\(\int \frac {(c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^2} \, dx\) [158]

Optimal result

Integrand size = 39, antiderivative size = 311 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {c^2 \left (3 c^2 C d-4 B c d^2+10 A d^3-2 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{16 d^3}+\frac {2 c \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4}-\frac {\left (3 c^2 C d-4 B c d^2+2 A d^3-2 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{3/2}}{8 d^3}-\frac {(C d-2 c D) x^3 \left (c^2-d^2 x^2\right )^{3/2}}{6 d}-\frac {\left (2 c C d-B d^2-3 c^2 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}+\frac {c^4 \left (3 c^2 C d-4 B c d^2+10 A d^3-2 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{16 d^4} \] Output:

1/16*c^2*(10*A*d^3-4*B*c*d^2+3*C*c^2*d-2*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/d^3 
+2/3*c*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(3/2)/d^4-1/8*(2*A*d^3 
-4*B*c*d^2+3*C*c^2*d-2*D*c^3)*x*(-d^2*x^2+c^2)^(3/2)/d^3-1/6*(C*d-2*D*c)*x 
^3*(-d^2*x^2+c^2)^(3/2)/d-1/5*(-B*d^2+2*C*c*d-3*D*c^2)*(-d^2*x^2+c^2)^(5/2 
)/d^4-1/7*D*(-d^2*x^2+c^2)^(7/2)/d^4+1/16*c^4*(10*A*d^3-4*B*c*d^2+3*C*c^2* 
d-2*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 1.96 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-352 c^6 D+14 c^5 d (32 C+15 D x)-c^4 d^2 (784 B+x (315 C+176 D x))-56 c d^5 x^2 (20 A+x (15 B+2 x (6 C+5 D x)))+28 c^3 d^3 (40 A+x (15 B+x (8 C+5 D x)))+4 d^6 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))+2 c^2 d^4 x (315 A+x (224 B+x (175 C+144 D x)))\right )+210 c^4 \left (-3 c^2 C d+4 B c d^2-10 A d^3+2 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{1680 d^4} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(-352*c^6*D + 14*c^5*d*(32*C + 15*D*x) - c^4*d^2*(784 
*B + x*(315*C + 176*D*x)) - 56*c*d^5*x^2*(20*A + x*(15*B + 2*x*(6*C + 5*D* 
x))) + 28*c^3*d^3*(40*A + x*(15*B + x*(8*C + 5*D*x))) + 4*d^6*x^3*(105*A + 
 2*x*(42*B + 5*x*(7*C + 6*D*x))) + 2*c^2*d^4*x*(315*A + x*(224*B + x*(175* 
C + 144*D*x)))) + 210*c^4*(-3*c^2*C*d + 4*B*c*d^2 - 10*A*d^3 + 2*c^3*D)*Ar 
cTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/(1680*d^4)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2170, 27, 2170, 27, 671, 466, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {-\frac {1}{6} d^2 \left (-\frac {\left (10 A d^3-4 B c d^2-2 c^3 D+3 c^2 C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{5/2}}{c+d x}dx}{c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{c d (c+d x)^2}\right )-\frac {d \left (c^2-d^2 x^2\right )^{7/2} (C d-2 c D)}{6 (c+d x)}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}\)

\(\Big \downarrow \) 466

\(\displaystyle \frac {-\frac {1}{6} d^2 \left (-\frac {\left (10 A d^3-4 B c d^2-2 c^3 D+3 c^2 C d\right ) \left (c \int \left (c^2-d^2 x^2\right )^{3/2}dx+\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{c d (c+d x)^2}\right )-\frac {d \left (c^2-d^2 x^2\right )^{7/2} (C d-2 c D)}{6 (c+d x)}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {-\frac {1}{6} d^2 \left (-\frac {\left (10 A d^3-4 B c d^2-2 c^3 D+3 c^2 C d\right ) \left (c \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{c d (c+d x)^2}\right )-\frac {d \left (c^2-d^2 x^2\right )^{7/2} (C d-2 c D)}{6 (c+d x)}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {-\frac {1}{6} d^2 \left (-\frac {\left (10 A d^3-4 B c d^2-2 c^3 D+3 c^2 C d\right ) \left (c \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{c d (c+d x)^2}\right )-\frac {d \left (c^2-d^2 x^2\right )^{7/2} (C d-2 c D)}{6 (c+d x)}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {1}{6} d^2 \left (-\frac {\left (10 A d^3-4 B c d^2-2 c^3 D+3 c^2 C d\right ) \left (c \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )}{c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{c d (c+d x)^2}\right )-\frac {d \left (c^2-d^2 x^2\right )^{7/2} (C d-2 c D)}{6 (c+d x)}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {-\frac {1}{6} d^2 \left (-\frac {\left (c \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right ) \left (10 A d^3-4 B c d^2-2 c^3 D+3 c^2 C d\right )}{c}-\frac {2 \left (c^2-d^2 x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{c d (c+d x)^2}\right )-\frac {d \left (c^2-d^2 x^2\right )^{7/2} (C d-2 c D)}{6 (c+d x)}}{d^5}-\frac {D \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}\)

