\(\int \frac {(c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3)}{(c+d x)^{7/2}} \, dx\) [208]

Optimal result

Integrand size = 41, antiderivative size = 315 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=-\frac {\left (11 c^2 C d-7 B c d^2+3 A d^3-15 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{d^4 (c+d x)^{5/2}}-\frac {2 \left (112 c C d-35 B d^2-207 c^2 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{105 d^4 (c+d x)^{3/2}}+\frac {2 (7 C d-27 c D) \left (c^2-d^2 x^2\right )^{3/2}}{35 d^4 \sqrt {c+d x}}+\frac {2 D \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{3/2}}{7 d^4}+\frac {\sqrt {2} \sqrt {c} \left (11 c^2 C d-7 B c d^2+3 A d^3-15 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{d^4} \] Output:

-(3*A*d^3-7*B*c*d^2+11*C*c^2*d-15*D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^ 
(1/2)-(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^(5/2) 
-2/105*(-35*B*d^2+112*C*c*d-207*D*c^2)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^(3 
/2)+2/35*(7*C*d-27*D*c)*(-d^2*x^2+c^2)^(3/2)/d^4/(d*x+c)^(1/2)+2/7*D*(d*x+ 
c)^(1/2)*(-d^2*x^2+c^2)^(3/2)/d^4+2^(1/2)*c^(1/2)*(3*A*d^3-7*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) 
)/d^4
 

Mathematica [A] (verified)

Time = 3.10 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.65 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\frac {\frac {2 \sqrt {c^2-d^2 x^2} \left (981 c^4 D+c^3 (-721 C d+684 d D x)+7 c^2 d^2 (65 B-18 x (4 C+D x))-d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))+c d^3 (-210 A+x (315 B+x (91 C+51 D x)))\right )}{(c+d x)^{3/2}}-105 \sqrt {2} \sqrt {c} \left (-11 c^2 C d+7 B c d^2-3 A d^3+15 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{105 d^4} \] Input:

Integrate[((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2 
),x]
 

Output:

((2*Sqrt[c^2 - d^2*x^2]*(981*c^4*D + c^3*(-721*C*d + 684*d*D*x) + 7*c^2*d^ 
2*(65*B - 18*x*(4*C + D*x)) - d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x))) 
 + c*d^3*(-210*A + x*(315*B + x*(91*C + 51*D*x)))))/(c + d*x)^(3/2) - 105* 
Sqrt[2]*Sqrt[c]*(-11*c^2*C*d + 7*B*c*d^2 - 3*A*d^3 + 15*c^3*D)*ArcTanh[(Sq 
rt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(105*d^4)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.95, 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, 466, 466, 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.

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

\(\Big \downarrow \) 2170

\(\displaystyle -\frac {2 \int -\frac {\left (c^2-d^2 x^2\right )^{3/2} \left ((7 C d-17 c D) x^2 d^4+\left (7 B d^2-13 c^2 D\right ) x d^3+\left (7 A d^3-3 c^3 D\right ) d^2\right )}{2 (c+d x)^{7/2}}dx}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left ((7 C d-17 c D) x^2 d^4+\left (7 B d^2-13 c^2 D\right ) x d^3+\left (7 A d^3-3 c^3 D\right ) d^2\right )}{(c+d x)^{7/2}}dx}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 2170

\(\displaystyle \frac {-\frac {2 \int \frac {35 d^6 \left (-2 D c^3+C d c^2-A d^3+d \left (-3 D c^2+2 C d c-B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{2 (c+d x)^{7/2}}dx}{5 d^4}-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-7 d^2 \int \frac {\left (-2 D c^3+C d c^2-A d^3+d \left (-3 D c^2+2 C d c-B d^2\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{7/2}}dx-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {-7 d^2 \left (\frac {\left (3 A d^3-7 B c d^2-15 c^3 D+11 c^2 C d\right ) \int \frac {\left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{5/2}}dx}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{7/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 466

