\(\int (A+B x) (c+d x)^3 (c^2-d^2 x^2)^{3/2} \, dx\) [12]

Optimal result

Integrand size = 29, antiderivative size = 223 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {9 c^5 (3 B c+8 A d) x \sqrt {c^2-d^2 x^2}}{128 d}+\frac {3 c^3 (3 B c+8 A d) x \left (c^2-d^2 x^2\right )^{3/2}}{64 d}-\frac {(3 B c+8 A d) (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{56 d^2}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}-\frac {3 c (3 B c+8 A d) (12 c+5 d x) \left (c^2-d^2 x^2\right )^{5/2}}{560 d^2}+\frac {9 c^7 (3 B c+8 A d) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{128 d^2} \] Output:

9/128*c^5*(8*A*d+3*B*c)*x*(-d^2*x^2+c^2)^(1/2)/d+3/64*c^3*(8*A*d+3*B*c)*x* 
(-d^2*x^2+c^2)^(3/2)/d-1/56*(8*A*d+3*B*c)*(d*x+c)^2*(-d^2*x^2+c^2)^(5/2)/d 
^2-1/8*B*(d*x+c)^3*(-d^2*x^2+c^2)^(5/2)/d^2-3/560*c*(8*A*d+3*B*c)*(5*d*x+1 
2*c)*(-d^2*x^2+c^2)^(5/2)/d^2+9/128*c^7*(8*A*d+3*B*c)*arctan(d*x/(-d^2*x^2 
+c^2)^(1/2))/d^2
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.96 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (8 A d \left (368 c^6-245 c^5 d x-656 c^4 d^2 x^2-350 c^3 d^3 x^3+208 c^2 d^4 x^4+280 c d^5 x^5+80 d^6 x^6\right )+B \left (1664 c^7+945 c^6 d x-1408 c^5 d^2 x^2-3850 c^4 d^3 x^3-2176 c^3 d^4 x^4+1400 c^2 d^5 x^5+1920 c d^6 x^6+560 d^7 x^7\right )\right )+630 c^7 (3 B c+8 A d) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{4480 d^2} \] Input:

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

Output:

-1/4480*(Sqrt[c^2 - d^2*x^2]*(8*A*d*(368*c^6 - 245*c^5*d*x - 656*c^4*d^2*x 
^2 - 350*c^3*d^3*x^3 + 208*c^2*d^4*x^4 + 280*c*d^5*x^5 + 80*d^6*x^6) + B*( 
1664*c^7 + 945*c^6*d*x - 1408*c^5*d^2*x^2 - 3850*c^4*d^3*x^3 - 2176*c^3*d^ 
4*x^4 + 1400*c^2*d^5*x^5 + 1920*c*d^6*x^6 + 560*d^7*x^7)) + 630*c^7*(3*B*c 
 + 8*A*d)*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^2
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {672, 469, 469, 455, 211, 211, 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.

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

\(\Big \downarrow \) 672

\(\displaystyle \frac {(8 A d+3 B c) \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}dx}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \left (\frac {7}{6} c \int (c+d x) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \left (\frac {7}{6} c \left (c \int \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )-\frac {(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \left (\frac {7}{6} c \left (c \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )-\frac {(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \left (\frac {7}{6} c \left (c \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )-\frac {(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \left (\frac {7}{6} c \left (c \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )-\frac {(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {(8 A d+3 B c) \left (\frac {9}{7} c \left (\frac {7}{6} c \left (c \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {\left (c^2-d^2 x^2\right )^{5/2}}{5 d}\right )-\frac {(c+d x) \left (c^2-d^2 x^2\right )^{5/2}}{6 d}\right )-\frac {(c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2}}{7 d}\right )}{8 d}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{5/2}}{8 d^2}\)

Input:

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

Output:

-1/8*(B*(c + d*x)^3*(c^2 - d^2*x^2)^(5/2))/d^2 + ((3*B*c + 8*A*d)*(-1/7*(( 
c + d*x)^2*(c^2 - d^2*x^2)^(5/2))/d + (9*c*(-1/6*((c + d*x)*(c^2 - d^2*x^2 
)^(5/2))/d + (7*c*(-1/5*(c^2 - d^2*x^2)^(5/2)/d + c*((x*(c^2 - d^2*x^2)^(3 
/2))/4 + (3*c^2*((x*Sqrt[c^2 - d^2*x^2])/2 + (c^2*ArcTan[(d*x)/Sqrt[c^2 - 
d^2*x^2]])/(2*d)))/4)))/6))/7))/(8*d)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 469

