\(\int \frac {(c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3)}{c+d x} \, dx\) [157]

Optimal result

Integrand size = 39, antiderivative size = 312 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {c^3 \left (8 c^2 C d-8 B c d^2+48 A d^3-3 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{128 d^3}+\frac {c \left (8 c^2 C d-8 B c d^2+48 A d^3-3 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{3/2}}{192 d^3}+\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4}-\frac {\left (8 c C d-8 B d^2-3 c^2 D\right ) x \left (c^2-d^2 x^2\right )^{5/2}}{48 d^3}+\frac {D x^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d}-\frac {(C d-c D) \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}+\frac {c^5 \left (8 c^2 C d-8 B c d^2+48 A d^3-3 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{128 d^4} \] Output:

1/128*c^3*(48*A*d^3-8*B*c*d^2+8*C*c^2*d-3*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/d^ 
3+1/192*c*(48*A*d^3-8*B*c*d^2+8*C*c^2*d-3*D*c^3)*x*(-d^2*x^2+c^2)^(3/2)/d^ 
3+1/5*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(5/2)/d^4-1/48*(-8*B*d^ 
2+8*C*c*d-3*D*c^2)*x*(-d^2*x^2+c^2)^(5/2)/d^3+1/8*D*x^3*(-d^2*x^2+c^2)^(5/ 
2)/d-1/7*(C*d-D*c)*(-d^2*x^2+c^2)^(7/2)/d^4+1/128*c^5*(48*A*d^3-8*B*c*d^2+ 
8*C*c^2*d-3*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-768 c^7 D+c^6 (768 C d+315 d D x)-32 c d^6 x^3 \left (105 A+84 B x+70 C x^2+60 D x^3\right )-24 c^5 d^2 (112 B+x (35 C+16 D x))+6 c^4 d^3 \left (448 A+x \left (140 B+64 C x+35 D x^2\right )\right )+16 c^3 d^4 x \left (525 A+x \left (336 B+245 C x+192 D x^2\right )\right )-8 c^2 d^5 x^2 \left (672 A+x \left (490 B+384 C x+315 D x^2\right )\right )+16 d^7 x^4 (168 A+5 x (28 B+3 x (8 C+7 D x)))\right )+210 c^5 \left (-8 c^2 C d+8 B c d^2-48 A d^3+3 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{13440 d^4} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(-768*c^7*D + c^6*(768*C*d + 315*d*D*x) - 32*c*d^6*x^ 
3*(105*A + 84*B*x + 70*C*x^2 + 60*D*x^3) - 24*c^5*d^2*(112*B + x*(35*C + 1 
6*D*x)) + 6*c^4*d^3*(448*A + x*(140*B + 64*C*x + 35*D*x^2)) + 16*c^3*d^4*x 
*(525*A + x*(336*B + 245*C*x + 192*D*x^2)) - 8*c^2*d^5*x^2*(672*A + x*(490 
*B + 384*C*x + 315*D*x^2)) + 16*d^7*x^4*(168*A + 5*x*(28*B + 3*x*(8*C + 7* 
D*x)))) + 210*c^5*(-8*c^2*C*d + 8*B*c*d^2 - 48*A*d^3 + 3*c^3*D)*ArcTan[(d* 
x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/(13440*d^4)
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2170, 25, 2170, 27, 667, 676, 211, 211, 224, 216}

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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {\int -\frac {\left (c^2-d^2 x^2\right )^{5/2} \left ((8 C d-15 c D) x^2 d^4+2 \left (4 B d^2-3 c^2 D\right ) x d^3+\left (D c^3+8 A d^3\right ) d^2\right )}{c+d x}dx}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left ((8 C d-15 c D) x^2 d^4+2 \left (4 B d^2-3 c^2 D\right ) x d^3+\left (D c^3+8 A d^3\right ) d^2\right )}{c+d x}dx}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {\int -\frac {7 d^6 \left (D c^3+8 A d^3-d \left (-9 D c^2+8 C d c-8 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}}{c+d x}dx}{7 d^4}-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \int \frac {\left (D c^3+8 A d^3-d \left (-9 D c^2+8 C d c-8 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}}{c+d x}dx-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 667

