\(\int \frac {(c+d x)^3 (A+B x+C x^2+D x^3)}{(c^2-d^2 x^2)^{3/2}} \, dx\) [177]

Optimal result

Integrand size = 39, antiderivative size = 269 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\left (c^2 C d+B c d^2+A d^3+c^3 D\right ) (c+d x)^3}{c d^4 \sqrt {c^2-d^2 x^2}}+\frac {2 \left (4 c^2 C d+3 B c d^2+2 A d^3+5 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{d^4}+\frac {\left (20 c^2 C d+12 B c d^2+8 A d^3+27 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{8 c d^3}+\frac {D x^3 \sqrt {c^2-d^2 x^2}}{4 d}-\frac {(C d+3 c D) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4}-\frac {c \left (44 c^2 C d+36 B c d^2+24 A d^3+51 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{8 d^4} \] Output:

(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(d*x+c)^3/c/d^4/(-d^2*x^2+c^2)^(1/2)+2*(2*A* 
d^3+3*B*c*d^2+4*C*c^2*d+5*D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4+1/8*(8*A*d^3+12* 
B*c*d^2+20*C*c^2*d+27*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/c/d^3+1/4*D*x^3*(-d^2* 
x^2+c^2)^(1/2)/d-1/3*(C*d+3*D*c)*(-d^2*x^2+c^2)^(3/2)/d^4-1/8*c*(24*A*d^3+ 
36*B*c*d^2+44*C*c^2*d+51*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 1.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.71 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {c^2-d^2 x^2} \left (240 c^4 D+c^3 d (208 C-87 D x)+c^2 d^2 (168 B-x (76 C+33 D x))-2 d^4 x \left (12 A+x \left (6 B+4 C x+3 D x^2\right )\right )+2 c d^3 \left (60 A-x \left (30 B+14 C x+9 D x^2\right )\right )\right )}{c-d x}+6 c \left (44 c^2 C d+36 B c d^2+24 A d^3+51 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{24 d^4} \] Input:

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

Output:

((Sqrt[c^2 - d^2*x^2]*(240*c^4*D + c^3*d*(208*C - 87*D*x) + c^2*d^2*(168*B 
 - x*(76*C + 33*D*x)) - 2*d^4*x*(12*A + x*(6*B + 4*C*x + 3*D*x^2)) + 2*c*d 
^3*(60*A - x*(30*B + 14*C*x + 9*D*x^2))))/(c - d*x) + 6*c*(44*c^2*C*d + 36 
*B*c*d^2 + 24*A*d^3 + 51*c^3*D)*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x 
^2])])/(24*d^4)
 

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {2166, 2346, 25, 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.

