\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}} \, dx\) [218]

Optimal result

Integrand size = 41, antiderivative size = 235 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c^2-d^2 x^2}}{4 c d^4 (c+d x)^{5/2}}+\frac {\left (13 c^2 C d-5 B c d^2-3 A d^3-21 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{16 c^2 d^4 (c+d x)^{3/2}}-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}-\frac {\left (19 c^2 C d+5 B c d^2+3 A d^3-75 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{16 \sqrt {2} c^{5/2} d^4} \] Output:

-1/4*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(1/2)/c/d^4/(d*x+c)^(5/2 
)+1/16*(-3*A*d^3-5*B*c*d^2+13*C*c^2*d-21*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)-1/32*(3*A*d^3+ 
5*B*c*d^2+19*C*c^2*d-75*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 = 1.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (49 c^4 D+3 A d^4 x+c d^3 (7 A+5 B x)+c^3 (-9 C d+85 d D x)+c^2 d^2 (B+x (-13 C+32 D x))\right )}{(c+d x)^{5/2}}+\sqrt {2} \left (-19 c^2 C d-5 B c d^2-3 A d^3+75 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{32 c^{5/2} d^4} \] Input:

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

Output:

((-2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(49*c^4*D + 3*A*d^4*x + c*d^3*(7*A + 5*B* 
x) + c^3*(-9*C*d + 85*d*D*x) + c^2*d^2*(B + x*(-13*C + 32*D*x))))/(c + d*x 
)^(5/2) + Sqrt[2]*(-19*c^2*C*d - 5*B*c*d^2 - 3*A*d^3 + 75*c^3*D)*ArcTanh[( 
Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(32*c^(5/2)*d^4)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2170, 27, 2170, 27, 671, 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.

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2170

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 671

\(\displaystyle \frac {d^2 \left (\frac {\left (3 A d^3+5 B c d^2-75 c^3 D+19 c^2 C d\right ) \int \frac {1}{(c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}dx}{8 c}-\frac {\sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{5/2}}\right )+\frac {2 d \sqrt {c^2-d^2 x^2} (C d-3 c D)}{(c+d x)^{3/2}}}{d^5}-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {d^2 \left (\frac {\left (3 A d^3+5 B c d^2-75 c^3 D+19 c^2 C d\right ) \left (\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{5/2}}\right )+\frac {2 d \sqrt {c^2-d^2 x^2} (C d-3 c D)}{(c+d x)^{3/2}}}{d^5}-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {d^2 \left (\frac {\left (3 A d^3+5 B c d^2-75 c^3 D+19 c^2 C d\right ) \left (\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{2 c}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{5/2}}\right )+\frac {2 d \sqrt {c^2-d^2 x^2} (C d-3 c D)}{(c+d x)^{3/2}}}{d^5}-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d^2 \left (\frac {\left (-\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{2 \sqrt {2} c^{3/2} d}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right ) \left (3 A d^3+5 B c d^2-75 c^3 D+19 c^2 C d\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{5/2}}\right )+\frac {2 d \sqrt {c^2-d^2 x^2} (C d-3 c D)}{(c+d x)^{3/2}}}{d^5}-\frac {2 D \sqrt {c^2-d^2 x^2}}{d^4 \sqrt {c+d x}}\)

Input:

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

Output:

(-2*D*Sqrt[c^2 - d^2*x^2])/(d^4*Sqrt[c + d*x]) + ((2*d*(C*d - 3*c*D)*Sqrt[ 
c^2 - d^2*x^2])/(c + d*x)^(3/2) + d^2*(-1/4*((c^2*C*d - B*c*d^2 + A*d^3 - 
c^3*D)*Sqrt[c^2 - d^2*x^2])/(c*d*(c + d*x)^(5/2)) + ((19*c^2*C*d + 5*B*c*d 
^2 + 3*A*d^3 - 75*c^3*D)*(-1/2*Sqrt[c^2 - d^2*x^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)))/(8*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 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. \(541\) vs. \(2(205)=410\).