\(\displaystyle \frac {d^2 \int (c-d x) \left (D c^3+8 A d^3-d \left (-9 D c^2+8 C d c-8 B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {d^2 \left (\frac {1}{6} c \left (48 A d^3-8 B c d^2-3 c^3 D+8 c^2 C d\right ) \int \left (c^2-d^2 x^2\right )^{3/2}dx+\frac {8 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d}-\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2} \left (-8 B d^2-9 c^2 D+8 c C d\right )\right )-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d^2 \left (\frac {1}{6} c \left (48 A d^3-8 B c d^2-3 c^3 D+8 c^2 C d\right ) \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {8 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d}-\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2} \left (-8 B d^2-9 c^2 D+8 c C d\right )\right )-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d^2 \left (\frac {1}{6} c \left (48 A d^3-8 B c d^2-3 c^3 D+8 c^2 C d\right ) \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {8 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d}-\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2} \left (-8 B d^2-9 c^2 D+8 c C d\right )\right )-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d^2 \left (\frac {1}{6} c \left (48 A d^3-8 B c d^2-3 c^3 D+8 c^2 C d\right ) \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {8 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d}-\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2} \left (-8 B d^2-9 c^2 D+8 c C d\right )\right )-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {d^2 \left (\frac {1}{6} c \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right ) \left (48 A d^3-8 B c d^2-3 c^3 D+8 c^2 C d\right )+\frac {8 \left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d}-\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2} \left (-8 B d^2-9 c^2 D+8 c C d\right )\right )-\frac {1}{7} d \left (c^2-d^2 x^2\right )^{7/2} (8 C d-15 c D)}{8 d^5}-\frac {D (c+d x) \left (c^2-d^2 x^2\right )^{7/2}}{8 d^4}\)

Input:

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

Output:

-1/8*(D*(c + d*x)*(c^2 - d^2*x^2)^(7/2))/d^4 + (-1/7*(d*(8*C*d - 15*c*D)*( 
c^2 - d^2*x^2)^(7/2)) + d^2*((8*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - 
 d^2*x^2)^(5/2))/(5*d) - ((8*c*C*d - 8*B*d^2 - 9*c^2*D)*x*(c^2 - d^2*x^2)^ 
(5/2))/6 + (c*(8*c^2*C*d - 8*B*c*d^2 + 48*A*d^3 - 3*c^3*D)*((x*(c^2 - d^2* 
x^2)^(3/2))/4 + (3*c^2*((x*Sqrt[c^2 - d^2*x^2])/2 + (c^2*ArcTan[(d*x)/Sqrt 
[c^2 - d^2*x^2]])/(2*d)))/4))/6))/(8*d^5)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

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

rule 667

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*( 
x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + c*x^2)^(p - 1), x] 
/; FreeQ[{a, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0]
 

rule 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

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. \(682\) vs. \(2(284)=568\).

Time = 0.42 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.19

method	result	size
default	\(\frac {B \,d^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )+D c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )-\frac {\left (C d -D c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{7 d}+D d^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{8 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{8 d^{2}}\right )-C c d \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{d^{3}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5}+c d \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{4 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )\right )}{d^{4}}\)	\(683\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.04 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {210 \, {\left (3 \, D c^{8} - 8 \, C c^{7} d + 8 \, B c^{6} d^{2} - 48 \, A c^{5} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (1680 \, D d^{7} x^{7} - 768 \, D c^{7} + 768 \, C c^{6} d - 2688 \, B c^{5} d^{2} + 2688 \, A c^{4} d^{3} - 1920 \, {\left (D c d^{6} - C d^{7}\right )} x^{6} - 280 \, {\left (9 \, D c^{2} d^{5} + 8 \, C c d^{6} - 8 \, B d^{7}\right )} x^{5} + 384 \, {\left (8 \, D c^{3} d^{4} - 8 \, C c^{2} d^{5} - 7 \, B c d^{6} + 7 \, A d^{7}\right )} x^{4} + 70 \, {\left (3 \, D c^{4} d^{3} + 56 \, C c^{3} d^{4} - 56 \, B c^{2} d^{5} - 48 \, A c d^{6}\right )} x^{3} - 384 \, {\left (D c^{5} d^{2} - C c^{4} d^{3} - 14 \, B c^{3} d^{4} + 14 \, A c^{2} d^{5}\right )} x^{2} + 105 \, {\left (3 \, D c^{6} d - 8 \, C c^{5} d^{2} + 8 \, B c^{4} d^{3} + 80 \, A c^{3} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{13440 \, d^{4}} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c),x, algorithm="f 
ricas")
 