Input:

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

Output:

-1/7*(D*(c^2 - d^2*x^2)^(7/2))/d^4 + (-1/6*(d*(C*d - 2*c*D)*(c^2 - d^2*x^2 
)^(7/2))/(c + d*x) - (d^2*((-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c^2 - 
d^2*x^2)^(7/2))/(c*d*(c + d*x)^2) - ((3*c^2*C*d - 4*B*c*d^2 + 10*A*d^3 - 2 
*c^3*D)*((c^2 - d^2*x^2)^(5/2)/(5*d) + c*((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)))/c))/6)/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 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 466

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

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. \(701\) vs. \(2(283)=566\).

Time = 0.43 (sec) , antiderivative size = 702, normalized size of antiderivative = 2.26

method	result	size
default	\(\frac {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 )-\frac {D \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{7 d}-2 D c \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 (B \,d^{2}-2 C c d +3 D c^{2}\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}}+\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 {7}{2}}}{3 c d \left (x +\frac {c}{d}\right )^{2}}+\frac {5 d \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 )}{3 c}\right )}{d^{5}}\)	\(702\)

Input:

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

Output:

1/d^3*(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)))))-1/7*D/d*(-d^2*x^2+c^2)^(7/2)-2*D*c*(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/d^4*(B*d^2-2*C*c*d+3*D*c^2)*(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)))))+1/d^5*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(1/3/c/d/(x+c/d)^2* 
(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(7/2)+5/3*d/c*(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 = 0.11 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {210 \, {\left (2 \, D c^{7} - 3 \, C c^{6} d + 4 \, B c^{5} d^{2} - 10 \, A c^{4} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (240 \, D d^{6} x^{6} - 352 \, D c^{6} + 448 \, C c^{5} d - 784 \, B c^{4} d^{2} + 1120 \, A c^{3} d^{3} - 280 \, {\left (2 \, D c d^{5} - C d^{6}\right )} x^{5} + 48 \, {\left (6 \, D c^{2} d^{4} - 14 \, C c d^{5} + 7 \, B d^{6}\right )} x^{4} + 70 \, {\left (2 \, D c^{3} d^{3} + 5 \, C c^{2} d^{4} - 12 \, B c d^{5} + 6 \, A d^{6}\right )} x^{3} - 16 \, {\left (11 \, D c^{4} d^{2} - 14 \, C c^{3} d^{3} - 28 \, B c^{2} d^{4} + 70 \, A c d^{5}\right )} x^{2} + 105 \, {\left (2 \, D c^{5} d - 3 \, C c^{4} d^{2} + 4 \, B c^{3} d^{3} + 6 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{1680 \, d^{4}} \] Input:

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

Output:

1/1680*(210*(2*D*c^7 - 3*C*c^6*d + 4*B*c^5*d^2 - 10*A*c^4*d^3)*arctan(-(c 
- sqrt(-d^2*x^2 + c^2))/(d*x)) + (240*D*d^6*x^6 - 352*D*c^6 + 448*C*c^5*d 
- 784*B*c^4*d^2 + 1120*A*c^3*d^3 - 280*(2*D*c*d^5 - C*d^6)*x^5 + 48*(6*D*c 
^2*d^4 - 14*C*c*d^5 + 7*B*d^6)*x^4 + 70*(2*D*c^3*d^3 + 5*C*c^2*d^4 - 12*B* 
c*d^5 + 6*A*d^6)*x^3 - 16*(11*D*c^4*d^2 - 14*C*c^3*d^3 - 28*B*c^2*d^4 + 70 
*A*c*d^5)*x^2 + 105*(2*D*c^5*d - 3*C*c^4*d^2 + 4*B*c^3*d^3 + 6*A*c^2*d^4)* 
x)*sqrt(-d^2*x^2 + c^2))/d^4
 