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

rule 672

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), 
 x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))   Int[(d + e*x 
)^m*(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] && NeQ[m + 2*p + 2, 0]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {\left (560 B \,d^{7} x^{7}+640 A \,d^{7} x^{6}+1920 B c \,d^{6} x^{6}+2240 A c \,d^{6} x^{5}+1400 B \,c^{2} d^{5} x^{5}+1664 A \,c^{2} d^{5} x^{4}-2176 B \,c^{3} d^{4} x^{4}-2800 A \,c^{3} d^{4} x^{3}-3850 B \,c^{4} d^{3} x^{3}-5248 c^{4} x^{2} A \,d^{3}-1408 c^{5} x^{2} B \,d^{2}-1960 A \,c^{5} d^{2} x +945 B \,c^{6} d x +2944 A \,c^{6} d +1664 B \,c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{4480 d^{2}}+\frac {9 c^{7} \left (8 A d +3 B c \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{128 d \sqrt {d^{2}}}\)	\(219\)
default	\(A \,c^{3} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )+d^{2} \left (A d +3 B c \right ) \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )+3 c d \left (A d +B c \right ) \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )-\frac {c^{2} \left (3 A d +B c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{5 d^{2}}+B \,d^{3} \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )\)	\(418\)

Input:

int((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4480/d^2*(560*B*d^7*x^7+640*A*d^7*x^6+1920*B*c*d^6*x^6+2240*A*c*d^6*x^5 
+1400*B*c^2*d^5*x^5+1664*A*c^2*d^5*x^4-2176*B*c^3*d^4*x^4-2800*A*c^3*d^4*x 
^3-3850*B*c^4*d^3*x^3-5248*A*c^4*d^3*x^2-1408*B*c^5*d^2*x^2-1960*A*c^5*d^2 
*x+945*B*c^6*d*x+2944*A*c^6*d+1664*B*c^7)*(-d^2*x^2+c^2)^(1/2)+9/128*c^7/d 
*(8*A*d+3*B*c)/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=-\frac {630 \, {\left (3 \, B c^{8} + 8 \, A c^{7} d\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (560 \, B d^{7} x^{7} + 1664 \, B c^{7} + 2944 \, A c^{6} d + 640 \, {\left (3 \, B c d^{6} + A d^{7}\right )} x^{6} + 280 \, {\left (5 \, B c^{2} d^{5} + 8 \, A c d^{6}\right )} x^{5} - 128 \, {\left (17 \, B c^{3} d^{4} - 13 \, A c^{2} d^{5}\right )} x^{4} - 350 \, {\left (11 \, B c^{4} d^{3} + 8 \, A c^{3} d^{4}\right )} x^{3} - 128 \, {\left (11 \, B c^{5} d^{2} + 41 \, A c^{4} d^{3}\right )} x^{2} + 35 \, {\left (27 \, B c^{6} d - 56 \, A c^{5} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{4480 \, d^{2}} \] Input:

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

Output:

-1/4480*(630*(3*B*c^8 + 8*A*c^7*d)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x 
)) + (560*B*d^7*x^7 + 1664*B*c^7 + 2944*A*c^6*d + 640*(3*B*c*d^6 + A*d^7)* 
x^6 + 280*(5*B*c^2*d^5 + 8*A*c*d^6)*x^5 - 128*(17*B*c^3*d^4 - 13*A*c^2*d^5 
)*x^4 - 350*(11*B*c^4*d^3 + 8*A*c^3*d^4)*x^3 - 128*(11*B*c^5*d^2 + 41*A*c^ 
4*d^3)*x^2 + 35*(27*B*c^6*d - 56*A*c^5*d^2)*x)*sqrt(-d^2*x^2 + c^2))/d^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (209) = 418\).

