\(\int \frac {\sqrt {c^2-d^2 x^2} (A+B x+C x^2+D x^3)}{(c+d x)^{9/2}} \, dx\) [200]

Optimal result

Integrand size = 41, antiderivative size = 292 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c^2-d^2 x^2}}{3 d^4 (c+d x)^{7/2}}+\frac {\left (25 c^2 C d-13 B c d^2+A d^3-37 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{24 c d^4 (c+d x)^{5/2}}-\frac {\left (39 c^2 C d-3 B c d^2-A d^3-107 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{32 c^2 d^4 (c+d x)^{3/2}}+\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}+\frac {\left (25 c^2 C d+3 B c d^2+A d^3-213 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{32 \sqrt {2} c^{5/2} d^4} \] Output:

-1/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(7/2)+ 
1/24*(A*d^3-13*B*c*d^2+25*C*c^2*d-37*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c/d^4/(d* 
x+c)^(5/2)-1/32*(-A*d^3-3*B*c*d^2+39*C*c^2*d-107*D*c^3)*(-d^2*x^2+c^2)^(1/ 
2)/c^2/d^4/(d*x+c)^(3/2)+2*D*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(1/2)+1/64*( 
A*d^3+3*B*c*d^2+25*C*c^2*d-213*D*c^3)*arctanh(2^(1/2)*c^(1/2)*(d*x+c)^(1/2 
)/(-d^2*x^2+c^2)^(1/2))*2^(1/2)/c^(5/2)/d^4
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (397 c^5 D+3 A d^5 x^2+c d^4 x (10 A+9 B x)+c^4 (-49 C d+1070 d D x)+c^3 d^2 (-11 B+x (-134 C+897 D x))-c^2 d^3 (25 A+x (34 B+3 x (39 C-64 D x)))\right )}{(c+d x)^{7/2}}-3 \sqrt {2} \left (-25 c^2 C d-3 B c d^2-A d^3+213 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{192 c^{5/2} d^4} \] Input:

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

Output:

((2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(397*c^5*D + 3*A*d^5*x^2 + c*d^4*x*(10*A + 
 9*B*x) + c^4*(-49*C*d + 1070*d*D*x) + c^3*d^2*(-11*B + x*(-134*C + 897*D* 
x)) - c^2*d^3*(25*A + x*(34*B + 3*x*(39*C - 64*D*x)))))/(c + d*x)^(7/2) - 
3*Sqrt[2]*(-25*c^2*C*d - 3*B*c*d^2 - A*d^3 + 213*c^3*D)*ArcTanh[(Sqrt[2]*S 
qrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(192*c^(5/2)*d^4)
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {2170, 27, 2170, 27, 671, 465, 470, 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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {\sqrt {c^2-d^2 x^2} \left ((C d-7 c D) x^2 d^4+\left (B d^2-11 c^2 D\right ) x d^3+\left (A d^3-5 c^3 D\right ) d^2\right )}{2 (c+d x)^{9/2}}dx}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c^2-d^2 x^2} \left ((C d-7 c D) x^2 d^4+\left (B d^2-11 c^2 D\right ) x d^3+\left (A d^3-5 c^3 D\right ) d^2\right )}{(c+d x)^{9/2}}dx}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {\frac {2 \int \frac {d^6 \left (-54 D c^3+7 C d c^2+A d^3+d \left (-53 D c^2+6 C d c+B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{2 (c+d x)^{9/2}}dx}{d^4}+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \int \frac {\left (-54 D c^3+7 C d c^2+A d^3+d \left (-53 D c^2+6 C d c+B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^{9/2}}dx+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {d^2 \left (\frac {\left (A d^3+3 B c d^2-213 c^3 D+25 c^2 C d\right ) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^{7/2}}dx}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{6 c d (c+d x)^{9/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 465