Sympy [A] (verification not implemented)

Time = 5.09 (sec) , antiderivative size = 1047, normalized size of antiderivative = 3.37 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

A*c**2*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)) - 2* 
A*c*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*d**2*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* 
*2*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)) - 2*B*c*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*d**2 
*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)) + C*c**2*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)) - 2*C*c*d*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, T...
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.56 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(-d^2*x^2 + c^2)^(5/2)*D*c^3/(d^5*x + c*d^4) - 1/2*I*D*c^7*arcsin(d*x 
/c + 2)/d^4 + 1/8*I*C*c^6*arcsin(d*x/c + 2)/d^3 + 1/4*I*B*c^5*arcsin(d*x/c 
 + 2)/d^2 - 5/8*I*A*c^4*arcsin(d*x/c + 2)/d - 5/8*D*c^7*arcsin(d*x/c)/d^4 
+ 5/16*C*c^6*arcsin(d*x/c)/d^3 + 1/4*(-d^2*x^2 + c^2)^(5/2)*C*c^2/(d^4*x + 
 c*d^3) + 5/8*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*A*c^2*x + 1/2*sqrt(d^2*x^2 + 
 4*c*d*x + 3*c^2)*D*c^5*x/d^3 - 5/8*sqrt(-d^2*x^2 + c^2)*D*c^5*x/d^3 - 1/8 
*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*C*c^4*x/d^2 + 5/16*sqrt(-d^2*x^2 + c^2)*C 
*c^4*x/d^2 - 1/4*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*B*c^3*x/d - 1/4*(-d^2*x^2 
 + c^2)^(5/2)*B*c/(d^3*x + c*d^2) + sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*D*c^6/ 
d^4 - 1/4*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*C*c^5/d^3 - 1/2*sqrt(d^2*x^2 + 4 
*c*d*x + 3*c^2)*B*c^4/d^2 + 5/4*sqrt(d^2*x^2 + 4*c*d*x + 3*c^2)*A*c^3/d + 
1/3*(-d^2*x^2 + c^2)^(3/2)*D*c^3*x/d^3 - 7/24*(-d^2*x^2 + c^2)^(3/2)*C*c^2 
*x/d^2 + 1/4*(-d^2*x^2 + c^2)^(3/2)*B*c*x/d + 1/4*(-d^2*x^2 + c^2)^(5/2)*A 
/(d^2*x + c*d) - 5/12*(-d^2*x^2 + c^2)^(3/2)*D*c^4/d^4 + 5/12*(-d^2*x^2 + 
c^2)^(3/2)*C*c^3/d^3 - 5/12*(-d^2*x^2 + c^2)^(3/2)*B*c^2/d^2 + 5/12*(-d^2* 
x^2 + c^2)^(3/2)*A*c/d - 1/3*(-d^2*x^2 + c^2)^(5/2)*D*c*x/d^3 + 1/6*(-d^2* 
x^2 + c^2)^(5/2)*C*x/d^2 + 3/5*(-d^2*x^2 + c^2)^(5/2)*D*c^2/d^4 - 2/5*(-d^ 
2*x^2 + c^2)^(5/2)*C*c/d^3 + 1/5*(-d^2*x^2 + c^2)^(5/2)*B/d^2 - 1/7*(-d^2* 
x^2 + c^2)^(7/2)*D/d^4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (283) = 566\).