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

\(\Big \downarrow \) 2166

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {-\frac {\int -\frac {4 c d^2 (C d+3 c D) x^3+d \left (27 D c^3+20 C d c^2+12 B d^2 c+8 A d^3\right ) x^2+4 c \left (7 D c^3+7 C d c^2+6 B d^2 c+4 A d^3\right ) x+\frac {4 c^2 \left (3 D c^3+3 C d c^2+3 B d^2 c+2 A d^3\right )}{d}}{\sqrt {c^2-d^2 x^2}}dx}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\int \frac {4 c d^2 (C d+3 c D) x^3+d \left (27 D c^3+20 C d c^2+12 B d^2 c+8 A d^3\right ) x^2+4 c \left (7 D c^3+7 C d c^2+6 B d^2 c+4 A d^3\right ) x+\frac {4 c^2 \left (3 D c^3+3 C d c^2+3 B d^2 c+2 A d^3\right )}{d}}{\sqrt {c^2-d^2 x^2}}dx}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {-\frac {\int -\frac {3 \left (27 D c^3+20 C d c^2+12 B d^2 c+8 A d^3\right ) x^2 d^3+4 c \left (27 D c^3+23 C d c^2+18 B d^2 c+12 A d^3\right ) x d^2+12 c^2 \left (3 D c^3+3 C d c^2+3 B d^2 c+2 A d^3\right ) d}{\sqrt {c^2-d^2 x^2}}dx}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {\int \frac {3 \left (27 D c^3+20 C d c^2+12 B d^2 c+8 A d^3\right ) x^2 d^3+4 c \left (27 D c^3+23 C d c^2+18 B d^2 c+12 A d^3\right ) x d^2+12 c^2 \left (3 D c^3+3 C d c^2+3 B d^2 c+2 A d^3\right ) d}{\sqrt {c^2-d^2 x^2}}dx}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {-\frac {\int -\frac {c d^3 \left (3 c \left (51 D c^3+44 C d c^2+36 B d^2 c+24 A d^3\right )+8 d \left (27 D c^3+23 C d c^2+18 B d^2 c+12 A d^3\right ) x\right )}{\sqrt {c^2-d^2 x^2}}dx}{2 d^2}-\frac {3}{2} d x \sqrt {c^2-d^2 x^2} \left (8 A d^3+12 B c d^2+27 c^3 D+20 c^2 C d\right )}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {\frac {\int \frac {c d^3 \left (3 c \left (51 D c^3+44 C d c^2+36 B d^2 c+24 A d^3\right )+8 d \left (27 D c^3+23 C d c^2+18 B d^2 c+12 A d^3\right ) x\right )}{\sqrt {c^2-d^2 x^2}}dx}{2 d^2}-\frac {3}{2} d x \sqrt {c^2-d^2 x^2} \left (8 A d^3+12 B c d^2+27 c^3 D+20 c^2 C d\right )}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {\frac {1}{2} c d \int \frac {3 c \left (51 D c^3+44 C d c^2+36 B d^2 c+24 A d^3\right )+8 d \left (27 D c^3+23 C d c^2+18 B d^2 c+12 A d^3\right ) x}{\sqrt {c^2-d^2 x^2}}dx-\frac {3}{2} d x \sqrt {c^2-d^2 x^2} \left (8 A d^3+12 B c d^2+27 c^3 D+20 c^2 C d\right )}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {\frac {1}{2} c d \left (3 c \left (24 A d^3+36 B c d^2+51 c^3 D+44 c^2 C d\right ) \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx-\frac {8 \sqrt {c^2-d^2 x^2} \left (12 A d^3+18 B c d^2+27 c^3 D+23 c^2 C d\right )}{d}\right )-\frac {3}{2} d x \sqrt {c^2-d^2 x^2} \left (8 A d^3+12 B c d^2+27 c^3 D+20 c^2 C d\right )}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {\frac {1}{2} c d \left (3 c \left (24 A d^3+36 B c d^2+51 c^3 D+44 c^2 C d\right ) \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}-\frac {8 \sqrt {c^2-d^2 x^2} \left (12 A d^3+18 B c d^2+27 c^3 D+23 c^2 C d\right )}{d}\right )-\frac {3}{2} d x \sqrt {c^2-d^2 x^2} \left (8 A d^3+12 B c d^2+27 c^3 D+20 c^2 C d\right )}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {(c+d x)^3 \left (A d^3+B c d^2+c^3 D+c^2 C d\right )}{c d^4 \sqrt {c^2-d^2 x^2}}-\frac {\frac {\frac {\frac {1}{2} c d \left (\frac {3 c \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right ) \left (24 A d^3+36 B c d^2+51 c^3 D+44 c^2 C d\right )}{d}-\frac {8 \sqrt {c^2-d^2 x^2} \left (12 A d^3+18 B c d^2+27 c^3 D+23 c^2 C d\right )}{d}\right )-\frac {3}{2} d x \sqrt {c^2-d^2 x^2} \left (8 A d^3+12 B c d^2+27 c^3 D+20 c^2 C d\right )}{3 d^2}-\frac {4}{3} c x^2 \sqrt {c^2-d^2 x^2} (3 c D+C d)}{4 d^2}-\frac {c D x^3 \sqrt {c^2-d^2 x^2}}{4 d}}{c}\)

Input:

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

Output:

((c^2*C*d + B*c*d^2 + A*d^3 + c^3*D)*(c + d*x)^3)/(c*d^4*Sqrt[c^2 - d^2*x^ 
2]) - (-1/4*(c*D*x^3*Sqrt[c^2 - d^2*x^2])/d + ((-4*c*(C*d + 3*c*D)*x^2*Sqr 
t[c^2 - d^2*x^2])/3 + ((-3*d*(20*c^2*C*d + 12*B*c*d^2 + 8*A*d^3 + 27*c^3*D 
)*x*Sqrt[c^2 - d^2*x^2])/2 + (c*d*((-8*(23*c^2*C*d + 18*B*c*d^2 + 12*A*d^3 
 + 27*c^3*D)*Sqrt[c^2 - d^2*x^2])/d + (3*c*(44*c^2*C*d + 36*B*c*d^2 + 24*A 
*d^3 + 51*c^3*D)*ArcTan[(d*x)/Sqrt[c^2 - d^2*x^2]])/d))/2)/(3*d^2))/(4*d^2 
))/c
 

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 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(499\) vs. \(2(249)=498\).

