\(\int \frac {(c+d x)^3 (A+B x+C x^2+D x^3)}{\sqrt {c^2-d^2 x^2}} \, dx\) [169]

Optimal result

Integrand size = 39, antiderivative size = 306 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {4 c^2 \left (c^2 C d+B c d^2+A d^3+c^3 D\right ) \sqrt {c^2-d^2 x^2}}{d^4}-\frac {c \left (26 c^2 C d+30 B c d^2+24 A d^3+23 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{16 d^3}-\frac {\left (18 c C d+6 B d^2+23 c^2 D\right ) x^3 \sqrt {c^2-d^2 x^2}}{24 d}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}+\frac {\left (5 c^2 C d+3 B c d^2+A d^3+7 c^3 D\right ) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4}-\frac {(C d+3 c D) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4}+\frac {c^3 \left (26 c^2 C d+30 B c d^2+40 A d^3+23 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{16 d^4} \] Output:

-4*c^2*(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4-1/16*c*(24*A 
*d^3+30*B*c*d^2+26*C*c^2*d+23*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/d^3-1/24*(6*B* 
d^2+18*C*c*d+23*D*c^2)*x^3*(-d^2*x^2+c^2)^(1/2)/d-1/6*d*D*x^5*(-d^2*x^2+c^ 
2)^(1/2)+1/3*(A*d^3+3*B*c*d^2+5*C*c^2*d+7*D*c^3)*(-d^2*x^2+c^2)^(3/2)/d^4- 
1/5*(C*d+3*D*c)*(-d^2*x^2+c^2)^(5/2)/d^4+1/16*c^3*(40*A*d^3+30*B*c*d^2+26* 
C*c^2*d+23*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.71 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (544 c^5 D+c^4 d (608 C+345 D x)+2 c^3 d^2 (360 B+x (195 C+136 D x))+12 c d^4 x (30 A+x (20 B+3 x (5 C+4 D x)))+4 d^5 x^2 (20 A+x (15 B+2 x (6 C+5 D x)))+2 c^2 d^3 (440 A+x (225 B+x (152 C+115 D x)))\right )+30 c^3 \left (26 c^2 C d+30 B c d^2+40 A d^3+23 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{240 d^4} \] Input:

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

Output:

-1/240*(Sqrt[c^2 - d^2*x^2]*(544*c^5*D + c^4*d*(608*C + 345*D*x) + 2*c^3*d 
^2*(360*B + x*(195*C + 136*D*x)) + 12*c*d^4*x*(30*A + x*(20*B + 3*x*(5*C + 
 4*D*x))) + 4*d^5*x^2*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 2*c^2*d^3*(4 
40*A + x*(225*B + x*(152*C + 115*D*x)))) + 30*c^3*(26*c^2*C*d + 30*B*c*d^2 
 + 40*A*d^3 + 23*c^3*D)*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d 
^4
 

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.15, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.359, Rules used = {2346, 25, 2346, 25, 2346, 27, 2346, 25, 2346, 25, 27, 455, 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.