Time = 0.23 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.35 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/107520*(13440*(2*D*c^8*d^8*sgn(1/(d*x + c))*sgn(d) - 3*C*c^7*d^9*sgn(1/( 
d*x + c))*sgn(d) + 4*B*c^6*d^10*sgn(1/(d*x + c))*sgn(d) - 10*A*c^5*d^11*sg 
n(1/(d*x + c))*sgn(d))*arctan(sqrt(2*c/(d*x + c) - 1)) + (210*D*c^8*d^8*(2 
*c/(d*x + c) - 1)^(13/2)*sgn(1/(d*x + c))*sgn(d) - 315*C*c^7*d^9*(2*c/(d*x 
 + c) - 1)^(13/2)*sgn(1/(d*x + c))*sgn(d) + 420*B*c^6*d^10*(2*c/(d*x + c) 
- 1)^(13/2)*sgn(1/(d*x + c))*sgn(d) - 1050*A*c^5*d^11*(2*c/(d*x + c) - 1)^ 
(13/2)*sgn(1/(d*x + c))*sgn(d) - 7560*D*c^8*d^8*(2*c/(d*x + c) - 1)^(11/2) 
*sgn(1/(d*x + c))*sgn(d) + 6860*C*c^7*d^9*(2*c/(d*x + c) - 1)^(11/2)*sgn(1 
/(d*x + c))*sgn(d) - 6160*B*c^6*d^10*(2*c/(d*x + c) - 1)^(11/2)*sgn(1/(d*x 
 + c))*sgn(d) + 1960*A*c^5*d^11*(2*c/(d*x + c) - 1)^(11/2)*sgn(1/(d*x + c) 
)*sgn(d) + 378*D*c^8*d^8*(2*c/(d*x + c) - 1)^(9/2)*sgn(1/(d*x + c))*sgn(d) 
 + 8393*C*c^7*d^9*(2*c/(d*x + c) - 1)^(9/2)*sgn(1/(d*x + c))*sgn(d) - 1716 
4*B*c^6*d^10*(2*c/(d*x + c) - 1)^(9/2)*sgn(1/(d*x + c))*sgn(d) + 16030*A*c 
^5*d^11*(2*c/(d*x + c) - 1)^(9/2)*sgn(1/(d*x + c))*sgn(d) - 9984*D*c^8*d^8 
*(2*c/(d*x + c) - 1)^(7/2)*sgn(1/(d*x + c))*sgn(d) + 5376*C*c^7*d^9*(2*c/( 
d*x + c) - 1)^(7/2)*sgn(1/(d*x + c))*sgn(d) - 16128*B*c^6*d^10*(2*c/(d*x + 
 c) - 1)^(7/2)*sgn(1/(d*x + c))*sgn(d) + 26880*A*c^5*d^11*(2*c/(d*x + c) - 
 1)^(7/2)*sgn(1/(d*x + c))*sgn(d) - 3962*D*c^8*d^8*(2*c/(d*x + c) - 1)^(5/ 
2)*sgn(1/(d*x + c))*sgn(d) + 5943*C*c^7*d^9*(2*c/(d*x + c) - 1)^(5/2)*sgn( 
1/(d*x + c))*sgn(d) - 7924*B*c^6*d^10*(2*c/(d*x + c) - 1)^(5/2)*sgn(1/(...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.37 \[ \int \frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {1050 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{4} d^{2}-420 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{5} d +105 \mathit {asin} \left (\frac {d x}{c}\right ) c^{7}+1120 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{2}+630 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{3} x -1120 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{4} x^{2}+420 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{5} x^{3}-784 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d +420 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{2} x +448 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{3} x^{2}-840 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{4} x^{3}+336 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{5} x^{4}+96 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}-105 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x +48 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}+490 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}-384 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}-280 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}+240 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}-1120 a \,c^{4} d^{2}+784 b \,c^{5} d -96 c^{7}}{1680 d^{3}} \] Input:

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

Output:

(1050*asin((d*x)/c)*a*c**4*d**2 - 420*asin((d*x)/c)*b*c**5*d + 105*asin((d 
*x)/c)*c**7 + 1120*sqrt(c**2 - d**2*x**2)*a*c**3*d**2 + 630*sqrt(c**2 - d* 
*2*x**2)*a*c**2*d**3*x - 1120*sqrt(c**2 - d**2*x**2)*a*c*d**4*x**2 + 420*s 
qrt(c**2 - d**2*x**2)*a*d**5*x**3 - 784*sqrt(c**2 - d**2*x**2)*b*c**4*d + 
420*sqrt(c**2 - d**2*x**2)*b*c**3*d**2*x + 448*sqrt(c**2 - d**2*x**2)*b*c* 
*2*d**3*x**2 - 840*sqrt(c**2 - d**2*x**2)*b*c*d**4*x**3 + 336*sqrt(c**2 - 
d**2*x**2)*b*d**5*x**4 + 96*sqrt(c**2 - d**2*x**2)*c**6 - 105*sqrt(c**2 - 
d**2*x**2)*c**5*d*x + 48*sqrt(c**2 - d**2*x**2)*c**4*d**2*x**2 + 490*sqrt( 
c**2 - d**2*x**2)*c**3*d**3*x**3 - 384*sqrt(c**2 - d**2*x**2)*c**2*d**4*x* 
*4 - 280*sqrt(c**2 - d**2*x**2)*c*d**5*x**5 + 240*sqrt(c**2 - d**2*x**2)*d 
**6*x**6 - 1120*a*c**4*d**2 + 784*b*c**5*d - 96*c**7)/(1680*d**3)