Time = 0.63 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.86

method	result	size
default	\(\frac {A c x}{\sqrt {-d^{2} x^{2}+c^{2}}}+\frac {c^{2} \left (3 A d +B c \right )}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+d^{2} \left (C d +3 D c \right ) \left (-\frac {x^{4}}{3 d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {4 c^{2} \left (-\frac {x^{2}}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {2 c^{2}}{d^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )}{3 d^{2}}\right )+c \left (3 A \,d^{2}+3 B c d +C \,c^{2}\right ) \left (\frac {x}{\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}-\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{2} \sqrt {d^{2}}}\right )+d \left (B \,d^{2}+3 C c d +3 D c^{2}\right ) \left (-\frac {x^{3}}{2 d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {3 c^{2} \left (\frac {x}{\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}-\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{2} \sqrt {d^{2}}}\right )}{2 d^{2}}\right )+\left (A \,d^{3}+3 B c \,d^{2}+3 C \,c^{2} d +D c^{3}\right ) \left (-\frac {x^{2}}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {2 c^{2}}{d^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )+d^{3} D \left (-\frac {x^{5}}{4 d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {5 c^{2} \left (-\frac {x^{3}}{2 d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {3 c^{2} \left (\frac {x}{\sqrt {-d^{2} x^{2}+c^{2}}\, d^{2}}-\frac {\arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d^{2} \sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{4 d^{2}}\right )\)	\(500\)

Input:

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

Output:

A*c*x/(-d^2*x^2+c^2)^(1/2)+c^2*(3*A*d+B*c)/d^2/(-d^2*x^2+c^2)^(1/2)+d^2*(C 
*d+3*D*c)*(-1/3*x^4/d^2/(-d^2*x^2+c^2)^(1/2)+4/3*c^2/d^2*(-x^2/d^2/(-d^2*x 
^2+c^2)^(1/2)+2*c^2/d^4/(-d^2*x^2+c^2)^(1/2)))+c*(3*A*d^2+3*B*c*d+C*c^2)*( 
1/(-d^2*x^2+c^2)^(1/2)/d^2*x-1/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)*(-1/2*x^3/d^2/(-d^2*x^2+c^2)^(1 
/2)+3/2*c^2/d^2*(1/(-d^2*x^2+c^2)^(1/2)/d^2*x-1/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)*(-x^2 
/d^2/(-d^2*x^2+c^2)^(1/2)+2*c^2/d^4/(-d^2*x^2+c^2)^(1/2))+d^3*D*(-1/4*x^5/ 
d^2/(-d^2*x^2+c^2)^(1/2)+5/4*c^2/d^2*(-1/2*x^3/d^2/(-d^2*x^2+c^2)^(1/2)+3/ 
2*c^2/d^2*(1/(-d^2*x^2+c^2)^(1/2)/d^2*x-1/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 = 314, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {240 \, D c^{5} + 208 \, C c^{4} d + 168 \, B c^{3} d^{2} + 120 \, A c^{2} d^{3} - 8 \, {\left (30 \, D c^{4} d + 26 \, C c^{3} d^{2} + 21 \, B c^{2} d^{3} + 15 \, A c d^{4}\right )} x + 6 \, {\left (51 \, D c^{5} + 44 \, C c^{4} d + 36 \, B c^{3} d^{2} + 24 \, A c^{2} d^{3} - {\left (51 \, D c^{4} d + 44 \, C c^{3} d^{2} + 36 \, B c^{2} d^{3} + 24 \, A c d^{4}\right )} x\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (6 \, D d^{4} x^{4} - 240 \, D c^{4} - 208 \, C c^{3} d - 168 \, B c^{2} d^{2} - 120 \, A c d^{3} + 2 \, {\left (9 \, D c d^{3} + 4 \, C d^{4}\right )} x^{3} + {\left (33 \, D c^{2} d^{2} + 28 \, C c d^{3} + 12 \, B d^{4}\right )} x^{2} + {\left (87 \, D c^{3} d + 76 \, C c^{2} d^{2} + 60 \, B c d^{3} + 24 \, A d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{24 \, {\left (d^{5} x - c d^{4}\right )}} \] Input:

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

Output:

-1/24*(240*D*c^5 + 208*C*c^4*d + 168*B*c^3*d^2 + 120*A*c^2*d^3 - 8*(30*D*c 
^4*d + 26*C*c^3*d^2 + 21*B*c^2*d^3 + 15*A*c*d^4)*x + 6*(51*D*c^5 + 44*C*c^ 
4*d + 36*B*c^3*d^2 + 24*A*c^2*d^3 - (51*D*c^4*d + 44*C*c^3*d^2 + 36*B*c^2* 
d^3 + 24*A*c*d^4)*x)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - (6*D*d^4* 
x^4 - 240*D*c^4 - 208*C*c^3*d - 168*B*c^2*d^2 - 120*A*c*d^3 + 2*(9*D*c*d^3 
 + 4*C*d^4)*x^3 + (33*D*c^2*d^2 + 28*C*c*d^3 + 12*B*d^4)*x^2 + (87*D*c^3*d 
 + 76*C*c^2*d^2 + 60*B*c*d^3 + 24*A*d^4)*x)*sqrt(-d^2*x^2 + c^2))/(d^5*x - 
 c*d^4)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (249) = 498\).

Time = 0.11 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.02 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=-\frac {D d x^{5}}{4 \, \sqrt {-d^{2} x^{2} + c^{2}}} - \frac {5 \, D c^{2} x^{3}}{8 \, \sqrt {-d^{2} x^{2} + c^{2}} d} + \frac {A c x}{\sqrt {-d^{2} x^{2} + c^{2}}} + \frac {15 \, D c^{4} x}{8 \, \sqrt {-d^{2} x^{2} + c^{2}} d^{3}} - \frac {{\left (3 \, D c d^{2} + C d^{3}\right )} x^{4}}{3 \, \sqrt {-d^{2} x^{2} + c^{2}} d^{2}} - \frac {15 \, D c^{4} \arcsin \left (\frac {d x}{c}\right )}{8 \, d^{4}} + \frac {B c^{3}}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} + \frac {3 \, A c^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} d} - \frac {{\left (3 \, D c^{2} d + 3 \, C c d^{2} + B d^{3}\right )} x^{3}}{2 \, \sqrt {-d^{2} x^{2} + c^{2}} d^{2}} - \frac {4 \, {\left (3 \, D c d^{2} + C d^{3}\right )} c^{2} x^{2}}{3 \, \sqrt {-d^{2} x^{2} + c^{2}} d^{4}} - \frac {{\left (D c^{3} + 3 \, C c^{2} d + 3 \, B c d^{2} + A d^{3}\right )} x^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} + \frac {3 \, {\left (3 \, D c^{2} d + 3 \, C c d^{2} + B d^{3}\right )} c^{2} x}{2 \, \sqrt {-d^{2} x^{2} + c^{2}} d^{4}} + \frac {{\left (C c^{3} + 3 \, B c^{2} d + 3 \, A c d^{2}\right )} x}{\sqrt {-d^{2} x^{2} + c^{2}} d^{2}} - \frac {3 \, {\left (3 \, D c^{2} d + 3 \, C c d^{2} + B d^{3}\right )} c^{2} \arcsin \left (\frac {d x}{c}\right )}{2 \, d^{5}} - \frac {{\left (C c^{3} + 3 \, B c^{2} d + 3 \, A c d^{2}\right )} \arcsin \left (\frac {d x}{c}\right )}{d^{3}} + \frac {8 \, {\left (3 \, D c d^{2} + C d^{3}\right )} c^{4}}{3 \, \sqrt {-d^{2} x^{2} + c^{2}} d^{6}} + \frac {2 \, {\left (D c^{3} + 3 \, C c^{2} d + 3 \, B c d^{2} + A d^{3}\right )} c^{2}}{\sqrt {-d^{2} x^{2} + c^{2}} d^{4}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {1}{24} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left (\frac {3 \, D x}{d} + \frac {4 \, {\left (3 \, D c d^{11} + C d^{12}\right )}}{d^{13}}\right )} x + \frac {3 \, {\left (19 \, D c^{2} d^{10} + 12 \, C c d^{11} + 4 \, B d^{12}\right )}}{d^{13}}\right )} x + \frac {8 \, {\left (18 \, D c^{3} d^{9} + 14 \, C c^{2} d^{10} + 9 \, B c d^{11} + 3 \, A d^{12}\right )}}{d^{13}}\right )} - \frac {{\left (51 \, D c^{4} + 44 \, C c^{3} d + 36 \, B c^{2} d^{2} + 24 \, A c d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{8 \, d^{3} {\left | d \right |}} + \frac {8 \, {\left (D c^{4} + C c^{3} d + B c^{2} d^{2} + A c d^{3}\right )}}{d^{3} {\left (\frac {c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}}{d^{2} x} - 1\right )} {\left | d \right |}} \] Input:

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

Output:

1/24*sqrt(-d^2*x^2 + c^2)*((2*(3*D*x/d + 4*(3*D*c*d^11 + C*d^12)/d^13)*x + 
 3*(19*D*c^2*d^10 + 12*C*c*d^11 + 4*B*d^12)/d^13)*x + 8*(18*D*c^3*d^9 + 14 
*C*c^2*d^10 + 9*B*c*d^11 + 3*A*d^12)/d^13) - 1/8*(51*D*c^4 + 44*C*c^3*d + 
36*B*c^2*d^2 + 24*A*c*d^3)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d)) + 8*(D 
*c^4 + C*c^3*d + B*c^2*d^2 + A*c*d^3)/(d^3*((c*d + sqrt(-d^2*x^2 + c^2)*ab 
s(d))/(d^2*x) - 1)*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.92 \[ \int \frac {(c+d x)^3 \left (A+B x+C x^2+D x^3\right )}{\left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {570 c^{5}+61 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{2} x^{2}-570 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4}+6 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{4} x^{4}-6 d^{5} x^{5}-12 b \,d^{4} x^{3}-72 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) a c \,d^{2}+72 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{2} d^{2}+24 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{3} x -216 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d +12 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{3} x^{2}+108 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{3} d -285 \mathit {asin} \left (\frac {d x}{c}\right ) c^{4} d x -192 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{2}+60 b \,c^{2} d^{2} x -72 b c \,d^{3} x^{2}+24 a c \,d^{3} x -285 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) c^{4}+192 a \,c^{2} d^{2}-24 a \,d^{4} x^{2}-108 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{2} d -108 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{2} d^{2} x +60 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{2} x +163 c^{4} d x -224 c^{3} d^{2} x^{2}-87 c^{2} d^{3} x^{3}-32 c \,d^{4} x^{4}+163 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d x +26 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{3} x^{3}+285 \mathit {asin} \left (\frac {d x}{c}\right ) c^{5}+216 b \,c^{3} d -72 \mathit {asin} \left (\frac {d x}{c}\right ) a c \,d^{3} x}{24 d^{3} \left (\sqrt {-d^{2} x^{2}+c^{2}}-c +d x \right )} \] Input:

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

Output:

( - 72*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)*a*c*d**2 - 108*sqrt(c**2 - d** 
2*x**2)*asin((d*x)/c)*b*c**2*d - 285*sqrt(c**2 - d**2*x**2)*asin((d*x)/c)* 
c**4 + 72*asin((d*x)/c)*a*c**2*d**2 - 72*asin((d*x)/c)*a*c*d**3*x + 108*as 
in((d*x)/c)*b*c**3*d - 108*asin((d*x)/c)*b*c**2*d**2*x + 285*asin((d*x)/c) 
*c**5 - 285*asin((d*x)/c)*c**4*d*x - 192*sqrt(c**2 - d**2*x**2)*a*c*d**2 + 
 24*sqrt(c**2 - d**2*x**2)*a*d**3*x - 216*sqrt(c**2 - d**2*x**2)*b*c**2*d 
+ 60*sqrt(c**2 - d**2*x**2)*b*c*d**2*x + 12*sqrt(c**2 - d**2*x**2)*b*d**3* 
x**2 - 570*sqrt(c**2 - d**2*x**2)*c**4 + 163*sqrt(c**2 - d**2*x**2)*c**3*d 
*x + 61*sqrt(c**2 - d**2*x**2)*c**2*d**2*x**2 + 26*sqrt(c**2 - d**2*x**2)* 
c*d**3*x**3 + 6*sqrt(c**2 - d**2*x**2)*d**4*x**4 + 192*a*c**2*d**2 + 24*a* 
c*d**3*x - 24*a*d**4*x**2 + 216*b*c**3*d + 60*b*c**2*d**2*x - 72*b*c*d**3* 
x**2 - 12*b*d**4*x**3 + 570*c**5 + 163*c**4*d*x - 224*c**3*d**2*x**2 - 87* 
c**2*d**3*x**3 - 32*c*d**4*x**4 - 6*d**5*x**5)/(24*d**3*(sqrt(c**2 - d**2* 
x**2) - c + d*x))