Output:

1/13440*(210*(3*D*c^8 - 8*C*c^7*d + 8*B*c^6*d^2 - 48*A*c^5*d^3)*arctan(-(c 
 - sqrt(-d^2*x^2 + c^2))/(d*x)) + (1680*D*d^7*x^7 - 768*D*c^7 + 768*C*c^6* 
d - 2688*B*c^5*d^2 + 2688*A*c^4*d^3 - 1920*(D*c*d^6 - C*d^7)*x^6 - 280*(9* 
D*c^2*d^5 + 8*C*c*d^6 - 8*B*d^7)*x^5 + 384*(8*D*c^3*d^4 - 8*C*c^2*d^5 - 7* 
B*c*d^6 + 7*A*d^7)*x^4 + 70*(3*D*c^4*d^3 + 56*C*c^3*d^4 - 56*B*c^2*d^5 - 4 
8*A*c*d^6)*x^3 - 384*(D*c^5*d^2 - C*c^4*d^3 - 14*B*c^3*d^4 + 14*A*c^2*d^5) 
*x^2 + 105*(3*D*c^6*d - 8*C*c^5*d^2 + 8*B*c^4*d^3 + 80*A*c^3*d^4)*x)*sqrt( 
-d^2*x^2 + c^2))/d^4
 

Sympy [A] (verification not implemented)

Time = 4.08 (sec) , antiderivative size = 1459, normalized size of antiderivative = 4.68 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\text {Too large to display} \] Input:

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

Output:

A*c**3*Piecewise((c**2*Piecewise((log(-2*d**2*x + 2*sqrt(-d**2)*sqrt(c**2 
- d**2*x**2))/sqrt(-d**2), Ne(c**2, 0)), (x*log(x)/sqrt(-d**2*x**2), True) 
)/2 + x*sqrt(c**2 - d**2*x**2)/2, Ne(d**2, 0)), (x*sqrt(c**2), True)) - A* 
c**2*d*Piecewise((sqrt(c**2 - d**2*x**2)*(-c**2/(3*d**2) + x**2/3), Ne(d** 
2, 0)), (x**2*sqrt(c**2)/2, True)) - A*c*d**2*Piecewise((c**4*Piecewise((l 
og(-2*d**2*x + 2*sqrt(-d**2)*sqrt(c**2 - d**2*x**2))/sqrt(-d**2), Ne(c**2, 
 0)), (x*log(x)/sqrt(-d**2*x**2), True))/(8*d**2) + sqrt(c**2 - d**2*x**2) 
*(-c**2*x/(8*d**2) + x**3/4), Ne(d**2, 0)), (x**3*sqrt(c**2)/3, True)) + A 
*d**3*Piecewise((sqrt(c**2 - d**2*x**2)*(-2*c**4/(15*d**4) - c**2*x**2/(15 
*d**2) + x**4/5), Ne(d**2, 0)), (x**4*sqrt(c**2)/4, True)) + B*c**3*Piecew 
ise((sqrt(c**2 - d**2*x**2)*(-c**2/(3*d**2) + x**2/3), Ne(d**2, 0)), (x**2 
*sqrt(c**2)/2, True)) - B*c**2*d*Piecewise((c**4*Piecewise((log(-2*d**2*x 
+ 2*sqrt(-d**2)*sqrt(c**2 - d**2*x**2))/sqrt(-d**2), Ne(c**2, 0)), (x*log( 
x)/sqrt(-d**2*x**2), True))/(8*d**2) + sqrt(c**2 - d**2*x**2)*(-c**2*x/(8* 
d**2) + x**3/4), Ne(d**2, 0)), (x**3*sqrt(c**2)/3, True)) - B*c*d**2*Piece 
wise((sqrt(c**2 - d**2*x**2)*(-2*c**4/(15*d**4) - c**2*x**2/(15*d**2) + x* 
*4/5), Ne(d**2, 0)), (x**4*sqrt(c**2)/4, True)) + B*d**3*Piecewise((c**6*P 
iecewise((log(-2*d**2*x + 2*sqrt(-d**2)*sqrt(c**2 - d**2*x**2))/sqrt(-d**2 
), Ne(c**2, 0)), (x*log(x)/sqrt(-d**2*x**2), True))/(16*d**4) + sqrt(c**2 
- d**2*x**2)*(-c**4*x/(16*d**4) - c**2*x**3/(24*d**2) + x**5/6), Ne(d**...
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.35 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx =\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c),x, algorithm="m 
axima")
 