\(\displaystyle \frac {-7 d^2 \left (\frac {\left (3 A d^3-7 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (2 c \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^{3/2}}dx+\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^{3/2}}\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{7/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 466

\(\displaystyle \frac {-7 d^2 \left (\frac {\left (3 A d^3-7 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (2 c \left (2 c \int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx+\frac {2 \sqrt {c^2-d^2 x^2}}{d \sqrt {c+d x}}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^{3/2}}\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{7/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {-7 d^2 \left (\frac {\left (3 A d^3-7 B c d^2-15 c^3 D+11 c^2 C d\right ) \left (2 c \left (4 c d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}+\frac {2 \sqrt {c^2-d^2 x^2}}{d \sqrt {c+d x}}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^{3/2}}\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{7/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-7 d^2 \left (\frac {\left (2 c \left (\frac {2 \sqrt {c^2-d^2 x^2}}{d \sqrt {c+d x}}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{d}\right )+\frac {2 \left (c^2-d^2 x^2\right )^{3/2}}{3 d (c+d x)^{3/2}}\right ) \left (3 A d^3-7 B c d^2-15 c^3 D+11 c^2 C d\right )}{4 c}+\frac {\left (c^2-d^2 x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 c d (c+d x)^{7/2}}\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{5/2} (7 C d-17 c D)}{5 (c+d x)^{5/2}}}{7 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{5/2}}{7 d^4 (c+d x)^{3/2}}\)

Input:

Int[((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2),x]
 

Output:

(-2*D*(c^2 - d^2*x^2)^(5/2))/(7*d^4*(c + d*x)^(3/2)) + ((-2*d*(7*C*d - 17* 
c*D)*(c^2 - d^2*x^2)^(5/2))/(5*(c + d*x)^(5/2)) - 7*d^2*(((c^2*C*d - B*c*d 
^2 + A*d^3 - c^3*D)*(c^2 - d^2*x^2)^(5/2))/(2*c*d*(c + d*x)^(7/2)) + ((11* 
c^2*C*d - 7*B*c*d^2 + 3*A*d^3 - 15*c^3*D)*((2*(c^2 - d^2*x^2)^(3/2))/(3*d* 
(c + d*x)^(3/2)) + 2*c*((2*Sqrt[c^2 - d^2*x^2])/(d*Sqrt[c + d*x]) - (2*Sqr 
t[2]*Sqrt[c]*ArcTanh[Sqrt[c^2 - d^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])]) 
/d)))/(4*c)))/(7*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 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 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 [A] (verified)

Time = 0.37 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (-30 D d^{4} x^{4} \sqrt {-d x +c}\, \sqrt {c}-42 C \,d^{4} x^{3} \sqrt {-d x +c}\, \sqrt {c}+102 D c^{\frac {3}{2}} d^{3} x^{3} \sqrt {-d x +c}+315 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{4} x -735 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3} x -70 B \,d^{4} x^{2} \sqrt {-d x +c}\, \sqrt {c}+1155 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2} x +182 C \sqrt {-d x +c}\, c^{\frac {3}{2}} d^{3} x^{2}-1575 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d x -252 D \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2} x^{2}+315 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3}-210 A \,d^{4} x \sqrt {-d x +c}\, \sqrt {c}-735 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2}+630 B \,c^{\frac {3}{2}} d^{3} x \sqrt {-d x +c}+1155 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d -1008 C \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2} x -1575 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}+1368 D \sqrt {-d x +c}\, c^{\frac {7}{2}} d x -420 A \sqrt {-d x +c}\, c^{\frac {3}{2}} d^{3}+910 B \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2}-1442 C \sqrt {-d x +c}\, c^{\frac {7}{2}} d +1962 D \sqrt {-d x +c}\, c^{\frac {9}{2}}\right )}{105 \left (d x +c \right )^{\frac {3}{2}} \sqrt {-d x +c}\, d^{4} \sqrt {c}}\)	\(519\)

Input:

int((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x,method=_RETUR 
NVERBOSE)
 

Output:

1/105*(-d^2*x^2+c^2)^(1/2)*(-30*D*d^4*x^4*(-d*x+c)^(1/2)*c^(1/2)-42*C*d^4* 
x^3*(-d*x+c)^(1/2)*c^(1/2)+102*D*c^(3/2)*d^3*x^3*(-d*x+c)^(1/2)+315*A*2^(1 
/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^4*x-735*B*2^(1/2)*arct 
anh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^3*x-70*B*d^4*x^2*(-d*x+c)^(1 
/2)*c^(1/2)+1155*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3 
*d^2*x+182*C*(-d*x+c)^(1/2)*c^(3/2)*d^3*x^2-1575*D*2^(1/2)*arctanh(1/2*(-d 
*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d*x-252*D*(-d*x+c)^(1/2)*c^(5/2)*d^2*x^2+ 
315*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^3-210*A*d^ 
4*x*(-d*x+c)^(1/2)*c^(1/2)-735*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2 
)/c^(1/2))*c^3*d^2+630*B*c^(3/2)*d^3*x*(-d*x+c)^(1/2)+1155*C*2^(1/2)*arcta 
nh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d-1008*C*(-d*x+c)^(1/2)*c^(5/2) 
*d^2*x-1575*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5+1368 
*D*(-d*x+c)^(1/2)*c^(7/2)*d*x-420*A*(-d*x+c)^(1/2)*c^(3/2)*d^3+910*B*(-d*x 
+c)^(1/2)*c^(5/2)*d^2-1442*C*(-d*x+c)^(1/2)*c^(7/2)*d+1962*D*(-d*x+c)^(1/2 
)*c^(9/2))/(d*x+c)^(3/2)/(-d*x+c)^(1/2)/d^4/c^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.14 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\left [\frac {105 \, \sqrt {2} {\left (15 \, D c^{5} - 11 \, C c^{4} d + 7 \, B c^{3} d^{2} - 3 \, A c^{2} d^{3} + {\left (15 \, D c^{3} d^{2} - 11 \, C c^{2} d^{3} + 7 \, B c d^{4} - 3 \, A d^{5}\right )} x^{2} + 2 \, {\left (15 \, D c^{4} d - 11 \, C c^{3} d^{2} + 7 \, B c^{2} d^{3} - 3 \, A c d^{4}\right )} x\right )} \sqrt {c} \log \left (-\frac {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}}\right ) - 4 \, {\left (15 \, D d^{4} x^{4} - 981 \, D c^{4} + 721 \, C c^{3} d - 455 \, B c^{2} d^{2} + 210 \, A c d^{3} - 3 \, {\left (17 \, D c d^{3} - 7 \, C d^{4}\right )} x^{3} + 7 \, {\left (18 \, D c^{2} d^{2} - 13 \, C c d^{3} + 5 \, B d^{4}\right )} x^{2} - 3 \, {\left (228 \, D c^{3} d - 168 \, C c^{2} d^{2} + 105 \, B c d^{3} - 35 \, A d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{210 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, \frac {105 \, \sqrt {2} {\left (15 \, D c^{5} - 11 \, C c^{4} d + 7 \, B c^{3} d^{2} - 3 \, A c^{2} d^{3} + {\left (15 \, D c^{3} d^{2} - 11 \, C c^{2} d^{3} + 7 \, B c d^{4} - 3 \, A d^{5}\right )} x^{2} + 2 \, {\left (15 \, D c^{4} d - 11 \, C c^{3} d^{2} + 7 \, B c^{2} d^{3} - 3 \, A c d^{4}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) - 2 \, {\left (15 \, D d^{4} x^{4} - 981 \, D c^{4} + 721 \, C c^{3} d - 455 \, B c^{2} d^{2} + 210 \, A c d^{3} - 3 \, {\left (17 \, D c d^{3} - 7 \, C d^{4}\right )} x^{3} + 7 \, {\left (18 \, D c^{2} d^{2} - 13 \, C c d^{3} + 5 \, B d^{4}\right )} x^{2} - 3 \, {\left (228 \, D c^{3} d - 168 \, C c^{2} d^{2} + 105 \, B c d^{3} - 35 \, A d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{105 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\right ] \] Input:

integrate((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algori 
thm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/210*(105*sqrt(2)*(15*D*c^5 - 11*C*c^4*d + 7*B*c^3*d^2 - 3*A*c^2*d^3 + ( 
15*D*c^3*d^2 - 11*C*c^2*d^3 + 7*B*c*d^4 - 3*A*d^5)*x^2 + 2*(15*D*c^4*d - 1 
1*C*c^3*d^2 + 7*B*c^2*d^3 - 3*A*c*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*(15*D*d^4*x^4 - 981*D*c^4 + 721*C*c^3*d - 455*B*c^2*d 
^2 + 210*A*c*d^3 - 3*(17*D*c*d^3 - 7*C*d^4)*x^3 + 7*(18*D*c^2*d^2 - 13*C*c 
*d^3 + 5*B*d^4)*x^2 - 3*(228*D*c^3*d - 168*C*c^2*d^2 + 105*B*c*d^3 - 35*A* 
d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(d^6*x^2 + 2*c*d^5*x + c^2*d^4 
), 1/105*(105*sqrt(2)*(15*D*c^5 - 11*C*c^4*d + 7*B*c^3*d^2 - 3*A*c^2*d^3 + 
 (15*D*c^3*d^2 - 11*C*c^2*d^3 + 7*B*c*d^4 - 3*A*d^5)*x^2 + 2*(15*D*c^4*d - 
 11*C*c^3*d^2 + 7*B*c^2*d^3 - 3*A*c*d^4)*x)*sqrt(-c)*arctan(1/2*sqrt(2)*sq 
rt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(-c)/(c*d*x + c^2)) - 2*(15*D*d^4*x^4 
 - 981*D*c^4 + 721*C*c^3*d - 455*B*c^2*d^2 + 210*A*c*d^3 - 3*(17*D*c*d^3 - 
 7*C*d^4)*x^3 + 7*(18*D*c^2*d^2 - 13*C*c*d^3 + 5*B*d^4)*x^2 - 3*(228*D*c^3 
*d - 168*C*c^2*d^2 + 105*B*c*d^3 - 35*A*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt( 
d*x + c))/(d^6*x^2 + 2*c*d^5*x + c^2*d^4)]
 

Sympy [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(7/2),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d 
*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (D x^{3} + C x^{2} + B x + A\right )}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algori 
thm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*(D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(7/2) 
, x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=-\frac {30 \, {\left (d x - c\right )}^{3} \sqrt {-d x + c} D - 42 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} D c - 210 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{2} - 1470 \, \sqrt {-d x + c} D c^{3} + 42 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} C d + 140 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c d + 1050 \, \sqrt {-d x + c} C c^{2} d - 70 \, {\left (-d x + c\right )}^{\frac {3}{2}} B d^{2} - 630 \, \sqrt {-d x + c} B c d^{2} + 210 \, \sqrt {-d x + c} A d^{3} - \frac {105 \, \sqrt {2} {\left (15 \, D c^{4} - 11 \, C c^{3} d + 7 \, B c^{2} d^{2} - 3 \, A c d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {210 \, {\left (\sqrt {-d x + c} D c^{4} - \sqrt {-d x + c} C c^{3} d + \sqrt {-d x + c} B c^{2} d^{2} - \sqrt {-d x + c} A c d^{3}\right )}}{d x + c}}{105 \, d^{4}} \] Input:

integrate((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algori 
thm="giac")
 

Output:

-1/105*(30*(d*x - c)^3*sqrt(-d*x + c)*D - 42*(d*x - c)^2*sqrt(-d*x + c)*D* 
c - 210*(-d*x + c)^(3/2)*D*c^2 - 1470*sqrt(-d*x + c)*D*c^3 + 42*(d*x - c)^ 
2*sqrt(-d*x + c)*C*d + 140*(-d*x + c)^(3/2)*C*c*d + 1050*sqrt(-d*x + c)*C* 
c^2*d - 70*(-d*x + c)^(3/2)*B*d^2 - 630*sqrt(-d*x + c)*B*c*d^2 + 210*sqrt( 
-d*x + c)*A*d^3 - 105*sqrt(2)*(15*D*c^4 - 11*C*c^3*d + 7*B*c^2*d^2 - 3*A*c 
*d^3)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/sqrt(-c) - 210*(sqrt(-d* 
x + c)*D*c^4 - sqrt(-d*x + c)*C*c^3*d + sqrt(-d*x + c)*B*c^2*d^2 - sqrt(-d 
*x + c)*A*c*d^3)/(d*x + c))/d^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2),x)
 

Output:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\frac {-336 \sqrt {-d x +c}\, a c \,d^{2}-168 \sqrt {-d x +c}\, a \,d^{3} x +728 \sqrt {-d x +c}\, b \,c^{2} d +504 \sqrt {-d x +c}\, b c \,d^{2} x -56 \sqrt {-d x +c}\, b \,d^{3} x^{2}+416 \sqrt {-d x +c}\, c^{4}+288 \sqrt {-d x +c}\, c^{3} d x -56 \sqrt {-d x +c}\, c^{2} d^{2} x^{2}+48 \sqrt {-d x +c}\, c \,d^{3} x^{3}-24 \sqrt {-d x +c}\, d^{4} x^{4}-252 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) a c \,d^{2}-252 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) a \,d^{3} x +588 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) b \,c^{2} d +588 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) b c \,d^{2} x +336 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) c^{4}+336 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) c^{3} d x +315 \sqrt {c}\, \sqrt {2}\, a c \,d^{2}+315 \sqrt {c}\, \sqrt {2}\, a \,d^{3} x -763 \sqrt {c}\, \sqrt {2}\, b \,c^{2} d -763 \sqrt {c}\, \sqrt {2}\, b c \,d^{2} x -640 \sqrt {c}\, \sqrt {2}\, c^{4}-640 \sqrt {c}\, \sqrt {2}\, c^{3} d x}{84 d^{3} \left (d x +c \right )} \] Input:

int((-d^2*x^2+c^2)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x)
 

Output:

( - 336*sqrt(c - d*x)*a*c*d**2 - 168*sqrt(c - d*x)*a*d**3*x + 728*sqrt(c - 
 d*x)*b*c**2*d + 504*sqrt(c - d*x)*b*c*d**2*x - 56*sqrt(c - d*x)*b*d**3*x* 
*2 + 416*sqrt(c - d*x)*c**4 + 288*sqrt(c - d*x)*c**3*d*x - 56*sqrt(c - d*x 
)*c**2*d**2*x**2 + 48*sqrt(c - d*x)*c*d**3*x**3 - 24*sqrt(c - d*x)*d**4*x* 
*4 - 252*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2)) 
*a*c*d**2 - 252*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2 
)))/2))*a*d**3*x + 588*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c) 
*sqrt(2)))/2))*b*c**2*d + 588*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/( 
sqrt(c)*sqrt(2)))/2))*b*c*d**2*x + 336*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c 
 + d*x)/(sqrt(c)*sqrt(2)))/2))*c**4 + 336*sqrt(c)*sqrt(2)*log(tan(asin(sqr 
t(c + d*x)/(sqrt(c)*sqrt(2)))/2))*c**3*d*x + 315*sqrt(c)*sqrt(2)*a*c*d**2 
+ 315*sqrt(c)*sqrt(2)*a*d**3*x - 763*sqrt(c)*sqrt(2)*b*c**2*d - 763*sqrt(c 
)*sqrt(2)*b*c*d**2*x - 640*sqrt(c)*sqrt(2)*c**4 - 640*sqrt(c)*sqrt(2)*c**3 
*d*x)/(84*d**3*(c + d*x))