Time = 0.97 (sec) , antiderivative size = 685, normalized size of antiderivative = 3.07 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {c^{2} - d^{2} x^{2}} \left (- \frac {B d^{5} x^{7}}{8} - \frac {x^{6} \left (A d^{7} + 3 B c d^{6}\right )}{7 d^{2}} - \frac {x^{5} \cdot \left (3 A c d^{6} + \frac {15 B c^{2} d^{5}}{8}\right )}{6 d^{2}} - \frac {x^{4} \left (A c^{2} d^{5} - 5 B c^{3} d^{4} + \frac {6 c^{2} \left (A d^{7} + 3 B c d^{6}\right )}{7 d^{2}}\right )}{5 d^{2}} - \frac {x^{3} \left (- 5 A c^{3} d^{4} - 5 B c^{4} d^{3} + \frac {5 c^{2} \cdot \left (3 A c d^{6} + \frac {15 B c^{2} d^{5}}{8}\right )}{6 d^{2}}\right )}{4 d^{2}} - \frac {x^{2} \left (- 5 A c^{4} d^{3} + B c^{5} d^{2} + \frac {4 c^{2} \left (A c^{2} d^{5} - 5 B c^{3} d^{4} + \frac {6 c^{2} \left (A d^{7} + 3 B c d^{6}\right )}{7 d^{2}}\right )}{5 d^{2}}\right )}{3 d^{2}} - \frac {x \left (A c^{5} d^{2} + 3 B c^{6} d + \frac {3 c^{2} \left (- 5 A c^{3} d^{4} - 5 B c^{4} d^{3} + \frac {5 c^{2} \cdot \left (3 A c d^{6} + \frac {15 B c^{2} d^{5}}{8}\right )}{6 d^{2}}\right )}{4 d^{2}}\right )}{2 d^{2}} - \frac {3 A c^{6} d + B c^{7} + \frac {2 c^{2} \left (- 5 A c^{4} d^{3} + B c^{5} d^{2} + \frac {4 c^{2} \left (A c^{2} d^{5} - 5 B c^{3} d^{4} + \frac {6 c^{2} \left (A d^{7} + 3 B c d^{6}\right )}{7 d^{2}}\right )}{5 d^{2}}\right )}{3 d^{2}}}{d^{2}}\right ) + \left (A c^{7} + \frac {c^{2} \left (A c^{5} d^{2} + 3 B c^{6} d + \frac {3 c^{2} \left (- 5 A c^{3} d^{4} - 5 B c^{4} d^{3} + \frac {5 c^{2} \cdot \left (3 A c d^{6} + \frac {15 B c^{2} d^{5}}{8}\right )}{6 d^{2}}\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 \\\left (A c^{3} x + \frac {B d^{3} x^{5}}{5} + \frac {x^{4} \left (A d^{3} + 3 B c d^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A c d^{2} + 3 B c^{2} d\right )}{3} + \frac {x^{2} \cdot \left (3 A c^{2} d + B c^{3}\right )}{2}\right ) \left (c^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(c**2 - d**2*x**2)*(-B*d**5*x**7/8 - x**6*(A*d**7 + 3*B*c*d 
**6)/(7*d**2) - x**5*(3*A*c*d**6 + 15*B*c**2*d**5/8)/(6*d**2) - x**4*(A*c* 
*2*d**5 - 5*B*c**3*d**4 + 6*c**2*(A*d**7 + 3*B*c*d**6)/(7*d**2))/(5*d**2) 
- x**3*(-5*A*c**3*d**4 - 5*B*c**4*d**3 + 5*c**2*(3*A*c*d**6 + 15*B*c**2*d* 
*5/8)/(6*d**2))/(4*d**2) - x**2*(-5*A*c**4*d**3 + B*c**5*d**2 + 4*c**2*(A* 
c**2*d**5 - 5*B*c**3*d**4 + 6*c**2*(A*d**7 + 3*B*c*d**6)/(7*d**2))/(5*d**2 
))/(3*d**2) - x*(A*c**5*d**2 + 3*B*c**6*d + 3*c**2*(-5*A*c**3*d**4 - 5*B*c 
**4*d**3 + 5*c**2*(3*A*c*d**6 + 15*B*c**2*d**5/8)/(6*d**2))/(4*d**2))/(2*d 
**2) - (3*A*c**6*d + B*c**7 + 2*c**2*(-5*A*c**4*d**3 + B*c**5*d**2 + 4*c** 
2*(A*c**2*d**5 - 5*B*c**3*d**4 + 6*c**2*(A*d**7 + 3*B*c*d**6)/(7*d**2))/(5 
*d**2))/(3*d**2))/d**2) + (A*c**7 + c**2*(A*c**5*d**2 + 3*B*c**6*d + 3*c** 
2*(-5*A*c**3*d**4 - 5*B*c**4*d**3 + 5*c**2*(3*A*c*d**6 + 15*B*c**2*d**5/8) 
/(6*d**2))/(4*d**2))/(2*d**2))*Piecewise((log(-2*d**2*x + 2*sqrt(-d**2)*sq 
rt(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 + B*d**3*x**5/5 + x**4*(A*d**3 + 3*B*c 
*d**2)/4 + x**3*(3*A*c*d**2 + 3*B*c**2*d)/3 + x**2*(3*A*c**2*d + B*c**3)/2 
)*(c**2)**(3/2), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (199) = 398\).