Time = 0.36 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.31

method	result	size
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{5} x^{2}+5 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{4} x^{2}+19 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3} x^{2}-75 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2} x^{2}+6 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{4} x +10 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3} x +38 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2} x -150 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d x +64 D \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2} x^{2}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3}+6 A \,d^{4} x \sqrt {-d x +c}\, \sqrt {c}+5 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2}+10 B \,c^{\frac {3}{2}} d^{3} x \sqrt {-d x +c}+19 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d -26 C \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2} x -75 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}+170 D \sqrt {-d x +c}\, c^{\frac {7}{2}} d x +14 A \sqrt {-d x +c}\, c^{\frac {3}{2}} d^{3}+2 B \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2}-18 C \sqrt {-d x +c}\, c^{\frac {7}{2}} d +98 D \sqrt {-d x +c}\, c^{\frac {9}{2}}\right )}{32 c^{\frac {5}{2}} \left (d x +c \right )^{\frac {5}{2}} \sqrt {-d x +c}\, d^{4}}\)	\(542\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-1/32*(-d^2*x^2+c^2)^(1/2)/c^(5/2)*(3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2) 
*2^(1/2)/c^(1/2))*d^5*x^2+5*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c 
^(1/2))*c*d^4*x^2+19*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2)) 
*c^2*d^3*x^2-75*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3* 
d^2*x^2+6*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^4*x+10 
*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^3*x+38*C*2^(1 
/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d^2*x-150*D*2^(1/2)*ar 
ctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d*x+64*D*(-d*x+c)^(1/2)*c^(5 
/2)*d^2*x^2+3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^ 
3+6*A*d^4*x*(-d*x+c)^(1/2)*c^(1/2)+5*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)* 
2^(1/2)/c^(1/2))*c^3*d^2+10*B*c^(3/2)*d^3*x*(-d*x+c)^(1/2)+19*C*2^(1/2)*ar 
ctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d-26*C*(-d*x+c)^(1/2)*c^(5/2 
)*d^2*x-75*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5+170*D 
*(-d*x+c)^(1/2)*c^(7/2)*d*x+14*A*(-d*x+c)^(1/2)*c^(3/2)*d^3+2*B*(-d*x+c)^( 
1/2)*c^(5/2)*d^2-18*C*(-d*x+c)^(1/2)*c^(7/2)*d+98*D*(-d*x+c)^(1/2)*c^(9/2) 
)/(d*x+c)^(5/2)/(-d*x+c)^(1/2)/d^4
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.01 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}} \, dx=\left [-\frac {\sqrt {2} {\left (75 \, D c^{6} - 19 \, C c^{5} d - 5 \, B c^{4} d^{2} - 3 \, A c^{3} d^{3} + {\left (75 \, D c^{3} d^{3} - 19 \, C c^{2} d^{4} - 5 \, B c d^{5} - 3 \, A d^{6}\right )} x^{3} + 3 \, {\left (75 \, D c^{4} d^{2} - 19 \, C c^{3} d^{3} - 5 \, B c^{2} d^{4} - 3 \, A c d^{5}\right )} x^{2} + 3 \, {\left (75 \, D c^{5} d - 19 \, C c^{4} d^{2} - 5 \, B c^{3} d^{3} - 3 \, A c^{2} d^{4}\right )} x\right )} \sqrt {c} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x + 2 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {c} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 4 \, {\left (32 \, D c^{3} d^{2} x^{2} + 49 \, D c^{5} - 9 \, C c^{4} d + B c^{3} d^{2} + 7 \, A c^{2} d^{3} + {\left (85 \, D c^{4} d - 13 \, C c^{3} d^{2} + 5 \, B c^{2} d^{3} + 3 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{64 \, {\left (c^{3} d^{7} x^{3} + 3 \, c^{4} d^{6} x^{2} + 3 \, c^{5} d^{5} x + c^{6} d^{4}\right )}}, -\frac {\sqrt {2} {\left (75 \, D c^{6} - 19 \, C c^{5} d - 5 \, B c^{4} d^{2} - 3 \, A c^{3} d^{3} + {\left (75 \, D c^{3} d^{3} - 19 \, C c^{2} d^{4} - 5 \, B c d^{5} - 3 \, A d^{6}\right )} x^{3} + 3 \, {\left (75 \, D c^{4} d^{2} - 19 \, C c^{3} d^{3} - 5 \, B c^{2} d^{4} - 3 \, A c d^{5}\right )} x^{2} + 3 \, {\left (75 \, D c^{5} d - 19 \, C c^{4} d^{2} - 5 \, B c^{3} d^{3} - 3 \, A c^{2} d^{4}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) + 2 \, {\left (32 \, D c^{3} d^{2} x^{2} + 49 \, D c^{5} - 9 \, C c^{4} d + B c^{3} d^{2} + 7 \, A c^{2} d^{3} + {\left (85 \, D c^{4} d - 13 \, C c^{3} d^{2} + 5 \, B c^{2} d^{3} + 3 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{32 \, {\left (c^{3} d^{7} x^{3} + 3 \, c^{4} d^{6} x^{2} + 3 \, c^{5} d^{5} x + c^{6} d^{4}\right )}}\right ] \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2),x, algori 
thm="fricas")
 

Output:

[-1/64*(sqrt(2)*(75*D*c^6 - 19*C*c^5*d - 5*B*c^4*d^2 - 3*A*c^3*d^3 + (75*D 
*c^3*d^3 - 19*C*c^2*d^4 - 5*B*c*d^5 - 3*A*d^6)*x^3 + 3*(75*D*c^4*d^2 - 19* 
C*c^3*d^3 - 5*B*c^2*d^4 - 3*A*c*d^5)*x^2 + 3*(75*D*c^5*d - 19*C*c^4*d^2 - 
5*B*c^3*d^3 - 3*A*c^2*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*(32*D*c^3*d^2*x^2 + 49*D*c^5 - 9*C*c^4*d + B*c^3*d^2 + 7*A*c^2*d^ 
3 + (85*D*c^4*d - 13*C*c^3*d^2 + 5*B*c^2*d^3 + 3*A*c*d^4)*x)*sqrt(-d^2*x^2 
 + c^2)*sqrt(d*x + c))/(c^3*d^7*x^3 + 3*c^4*d^6*x^2 + 3*c^5*d^5*x + c^6*d^ 
4), -1/32*(sqrt(2)*(75*D*c^6 - 19*C*c^5*d - 5*B*c^4*d^2 - 3*A*c^3*d^3 + (7 
5*D*c^3*d^3 - 19*C*c^2*d^4 - 5*B*c*d^5 - 3*A*d^6)*x^3 + 3*(75*D*c^4*d^2 - 
19*C*c^3*d^3 - 5*B*c^2*d^4 - 3*A*c*d^5)*x^2 + 3*(75*D*c^5*d - 19*C*c^4*d^2 
 - 5*B*c^3*d^3 - 3*A*c^2*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*(32*D*c^3*d^2*x^2 + 49*D 
*c^5 - 9*C*c^4*d + B*c^3*d^2 + 7*A*c^2*d^3 + (85*D*c^4*d - 13*C*c^3*d^2 + 
5*B*c^2*d^3 + 3*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^3*d^7*x 
^3 + 3*c^4*d^6*x^2 + 3*c^5*d^5*x + c^6*d^4)]
 

Sympy [F]

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

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2)/(-d**2*x**2+c**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2),x, algori 
thm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {64 \, \sqrt {-d x + c} D + \frac {\sqrt {2} {\left (75 \, D c^{3} - 19 \, C c^{2} d - 5 \, B c d^{2} - 3 \, A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{2}} - \frac {2 \, {\left (21 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{3} - 38 \, \sqrt {-d x + c} D c^{4} - 13 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{2} d + 22 \, \sqrt {-d x + c} C c^{3} d + 5 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c d^{2} - 6 \, \sqrt {-d x + c} B c^{2} d^{2} + 3 \, {\left (-d x + c\right )}^{\frac {3}{2}} A d^{3} - 10 \, \sqrt {-d x + c} A c d^{3}\right )}}{{\left (d x + c\right )}^{2} c^{2}}}{32 \, d^{4}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2),x, algori 
thm="giac")
 

Output:

-1/32*(64*sqrt(-d*x + c)*D + sqrt(2)*(75*D*c^3 - 19*C*c^2*d - 5*B*c*d^2 - 
3*A*d^3)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c^2) - 2*(2 
1*(-d*x + c)^(3/2)*D*c^3 - 38*sqrt(-d*x + c)*D*c^4 - 13*(-d*x + c)^(3/2)*C 
*c^2*d + 22*sqrt(-d*x + c)*C*c^3*d + 5*(-d*x + c)^(3/2)*B*c*d^2 - 6*sqrt(- 
d*x + c)*B*c^2*d^2 + 3*(-d*x + c)^(3/2)*A*d^3 - 10*sqrt(-d*x + c)*A*c*d^3) 
/((d*x + c)^2*c^2))/d^4
 

Mupad [F(-1)]

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

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(1/2)*(c + d*x)^(5/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((c^2 - d^2*x^2)^(1/2)*(c + d*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.63 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2)/(-d^2*x^2+c^2)^(1/2),x)
 

Output:

( - 28*sqrt(c - d*x)*a*c**2*d**2 - 12*sqrt(c - d*x)*a*c*d**3*x - 4*sqrt(c 
- d*x)*b*c**3*d - 20*sqrt(c - d*x)*b*c**2*d**2*x - 160*sqrt(c - d*x)*c**5 
- 288*sqrt(c - d*x)*c**4*d*x - 128*sqrt(c - d*x)*c**3*d**2*x**2 + 3*sqrt(c 
)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*c**2*d**2 + 6*sqrt(c)*sqr 
t(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*c*d**3*x + 3*sqrt(c)*sqrt(2)*l 
og(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*d**4*x**2 + 5*sqrt(c)*sqrt(2)*log(sq 
rt(c - d*x) - sqrt(c)*sqrt(2))*b*c**3*d + 10*sqrt(c)*sqrt(2)*log(sqrt(c - 
d*x) - sqrt(c)*sqrt(2))*b*c**2*d**2*x + 5*sqrt(c)*sqrt(2)*log(sqrt(c - d*x 
) - sqrt(c)*sqrt(2))*b*c*d**3*x**2 - 56*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) 
- sqrt(c)*sqrt(2))*c**5 - 112*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)* 
sqrt(2))*c**4*d*x - 56*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2) 
)*c**3*d**2*x**2 - 3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))* 
a*c**2*d**2 - 6*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c*d 
**3*x - 3*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*d**4*x**2 
 - 5*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**3*d - 10*sq 
rt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**2*d**2*x - 5*sqrt( 
c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c*d**3*x**2 + 56*sqrt(c) 
*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**5 + 112*sqrt(c)*sqrt(2)*l 
og(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**4*d*x + 56*sqrt(c)*sqrt(2)*log(sqrt 
(c - d*x) + sqrt(c)*sqrt(2))*c**3*d**2*x**2)/(64*c**3*d**3*(c**2 + 2*c*...