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

\(\Big \downarrow \) 2346

\(\displaystyle -\frac {\int -\frac {6 d^4 (C d+3 c D) x^5+d^3 \left (23 D c^2+18 C d c+6 B d^2\right ) x^4+6 d^2 \left (D c^3+3 C d c^2+3 B d^2 c+A d^3\right ) x^3+6 c d^2 \left (C c^2+3 B d c+3 A d^2\right ) x^2+6 c^2 d^2 (B c+3 A d) x+6 A c^3 d^2}{\sqrt {c^2-d^2 x^2}}dx}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {6 d^4 (C d+3 c D) x^5+d^3 \left (23 D c^2+18 C d c+6 B d^2\right ) x^4+6 d^2 \left (D c^3+3 C d c^2+3 B d^2 c+A d^3\right ) x^3+6 c d^2 \left (C c^2+3 B d c+3 A d^2\right ) x^2+6 c^2 d^2 (B c+3 A d) x+6 A c^3 d^2}{\sqrt {c^2-d^2 x^2}}dx}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -\frac {5 \left (23 D c^2+18 C d c+6 B d^2\right ) x^4 d^5+30 A c^3 d^4+6 \left (17 D c^3+19 C d c^2+15 B d^2 c+5 A d^3\right ) x^3 d^4+30 c \left (C c^2+3 B d c+3 A d^2\right ) x^2 d^4+30 c^2 (B c+3 A d) x d^4}{\sqrt {c^2-d^2 x^2}}dx}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {5 \left (23 D c^2+18 C d c+6 B d^2\right ) x^4 d^5+30 A c^3 d^4+6 \left (17 D c^3+19 C d c^2+15 B d^2 c+5 A d^3\right ) x^3 d^4+30 c \left (C c^2+3 B d c+3 A d^2\right ) x^2 d^4+30 c^2 (B c+3 A d) x d^4}{\sqrt {c^2-d^2 x^2}}dx}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (40 A c^3 d^6+8 \left (17 D c^3+19 C d c^2+15 B d^2 c+5 A d^3\right ) x^3 d^6+40 c^2 (B c+3 A d) x d^6+5 c \left (23 D c^3+26 C d c^2+30 B d^2 c+24 A d^3\right ) x^2 d^5\right )}{\sqrt {c^2-d^2 x^2}}dx}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {40 A c^3 d^6+8 \left (17 D c^3+19 C d c^2+15 B d^2 c+5 A d^3\right ) x^3 d^6+40 c^2 (B c+3 A d) x d^6+5 c \left (23 D c^3+26 C d c^2+30 B d^2 c+24 A d^3\right ) x^2 d^5}{\sqrt {c^2-d^2 x^2}}dx}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int -\frac {120 A c^3 d^8+15 c \left (23 D c^3+26 C d c^2+30 B d^2 c+24 A d^3\right ) x^2 d^7+8 c^2 \left (34 D c^3+38 C d c^2+45 B d^2 c+55 A d^3\right ) x d^6}{\sqrt {c^2-d^2 x^2}}dx}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {120 A c^3 d^8+15 c \left (23 D c^3+26 C d c^2+30 B d^2 c+24 A d^3\right ) x^2 d^7+8 c^2 \left (34 D c^3+38 C d c^2+45 B d^2 c+55 A d^3\right ) x d^6}{\sqrt {c^2-d^2 x^2}}dx}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\int -\frac {c^2 d^7 \left (15 c \left (23 D c^3+26 C d c^2+30 B d^2 c+40 A d^3\right )+16 d \left (34 D c^3+38 C d c^2+45 B d^2 c+55 A d^3\right ) x\right )}{\sqrt {c^2-d^2 x^2}}dx}{2 d^2}-\frac {15}{2} c d^5 x \sqrt {c^2-d^2 x^2} \left (24 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\int \frac {c^2 d^7 \left (15 c \left (23 D c^3+26 C d c^2+30 B d^2 c+40 A d^3\right )+16 d \left (34 D c^3+38 C d c^2+45 B d^2 c+55 A d^3\right ) x\right )}{\sqrt {c^2-d^2 x^2}}dx}{2 d^2}-\frac {15}{2} c d^5 x \sqrt {c^2-d^2 x^2} \left (24 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {1}{2} c^2 d^5 \int \frac {15 c \left (23 D c^3+26 C d c^2+30 B d^2 c+40 A d^3\right )+16 d \left (34 D c^3+38 C d c^2+45 B d^2 c+55 A d^3\right ) x}{\sqrt {c^2-d^2 x^2}}dx-\frac {15}{2} c d^5 x \sqrt {c^2-d^2 x^2} \left (24 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {1}{2} c^2 d^5 \left (15 c \left (40 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right ) \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx-\frac {16 \sqrt {c^2-d^2 x^2} \left (55 A d^3+45 B c d^2+34 c^3 D+38 c^2 C d\right )}{d}\right )-\frac {15}{2} c d^5 x \sqrt {c^2-d^2 x^2} \left (24 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {1}{2} c^2 d^5 \left (15 c \left (40 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right ) \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}-\frac {16 \sqrt {c^2-d^2 x^2} \left (55 A d^3+45 B c d^2+34 c^3 D+38 c^2 C d\right )}{d}\right )-\frac {15}{2} c d^5 x \sqrt {c^2-d^2 x^2} \left (24 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {1}{2} c^2 d^5 \left (\frac {15 c \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right ) \left (40 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{d}-\frac {16 \sqrt {c^2-d^2 x^2} \left (55 A d^3+45 B c d^2+34 c^3 D+38 c^2 C d\right )}{d}\right )-\frac {15}{2} c d^5 x \sqrt {c^2-d^2 x^2} \left (24 A d^3+30 B c d^2+23 c^3 D+26 c^2 C d\right )}{3 d^2}-\frac {8}{3} d^4 x^2 \sqrt {c^2-d^2 x^2} \left (5 A d^3+15 B c d^2+17 c^3 D+19 c^2 C d\right )\right )}{4 d^2}-\frac {5}{4} d^3 x^3 \sqrt {c^2-d^2 x^2} \left (6 B d^2+23 c^2 D+18 c C d\right )}{5 d^2}-\frac {6}{5} d^2 x^4 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{6 d^2}-\frac {1}{6} d D x^5 \sqrt {c^2-d^2 x^2}\)