\(\displaystyle \frac {d^2 \left (\frac {\left (A d^3+3 B c d^2-213 c^3 D+25 c^2 C d\right ) \left (-\frac {1}{4} \int \frac {1}{(c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}dx-\frac {\sqrt {c^2-d^2 x^2}}{2 d (c+d x)^{5/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{6 c d (c+d x)^{9/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {d^2 \left (\frac {\left (A d^3+3 B c d^2-213 c^3 D+25 c^2 C d\right ) \left (\frac {1}{4} \left (\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}-\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{4 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{2 d (c+d x)^{5/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{6 c d (c+d x)^{9/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {d^2 \left (\frac {\left (A d^3+3 B c d^2-213 c^3 D+25 c^2 C d\right ) \left (\frac {1}{4} \left (\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}-\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{2 c}\right )-\frac {\sqrt {c^2-d^2 x^2}}{2 d (c+d x)^{5/2}}\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{6 c d (c+d x)^{9/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \left (\frac {\left (\frac {1}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{2 \sqrt {2} c^{3/2} d}+\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )-\frac {\sqrt {c^2-d^2 x^2}}{2 d (c+d x)^{5/2}}\right ) \left (A d^3+3 B c d^2-213 c^3 D+25 c^2 C d\right )}{4 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{6 c d (c+d x)^{9/2}}\right )+\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-7 c D)}{(c+d x)^{7/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}\)

Input:

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

Output:

(-2*D*(c^2 - d^2*x^2)^(3/2))/(d^4*(c + d*x)^(5/2)) + ((2*d*(C*d - 7*c*D)*( 
c^2 - d^2*x^2)^(3/2))/(c + d*x)^(7/2) + d^2*(-1/6*((c^2*C*d - B*c*d^2 + A* 
d^3 - c^3*D)*(c^2 - d^2*x^2)^(3/2))/(c*d*(c + d*x)^(9/2)) + ((25*c^2*C*d + 
 3*B*c*d^2 + A*d^3 - 213*c^3*D)*(-1/2*Sqrt[c^2 - d^2*x^2]/(d*(c + d*x)^(5/ 
2)) + (Sqrt[c^2 - d^2*x^2]/(2*c*d*(c + d*x)^(3/2)) + ArcTanh[Sqrt[c^2 - d^ 
2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])]/(2*Sqrt[2]*c^(3/2)*d))/4))/(4*c))) 
/d^5
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 465

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 + p + 1))), x] - Simp[b*(p/(d^2*(n + 
 p + 1)))   Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, 
b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LtQ[n, -2] || EqQ[n 
+ 2*p + 1, 0]) && NeQ[n + p + 1, 0] && 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. \(749\) vs. \(2(256)=512\).

Time = 0.40 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.57

method	result	size
default	\(\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (9 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{5} x^{3}+75 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x^{3}-639 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x^{3}+225 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2} x +2140 D \sqrt {-d x +c}\, c^{\frac {9}{2}} d x -1917 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5} d x +9 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2}-98 C \sqrt {-d x +c}\, c^{\frac {9}{2}} d +75 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5} d -1917 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d^{2} x^{2}+9 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x +27 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x +794 D \sqrt {-d x +c}\, c^{\frac {11}{2}}+27 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{4} x^{2}+9 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{5} x^{2}+18 B \,c^{\frac {3}{2}} d^{4} x^{2} \sqrt {-d x +c}+20 A \,c^{\frac {3}{2}} d^{4} x \sqrt {-d x +c}-268 C \sqrt {-d x +c}\, c^{\frac {7}{2}} d^{2} x -234 C \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{3} x^{2}-68 B \,c^{\frac {5}{2}} d^{3} x \sqrt {-d x +c}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{6} x^{3}-50 A \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{3}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3}-22 B \sqrt {-d x +c}\, c^{\frac {7}{2}} d^{2}+1794 D \sqrt {-d x +c}\, c^{\frac {7}{2}} d^{2} x^{2}+225 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{3} x^{2}-639 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}+6 A \,d^{5} x^{2} \sqrt {-d x +c}\, \sqrt {c}+384 D \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{3} x^{3}\right )}{192 c^{\frac {5}{2}} \left (d x +c \right )^{\frac {7}{2}} \sqrt {-d x +c}\, d^{4}}\)	\(750\)

Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x,method=_RETUR 
NVERBOSE)
 

Output:

1/192*(-d^2*x^2+c^2)^(1/2)/c^(5/2)*(9*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2) 
*2^(1/2)/c^(1/2))*c*d^5*x^3+75*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2 
)/c^(1/2))*c^2*d^4*x^3-639*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^ 
(1/2))*c^3*d^3*x^3+225*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2 
))*c^4*d^2*x+2140*D*(-d*x+c)^(1/2)*c^(9/2)*d*x-1917*D*2^(1/2)*arctanh(1/2* 
(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d*x+9*B*2^(1/2)*arctanh(1/2*(-d*x+c)^( 
1/2)*2^(1/2)/c^(1/2))*c^4*d^2-98*C*(-d*x+c)^(1/2)*c^(9/2)*d+75*C*2^(1/2)*a 
rctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5*d-1917*D*2^(1/2)*arctanh(1/ 
2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d^2*x^2+9*A*2^(1/2)*arctanh(1/2*(-d* 
x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^4*x+27*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1 
/2)*2^(1/2)/c^(1/2))*c^3*d^3*x+794*D*(-d*x+c)^(1/2)*c^(11/2)+27*B*2^(1/2)* 
arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^4*x^2+9*A*2^(1/2)*arctan 
h(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^5*x^2+18*B*c^(3/2)*d^4*x^2*(-d*x 
+c)^(1/2)+20*A*c^(3/2)*d^4*x*(-d*x+c)^(1/2)-268*C*(-d*x+c)^(1/2)*c^(7/2)*d 
^2*x-234*C*(-d*x+c)^(1/2)*c^(5/2)*d^3*x^2-68*B*c^(5/2)*d^3*x*(-d*x+c)^(1/2 
)+3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*d^6*x^3-50*A*(-d 
*x+c)^(1/2)*c^(5/2)*d^3+3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^( 
1/2))*c^3*d^3-22*B*(-d*x+c)^(1/2)*c^(7/2)*d^2+1794*D*(-d*x+c)^(1/2)*c^(7/2 
)*d^2*x^2+225*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d^ 
3*x^2-639*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^6+6*A...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x, algori 
thm="fricas")
 

Output:

[-1/384*(3*sqrt(2)*(213*D*c^7 - 25*C*c^6*d - 3*B*c^5*d^2 - A*c^4*d^3 + (21 
3*D*c^3*d^4 - 25*C*c^2*d^5 - 3*B*c*d^6 - A*d^7)*x^4 + 4*(213*D*c^4*d^3 - 2 
5*C*c^3*d^4 - 3*B*c^2*d^5 - A*c*d^6)*x^3 + 6*(213*D*c^5*d^2 - 25*C*c^4*d^3 
 - 3*B*c^3*d^4 - A*c^2*d^5)*x^2 + 4*(213*D*c^6*d - 25*C*c^5*d^2 - 3*B*c^4* 
d^3 - A*c^3*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*( 
192*D*c^3*d^3*x^3 + 397*D*c^6 - 49*C*c^5*d - 11*B*c^4*d^2 - 25*A*c^3*d^3 + 
 3*(299*D*c^4*d^2 - 39*C*c^3*d^3 + 3*B*c^2*d^4 + A*c*d^5)*x^2 + 2*(535*D*c 
^5*d - 67*C*c^4*d^2 - 17*B*c^3*d^3 + 5*A*c^2*d^4)*x)*sqrt(-d^2*x^2 + c^2)* 
sqrt(d*x + c))/(c^3*d^8*x^4 + 4*c^4*d^7*x^3 + 6*c^5*d^6*x^2 + 4*c^6*d^5*x 
+ c^7*d^4), 1/192*(3*sqrt(2)*(213*D*c^7 - 25*C*c^6*d - 3*B*c^5*d^2 - A*c^4 
*d^3 + (213*D*c^3*d^4 - 25*C*c^2*d^5 - 3*B*c*d^6 - A*d^7)*x^4 + 4*(213*D*c 
^4*d^3 - 25*C*c^3*d^4 - 3*B*c^2*d^5 - A*c*d^6)*x^3 + 6*(213*D*c^5*d^2 - 25 
*C*c^4*d^3 - 3*B*c^3*d^4 - A*c^2*d^5)*x^2 + 4*(213*D*c^6*d - 25*C*c^5*d^2 
- 3*B*c^4*d^3 - A*c^3*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*(192*D*c^3*d^3*x^3 + 397*D* 
c^6 - 49*C*c^5*d - 11*B*c^4*d^2 - 25*A*c^3*d^3 + 3*(299*D*c^4*d^2 - 39*C*c 
^3*d^3 + 3*B*c^2*d^4 + A*c*d^5)*x^2 + 2*(535*D*c^5*d - 67*C*c^4*d^2 - 17*B 
*c^3*d^3 + 5*A*c^2*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^3*d^8*x^ 
4 + 4*c^4*d^7*x^3 + 6*c^5*d^6*x^2 + 4*c^6*d^5*x + c^7*d^4)]
 