Output:

3/8*I*D*c^8*arcsin(d*x/c + 2)/d^4 - 3/8*I*C*c^7*arcsin(d*x/c + 2)/d^3 + 3/ 
8*I*B*c^6*arcsin(d*x/c + 2)/d^2 - 3/8*I*A*c^5*arcsin(d*x/c + 2)/d + 45/128 
*D*c^8*arcsin(d*x/c)/d^4 - 5/16*C*c^7*arcsin(d*x/c)/d^3 + 5/16*B*c^6*arcsi 
n(d*x/c)/d^2 + 3/8*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*A*c^3*x - 3/8*sqrt(d^2* 
x^2 + 4*c*d*x + 3*c^2)*D*c^6*x/d^3 + 45/128*sqrt(-d^2*x^2 + c^2)*D*c^6*x/d 
^3 + 3/8*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*C*c^5*x/d^2 - 5/16*sqrt(-d^2*x^2 
+ c^2)*C*c^5*x/d^2 - 3/8*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*B*c^4*x/d + 5/16* 
sqrt(-d^2*x^2 + c^2)*B*c^4*x/d - 3/4*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*D*c^7 
/d^4 + 3/4*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*C*c^6/d^3 - 3/4*sqrt(d^2*x^2 + 
4*c*d*x + 3*c^2)*B*c^5/d^2 + 3/4*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*A*c^4/d + 
 1/4*(-d^2*x^2 + c^2)^(3/2)*A*c*x - 1/64*(-d^2*x^2 + c^2)^(3/2)*D*c^4*x/d^ 
3 + 1/24*(-d^2*x^2 + c^2)^(3/2)*C*c^3*x/d^2 - 1/24*(-d^2*x^2 + c^2)^(3/2)* 
B*c^2*x/d + 3/16*(-d^2*x^2 + c^2)^(5/2)*D*c^2*x/d^3 - 1/6*(-d^2*x^2 + c^2) 
^(5/2)*C*c*x/d^2 + 1/6*(-d^2*x^2 + c^2)^(5/2)*B*x/d - 1/5*(-d^2*x^2 + c^2) 
^(5/2)*D*c^3/d^4 + 1/5*(-d^2*x^2 + c^2)^(5/2)*C*c^2/d^3 - 1/5*(-d^2*x^2 + 
c^2)^(5/2)*B*c/d^2 + 1/5*(-d^2*x^2 + c^2)^(5/2)*A/d - 1/8*(-d^2*x^2 + c^2) 
^(7/2)*D*x/d^3 + 1/7*(-d^2*x^2 + c^2)^(7/2)*D*c/d^4 - 1/7*(-d^2*x^2 + c^2) 
^(7/2)*C/d^3
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {1}{13440} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, D d^{3} x - \frac {8 \, {\left (D c d^{14} - C d^{15}\right )}}{d^{12}}\right )} x - \frac {7 \, {\left (9 \, D c^{2} d^{13} + 8 \, C c d^{14} - 8 \, B d^{15}\right )}}{d^{12}}\right )} x + \frac {48 \, {\left (8 \, D c^{3} d^{12} - 8 \, C c^{2} d^{13} - 7 \, B c d^{14} + 7 \, A d^{15}\right )}}{d^{12}}\right )} x + \frac {35 \, {\left (3 \, D c^{4} d^{11} + 56 \, C c^{3} d^{12} - 56 \, B c^{2} d^{13} - 48 \, A c d^{14}\right )}}{d^{12}}\right )} x - \frac {192 \, {\left (D c^{5} d^{10} - C c^{4} d^{11} - 14 \, B c^{3} d^{12} + 14 \, A c^{2} d^{13}\right )}}{d^{12}}\right )} x + \frac {105 \, {\left (3 \, D c^{6} d^{9} - 8 \, C c^{5} d^{10} + 8 \, B c^{4} d^{11} + 80 \, A c^{3} d^{12}\right )}}{d^{12}}\right )} x - \frac {384 \, {\left (2 \, D c^{7} d^{8} - 2 \, C c^{6} d^{9} + 7 \, B c^{5} d^{10} - 7 \, A c^{4} d^{11}\right )}}{d^{12}}\right )} - \frac {{\left (3 \, D c^{8} - 8 \, C c^{7} d + 8 \, B c^{6} d^{2} - 48 \, A c^{5} d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{128 \, d^{3} {\left | d \right |}} \] Input:

integrate((-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c),x, algorithm="g 
iac")
 

Output:

1/13440*sqrt(-d^2*x^2 + c^2)*((2*((4*(5*(6*(7*D*d^3*x - 8*(D*c*d^14 - C*d^ 
15)/d^12)*x - 7*(9*D*c^2*d^13 + 8*C*c*d^14 - 8*B*d^15)/d^12)*x + 48*(8*D*c 
^3*d^12 - 8*C*c^2*d^13 - 7*B*c*d^14 + 7*A*d^15)/d^12)*x + 35*(3*D*c^4*d^11 
 + 56*C*c^3*d^12 - 56*B*c^2*d^13 - 48*A*c*d^14)/d^12)*x - 192*(D*c^5*d^10 
- C*c^4*d^11 - 14*B*c^3*d^12 + 14*A*c^2*d^13)/d^12)*x + 105*(3*D*c^6*d^9 - 
 8*C*c^5*d^10 + 8*B*c^4*d^11 + 80*A*c^3*d^12)/d^12)*x - 384*(2*D*c^7*d^8 - 
 2*C*c^6*d^9 + 7*B*c^5*d^10 - 7*A*c^4*d^11)/d^12) - 1/128*(3*D*c^8 - 8*C*c 
^7*d + 8*B*c^6*d^2 - 48*A*c^5*d^3)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d) 
)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {720 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{5} d^{2}-120 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{6} d +75 \mathit {asin} \left (\frac {d x}{c}\right ) c^{8}+384 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{2}+1200 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{3} x -768 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{4} x^{2}-480 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{5} x^{3}+384 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{6} x^{4}-384 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d +120 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{2} x +768 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{3} x^{2}-560 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{4} x^{3}-384 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{5} x^{4}+320 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{6} x^{5}-75 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d x +590 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{3} x^{3}-680 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{5} x^{5}+240 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{7} x^{7}-384 a \,c^{5} d^{2}+384 b \,c^{6} d}{1920 d^{3}} \] Input:

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

Output:

(720*asin((d*x)/c)*a*c**5*d**2 - 120*asin((d*x)/c)*b*c**6*d + 75*asin((d*x 
)/c)*c**8 + 384*sqrt(c**2 - d**2*x**2)*a*c**4*d**2 + 1200*sqrt(c**2 - d**2 
*x**2)*a*c**3*d**3*x - 768*sqrt(c**2 - d**2*x**2)*a*c**2*d**4*x**2 - 480*s 
qrt(c**2 - d**2*x**2)*a*c*d**5*x**3 + 384*sqrt(c**2 - d**2*x**2)*a*d**6*x* 
*4 - 384*sqrt(c**2 - d**2*x**2)*b*c**5*d + 120*sqrt(c**2 - d**2*x**2)*b*c* 
*4*d**2*x + 768*sqrt(c**2 - d**2*x**2)*b*c**3*d**3*x**2 - 560*sqrt(c**2 - 
d**2*x**2)*b*c**2*d**4*x**3 - 384*sqrt(c**2 - d**2*x**2)*b*c*d**5*x**4 + 3 
20*sqrt(c**2 - d**2*x**2)*b*d**6*x**5 - 75*sqrt(c**2 - d**2*x**2)*c**6*d*x 
 + 590*sqrt(c**2 - d**2*x**2)*c**4*d**3*x**3 - 680*sqrt(c**2 - d**2*x**2)* 
c**2*d**5*x**5 + 240*sqrt(c**2 - d**2*x**2)*d**7*x**7 - 384*a*c**5*d**2 + 
384*b*c**6*d)/(1920*d**3)