Input:

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

Output:

-1/6*(d*D*x^5*Sqrt[c^2 - d^2*x^2]) + ((-6*d^2*(C*d + 3*c*D)*x^4*Sqrt[c^2 - 
 d^2*x^2])/5 + ((-5*d^3*(18*c*C*d + 6*B*d^2 + 23*c^2*D)*x^3*Sqrt[c^2 - d^2 
*x^2])/4 + (3*((-8*d^4*(19*c^2*C*d + 15*B*c*d^2 + 5*A*d^3 + 17*c^3*D)*x^2* 
Sqrt[c^2 - d^2*x^2])/3 + ((-15*c*d^5*(26*c^2*C*d + 30*B*c*d^2 + 24*A*d^3 + 
 23*c^3*D)*x*Sqrt[c^2 - d^2*x^2])/2 + (c^2*d^5*((-16*(38*c^2*C*d + 45*B*c* 
d^2 + 55*A*d^3 + 34*c^3*D)*Sqrt[c^2 - d^2*x^2])/d + (15*c*(26*c^2*C*d + 30 
*B*c*d^2 + 40*A*d^3 + 23*c^3*D)*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d))/2)/ 
(3*d^2)))/(4*d^2))/(5*d^2))/(6*d^2)
 

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 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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.72

method	result	size
default	\(\frac {A \,c^{3} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{\sqrt {d^{2}}}-\frac {c^{2} \left (3 A d +B c \right ) \sqrt {-d^{2} x^{2}+c^{2}}}{d^{2}}+d^{2} \left (C d +3 D c \right ) \left (-\frac {x^{4} \sqrt {-d^{2} x^{2}+c^{2}}}{5 d^{2}}+\frac {4 c^{2} \left (-\frac {x^{2} \sqrt {-d^{2} x^{2}+c^{2}}}{3 d^{2}}-\frac {2 c^{2} \sqrt {-d^{2} x^{2}+c^{2}}}{3 d^{4}}\right )}{5 d^{2}}\right )+c \left (3 A \,d^{2}+3 B c d +C \,c^{2}\right ) \left (-\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{2} \sqrt {d^{2}}}\right )+d \left (B \,d^{2}+3 C c d +3 D c^{2}\right ) \left (-\frac {x^{3} \sqrt {-d^{2} x^{2}+c^{2}}}{4 d^{2}}+\frac {3 c^{2} \left (-\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{2} \sqrt {d^{2}}}\right )}{4 d^{2}}\right )+\left (A \,d^{3}+3 B c \,d^{2}+3 C \,c^{2} d +D c^{3}\right ) \left (-\frac {x^{2} \sqrt {-d^{2} x^{2}+c^{2}}}{3 d^{2}}-\frac {2 c^{2} \sqrt {-d^{2} x^{2}+c^{2}}}{3 d^{4}}\right )+d^{3} D \left (-\frac {x^{5} \sqrt {-d^{2} x^{2}+c^{2}}}{6 d^{2}}+\frac {5 c^{2} \left (-\frac {x^{3} \sqrt {-d^{2} x^{2}+c^{2}}}{4 d^{2}}+\frac {3 c^{2} \left (-\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 d^{2} \sqrt {d^{2}}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )\)	\(527\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {30 \, {\left (23 \, D c^{6} + 26 \, C c^{5} d + 30 \, B c^{4} d^{2} + 40 \, A c^{3} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (40 \, D d^{5} x^{5} + 544 \, D c^{5} + 608 \, C c^{4} d + 720 \, B c^{3} d^{2} + 880 \, A c^{2} d^{3} + 48 \, {\left (3 \, D c d^{4} + C d^{5}\right )} x^{4} + 10 \, {\left (23 \, D c^{2} d^{3} + 18 \, C c d^{4} + 6 \, B d^{5}\right )} x^{3} + 16 \, {\left (17 \, D c^{3} d^{2} + 19 \, C c^{2} d^{3} + 15 \, B c d^{4} + 5 \, A d^{5}\right )} x^{2} + 15 \, {\left (23 \, D c^{4} d + 26 \, C c^{3} d^{2} + 30 \, B c^{2} d^{3} + 24 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{240 \, d^{4}} \] Input:

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

Output:

-1/240*(30*(23*D*c^6 + 26*C*c^5*d + 30*B*c^4*d^2 + 40*A*c^3*d^3)*arctan(-( 
c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (40*D*d^5*x^5 + 544*D*c^5 + 608*C*c^4*d 
 + 720*B*c^3*d^2 + 880*A*c^2*d^3 + 48*(3*D*c*d^4 + C*d^5)*x^4 + 10*(23*D*c 
^2*d^3 + 18*C*c*d^4 + 6*B*d^5)*x^3 + 16*(17*D*c^3*d^2 + 19*C*c^2*d^3 + 15* 
B*c*d^4 + 5*A*d^5)*x^2 + 15*(23*D*c^4*d + 26*C*c^3*d^2 + 30*B*c^2*d^3 + 24 
*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2))/d^4
 

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.82 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=\begin {cases} \sqrt {c^{2} - d^{2} x^{2}} \left (- \frac {D d x^{5}}{6} - \frac {x^{4} \left (C d^{3} + 3 D c d^{2}\right )}{5 d^{2}} - \frac {x^{3} \left (B d^{3} + 3 C c d^{2} + \frac {23 D c^{2} d}{6}\right )}{4 d^{2}} - \frac {x^{2} \left (A d^{3} + 3 B c d^{2} + 3 C c^{2} d + D c^{3} + \frac {4 c^{2} \left (C d^{3} + 3 D c d^{2}\right )}{5 d^{2}}\right )}{3 d^{2}} - \frac {x \left (3 A c d^{2} + 3 B c^{2} d + C c^{3} + \frac {3 c^{2} \left (B d^{3} + 3 C c d^{2} + \frac {23 D c^{2} d}{6}\right )}{4 d^{2}}\right )}{2 d^{2}} - \frac {3 A c^{2} d + B c^{3} + \frac {2 c^{2} \left (A d^{3} + 3 B c d^{2} + 3 C c^{2} d + D c^{3} + \frac {4 c^{2} \left (C d^{3} + 3 D c d^{2}\right )}{5 d^{2}}\right )}{3 d^{2}}}{d^{2}}\right ) + \left (A c^{3} + \frac {c^{2} \cdot \left (3 A c d^{2} + 3 B c^{2} d + C c^{3} + \frac {3 c^{2} \left (B d^{3} + 3 C c d^{2} + \frac {23 D c^{2} d}{6}\right )}{4 d^{2}}\right )}{2 d^{2}}\right ) \left (\begin {cases} \frac {\log {\left (- 2 d^{2} x + 2 \sqrt {- d^{2}} \sqrt {c^{2} - d^{2} x^{2}} \right )}}{\sqrt {- d^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- d^{2} x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: d^{2} \neq 0 \\\frac {A c^{3} x + \frac {D d^{3} x^{7}}{7} + \frac {x^{6} \left (C d^{3} + 3 D c d^{2}\right )}{6} + \frac {x^{5} \left (B d^{3} + 3 C c d^{2} + 3 D c^{2} d\right )}{5} + \frac {x^{4} \left (A d^{3} + 3 B c d^{2} + 3 C c^{2} d + D c^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 A c d^{2} + 3 B c^{2} d + C c^{3}\right )}{3} + \frac {x^{2} \cdot \left (3 A c^{2} d + B c^{3}\right )}{2}}{\sqrt {c^{2}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(c**2 - d**2*x**2)*(-D*d*x**5/6 - x**4*(C*d**3 + 3*D*c*d**2 
)/(5*d**2) - x**3*(B*d**3 + 3*C*c*d**2 + 23*D*c**2*d/6)/(4*d**2) - x**2*(A 
*d**3 + 3*B*c*d**2 + 3*C*c**2*d + D*c**3 + 4*c**2*(C*d**3 + 3*D*c*d**2)/(5 
