} A c^{2} d^{3}\right )}}{{\left (d x + c\right )}^{3} c^{4} d^{3}}}{768 \, d} \] Input:

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

Output:

-1/768*(3*sqrt(2)*(15*D*c^3 - 11*C*c^2*d - 25*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) - 96*(D*c^3 + C*c^2 
*d + B*c*d^2 + A*d^3)/(sqrt(-d*x + c)*c^4*d^3) - 2*(93*(d*x - c)^2*sqrt(-d 
*x + c)*D*c^3 - 272*(-d*x + c)^(3/2)*D*c^4 + 204*sqrt(-d*x + c)*D*c^5 + 15 
*(d*x - c)^2*sqrt(-d*x + c)*C*c^2*d - 112*(-d*x + c)^(3/2)*C*c^3*d + 132*s 
qrt(-d*x + c)*C*c^4*d - 27*(d*x - c)^2*sqrt(-d*x + c)*B*c*d^2 + 112*(-d*x 
+ c)^(3/2)*B*c^2*d^2 - 84*sqrt(-d*x + c)*B*c^3*d^2 - 57*(d*x - c)^2*sqrt(- 
d*x + c)*A*d^3 + 272*(-d*x + c)^(3/2)*A*c*d^3 - 348*sqrt(-d*x + c)*A*c^2*d 
^3)/((d*x + c)^3*c^4*d^3))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.04 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \left (c^2-d^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(3/2),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 + 315*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)* 
sqrt(2))*a*c**2*d**3*x + 315*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 + 75*sqrt(c)*sqrt(c - d*x)* 
sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**4*d + 225*sqrt(c)*sqrt(c 
 - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**3*d**2*x + 225*s 
qrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**2*d 
**3*x**2 + 75*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sq 
rt(2))*b*c*d**4*x**3 - 12*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) 
- sqrt(c)*sqrt(2))*c**6 - 36*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d* 
x) - sqrt(c)*sqrt(2))*c**5*d*x - 36*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt 
(c - d*x) - sqrt(c)*sqrt(2))*c**4*d**2*x**2 - 12*sqrt(c)*sqrt(c - d*x)*sqr 
t(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*c**3*d**3*x**3 - 105*sqrt(c)*sqr 
t(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c**3*d**2 - 315* 
sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c**2* 
d**3*x - 315*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqr 
t(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 - 75*sqrt(c)*sqrt(c - d*x)*sqrt(2)*log(sqrt 
(c - d*x) + sqrt(c)*sqrt(2))*b*c**4*d - 225*sqrt(c)*sqrt(c - d*x)*sqrt(...