Sympy [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int \frac {\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(9/2),x)
 

Output:

Integral(sqrt(-(-c + d*x)*(c + d*x))*(A + B*x + C*x**2 + D*x**3)/(c + d*x) 
**(9/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int { \frac {\sqrt {-d^{2} x^{2} + c^{2}} {\left (D x^{3} + C x^{2} + B x + A\right )}}{{\left (d x + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x, algori 
thm="maxima")
 

Output:

integrate(sqrt(-d^2*x^2 + c^2)*(D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(9/2), 
x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\frac {384 \, \sqrt {-d x + c} D + \frac {3 \, \sqrt {2} {\left (213 \, D c^{3} - 25 \, C c^{2} d - 3 \, B c d^{2} - A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{2}} + \frac {2 \, {\left (321 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} D c^{3} - 1136 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{4} + 1020 \, \sqrt {-d x + c} D c^{5} - 117 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} C c^{2} d + 368 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{3} d - 300 \, \sqrt {-d x + c} C c^{4} d + 9 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} B c d^{2} + 16 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c^{2} d^{2} - 36 \, \sqrt {-d x + c} B c^{3} d^{2} + 3 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} A d^{3} - 16 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c d^{3} - 12 \, \sqrt {-d x + c} A c^{2} d^{3}\right )}}{{\left (d x + c\right )}^{3} c^{2}}}{192 \, d^{4}} \] Input:

integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x, algori 
thm="giac")
 

Output:

1/192*(384*sqrt(-d*x + c)*D + 3*sqrt(2)*(213*D*c^3 - 25*C*c^2*d - 3*B*c*d^ 
2 - A*d^3)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c^2) + 2* 
(321*(d*x - c)^2*sqrt(-d*x + c)*D*c^3 - 1136*(-d*x + c)^(3/2)*D*c^4 + 1020 
*sqrt(-d*x + c)*D*c^5 - 117*(d*x - c)^2*sqrt(-d*x + c)*C*c^2*d + 368*(-d*x 
 + c)^(3/2)*C*c^3*d - 300*sqrt(-d*x + c)*C*c^4*d + 9*(d*x - c)^2*sqrt(-d*x 
 + c)*B*c*d^2 + 16*(-d*x + c)^(3/2)*B*c^2*d^2 - 36*sqrt(-d*x + c)*B*c^3*d^ 
2 + 3*(d*x - c)^2*sqrt(-d*x + c)*A*d^3 - 16*(-d*x + c)^(3/2)*A*c*d^3 - 12* 
sqrt(-d*x + c)*A*c^2*d^3)/((d*x + c)^3*c^2))/d^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{9/2}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(9/2),x)
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{9/2}} \, dx=\int \frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {9}{2}}}d x \] Input:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x)
 

Output:

int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(9/2),x)