*d**2))/(3*d**2) - x*(3*A*c*d**2 + 3*B*c**2*d + C*c**3 + 3*c**2*(B*d**3 + 
3*C*c*d**2 + 23*D*c**2*d/6)/(4*d**2))/(2*d**2) - (3*A*c**2*d + B*c**3 + 2* 
c**2*(A*d**3 + 3*B*c*d**2 + 3*C*c**2*d + D*c**3 + 4*c**2*(C*d**3 + 3*D*c*d 
**2)/(5*d**2))/(3*d**2))/d**2) + (A*c**3 + c**2*(3*A*c*d**2 + 3*B*c**2*d + 
 C*c**3 + 3*c**2*(B*d**3 + 3*C*c*d**2 + 23*D*c**2*d/6)/(4*d**2))/(2*d**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)), Ne(d**2, 0)), ((A*c 
**3*x + D*d**3*x**7/7 + x**6*(C*d**3 + 3*D*c*d**2)/6 + x**5*(B*d**3 + 3*C* 
c*d**2 + 3*D*c**2*d)/5 + x**4*(A*d**3 + 3*B*c*d**2 + 3*C*c**2*d + D*c**3)/ 
4 + x**3*(3*A*c*d**2 + 3*B*c**2*d + C*c**3)/3 + x**2*(3*A*c**2*d + B*c**3) 
/2)/sqrt(c**2), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.78 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {1}{6} \, \sqrt {-d^{2} x^{2} + c^{2}} D d x^{5} - \frac {5 \, \sqrt {-d^{2} x^{2} + c^{2}} D c^{2} x^{3}}{24 \, d} + \frac {5 \, D c^{6} \arcsin \left (\frac {d x}{c}\right )}{16 \, d^{4}} + \frac {A c^{3} \arcsin \left (\frac {d x}{c}\right )}{d} - \frac {5 \, \sqrt {-d^{2} x^{2} + c^{2}} D c^{4} x}{16 \, d^{3}} - \frac {{\left (3 \, D c d^{2} + C d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} x^{4}}{5 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} B c^{3}}{d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{2}}{d} - \frac {{\left (3 \, D c^{2} d + 3 \, C c d^{2} + B d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} x^{3}}{4 \, d^{2}} - \frac {4 \, {\left (3 \, D c d^{2} + C d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} c^{2} x^{2}}{15 \, d^{4}} - \frac {{\left (D c^{3} + 3 \, C c^{2} d + 3 \, B c d^{2} + A d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} x^{2}}{3 \, d^{2}} + \frac {3 \, {\left (3 \, D c^{2} d + 3 \, C c d^{2} + B d^{3}\right )} c^{4} \arcsin \left (\frac {d x}{c}\right )}{8 \, d^{5}} + \frac {{\left (C c^{3} + 3 \, B c^{2} d + 3 \, A c d^{2}\right )} c^{2} \arcsin \left (\frac {d x}{c}\right )}{2 \, d^{3}} - \frac {3 \, {\left (3 \, D c^{2} d + 3 \, C c d^{2} + B d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} c^{2} x}{8 \, d^{4}} - \frac {{\left (C c^{3} + 3 \, B c^{2} d + 3 \, A c d^{2}\right )} \sqrt {-d^{2} x^{2} + c^{2}} x}{2 \, d^{2}} - \frac {8 \, {\left (3 \, D c d^{2} + C d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} c^{4}}{15 \, d^{6}} - \frac {2 \, {\left (D c^{3} + 3 \, C c^{2} d + 3 \, B c d^{2} + A d^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} c^{2}}{3 \, d^{4}} \] Input:

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

Output:

-1/6*sqrt(-d^2*x^2 + c^2)*D*d*x^5 - 5/24*sqrt(-d^2*x^2 + c^2)*D*c^2*x^3/d 
+ 5/16*D*c^6*arcsin(d*x/c)/d^4 + A*c^3*arcsin(d*x/c)/d - 5/16*sqrt(-d^2*x^ 
2 + c^2)*D*c^4*x/d^3 - 1/5*(3*D*c*d^2 + C*d^3)*sqrt(-d^2*x^2 + c^2)*x^4/d^ 
2 - sqrt(-d^2*x^2 + c^2)*B*c^3/d^2 - 3*sqrt(-d^2*x^2 + c^2)*A*c^2/d - 1/4* 
(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*sqrt(-d^2*x^2 + c^2)*x^3/d^2 - 4/15*(3*D*c 
*d^2 + C*d^3)*sqrt(-d^2*x^2 + c^2)*c^2*x^2/d^4 - 1/3*(D*c^3 + 3*C*c^2*d + 
3*B*c*d^2 + A*d^3)*sqrt(-d^2*x^2 + c^2)*x^2/d^2 + 3/8*(3*D*c^2*d + 3*C*c*d 
^2 + B*d^3)*c^4*arcsin(d*x/c)/d^5 + 1/2*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*c^ 
2*arcsin(d*x/c)/d^3 - 3/8*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*sqrt(-d^2*x^2 + 
c^2)*c^2*x/d^4 - 1/2*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*sqrt(-d^2*x^2 + c^2)* 
x/d^2 - 8/15*(3*D*c*d^2 + C*d^3)*sqrt(-d^2*x^2 + c^2)*c^4/d^6 - 2/3*(D*c^3 
 + 3*C*c^2*d + 3*B*c*d^2 + A*d^3)*sqrt(-d^2*x^2 + c^2)*c^2/d^4
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {1}{240} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, D d x + \frac {6 \, {\left (3 \, D c d^{8} + C d^{9}\right )}}{d^{8}}\right )} x + \frac {5 \, {\left (23 \, D c^{2} d^{7} + 18 \, C c d^{8} + 6 \, B d^{9}\right )}}{d^{8}}\right )} x + \frac {8 \, {\left (17 \, D c^{3} d^{6} + 19 \, C c^{2} d^{7} + 15 \, B c d^{8} + 5 \, A d^{9}\right )}}{d^{8}}\right )} x + \frac {15 \, {\left (23 \, D c^{4} d^{5} + 26 \, C c^{3} d^{6} + 30 \, B c^{2} d^{7} + 24 \, A c d^{8}\right )}}{d^{8}}\right )} x + \frac {16 \, {\left (34 \, D c^{5} d^{4} + 38 \, C c^{4} d^{5} + 45 \, B c^{3} d^{6} + 55 \, A c^{2} d^{7}\right )}}{d^{8}}\right )} + \frac {{\left (23 \, D c^{6} + 26 \, C c^{5} d + 30 \, B c^{4} d^{2} + 40 \, A c^{3} d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{16 \, d^{3} {\left | d \right |}} \] Input:

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

Output:

-1/240*sqrt(-d^2*x^2 + c^2)*((2*((4*(5*D*d*x + 6*(3*D*c*d^8 + C*d^9)/d^8)* 
x + 5*(23*D*c^2*d^7 + 18*C*c*d^8 + 6*B*d^9)/d^8)*x + 8*(17*D*c^3*d^6 + 19* 
C*c^2*d^7 + 15*B*c*d^8 + 5*A*d^9)/d^8)*x + 15*(23*D*c^4*d^5 + 26*C*c^3*d^6 
 + 30*B*c^2*d^7 + 24*A*c*d^8)/d^8)*x + 16*(34*D*c^5*d^4 + 38*C*c^4*d^5 + 4 
5*B*c^3*d^6 + 55*A*c^2*d^7)/d^8) + 1/16*(23*D*c^6 + 26*C*c^5*d + 30*B*c^4* 
d^2 + 40*A*c^3*d^3)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.15 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt {c^2-d^2 x^2}} \, dx=\frac {600 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{3} d^{2}+450 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{4} d +735 \mathit {asin} \left (\frac {d x}{c}\right ) c^{6}-880 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{2}-360 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{3} x -80 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{4} x^{2}-720 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d -450 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{2} x -240 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{3} x^{2}-60 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{4} x^{3}-1152 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}-735 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x -576 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}-410 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}-192 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}-40 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}+880 a \,c^{3} d^{2}+720 b \,c^{4} d +1152 c^{6}}{240 d^{3}} \] Input:

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

Output:

(600*asin((d*x)/c)*a*c**3*d**2 + 450*asin((d*x)/c)*b*c**4*d + 735*asin((d* 
x)/c)*c**6 - 880*sqrt(c**2 - d**2*x**2)*a*c**2*d**2 - 360*sqrt(c**2 - d**2 
*x**2)*a*c*d**3*x - 80*sqrt(c**2 - d**2*x**2)*a*d**4*x**2 - 720*sqrt(c**2 
- d**2*x**2)*b*c**3*d - 450*sqrt(c**2 - d**2*x**2)*b*c**2*d**2*x - 240*sqr 
t(c**2 - d**2*x**2)*b*c*d**3*x**2 - 60*sqrt(c**2 - d**2*x**2)*b*d**4*x**3 
- 1152*sqrt(c**2 - d**2*x**2)*c**5 - 735*sqrt(c**2 - d**2*x**2)*c**4*d*x - 
 576*sqrt(c**2 - d**2*x**2)*c**3*d**2*x**2 - 410*sqrt(c**2 - d**2*x**2)*c* 
*2*d**3*x**3 - 192*sqrt(c**2 - d**2*x**2)*c*d**4*x**4 - 40*sqrt(c**2 - d** 
2*x**2)*d**5*x**5 + 880*a*c**3*d**2 + 720*b*c**4*d + 1152*c**6)/(240*d**3)