Time = 0.12 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.87 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {3 \, B c^{8} \arcsin \left (\frac {d x}{c}\right )}{128 \, d^{2}} + \frac {3 \, A c^{7} \arcsin \left (\frac {d x}{c}\right )}{8 \, d} + \frac {3}{8} \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{5} x + \frac {3 \, \sqrt {-d^{2} x^{2} + c^{2}} B c^{6} x}{128 \, d} - \frac {1}{8} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B d x^{3} + \frac {1}{4} \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A c^{3} x + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} B c^{4} x}{64 \, d} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B c^{2} x}{16 \, d} + \frac {3 \, {\left (B c^{2} d + A c d^{2}\right )} c^{6} \arcsin \left (\frac {d x}{c}\right )}{16 \, d^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B c^{3}}{5 \, d^{2}} - \frac {3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} A c^{2}}{5 \, d} + \frac {3 \, {\left (B c^{2} d + A c d^{2}\right )} \sqrt {-d^{2} x^{2} + c^{2}} c^{4} x}{16 \, d^{2}} + \frac {{\left (B c^{2} d + A c d^{2}\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} c^{2} x}{8 \, d^{2}} - \frac {{\left (3 \, B c d^{2} + A d^{3}\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, d^{2}} - \frac {{\left (B c^{2} d + A c d^{2}\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} x}{2 \, d^{2}} - \frac {2 \, {\left (3 \, B c d^{2} + A d^{3}\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c^{2}}{35 \, d^{4}} \] Input:

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

Output:

3/128*B*c^8*arcsin(d*x/c)/d^2 + 3/8*A*c^7*arcsin(d*x/c)/d + 3/8*sqrt(-d^2* 
x^2 + c^2)*A*c^5*x + 3/128*sqrt(-d^2*x^2 + c^2)*B*c^6*x/d - 1/8*(-d^2*x^2 
+ c^2)^(5/2)*B*d*x^3 + 1/4*(-d^2*x^2 + c^2)^(3/2)*A*c^3*x + 1/64*(-d^2*x^2 
 + c^2)^(3/2)*B*c^4*x/d - 1/16*(-d^2*x^2 + c^2)^(5/2)*B*c^2*x/d + 3/16*(B* 
c^2*d + A*c*d^2)*c^6*arcsin(d*x/c)/d^3 - 1/5*(-d^2*x^2 + c^2)^(5/2)*B*c^3/ 
d^2 - 3/5*(-d^2*x^2 + c^2)^(5/2)*A*c^2/d + 3/16*(B*c^2*d + A*c*d^2)*sqrt(- 
d^2*x^2 + c^2)*c^4*x/d^2 + 1/8*(B*c^2*d + A*c*d^2)*(-d^2*x^2 + c^2)^(3/2)* 
c^2*x/d^2 - 1/7*(3*B*c*d^2 + A*d^3)*(-d^2*x^2 + c^2)^(5/2)*x^2/d^2 - 1/2*( 
B*c^2*d + A*c*d^2)*(-d^2*x^2 + c^2)^(5/2)*x/d^2 - 2/35*(3*B*c*d^2 + A*d^3) 
*(-d^2*x^2 + c^2)^(5/2)*c^2/d^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.08 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {9 \, {\left (3 \, B c^{8} + 8 \, A c^{7} d\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{128 \, d {\left | d \right |}} - \frac {1}{4480} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, B d^{5} x + \frac {8 \, {\left (3 \, B c d^{16} + A d^{17}\right )}}{d^{12}}\right )} x + \frac {7 \, {\left (5 \, B c^{2} d^{15} + 8 \, A c d^{16}\right )}}{d^{12}}\right )} x - \frac {16 \, {\left (17 \, B c^{3} d^{14} - 13 \, A c^{2} d^{15}\right )}}{d^{12}}\right )} x - \frac {175 \, {\left (11 \, B c^{4} d^{13} + 8 \, A c^{3} d^{14}\right )}}{d^{12}}\right )} x - \frac {64 \, {\left (11 \, B c^{5} d^{12} + 41 \, A c^{4} d^{13}\right )}}{d^{12}}\right )} x + \frac {35 \, {\left (27 \, B c^{6} d^{11} - 56 \, A c^{5} d^{12}\right )}}{d^{12}}\right )} x + \frac {128 \, {\left (13 \, B c^{7} d^{10} + 23 \, A c^{6} d^{11}\right )}}{d^{12}}\right )} \] Input:

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

Output:

9/128*(3*B*c^8 + 8*A*c^7*d)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d*abs(d)) - 1/448 
0*sqrt(-d^2*x^2 + c^2)*((2*((4*(5*(2*(7*B*d^5*x + 8*(3*B*c*d^16 + A*d^17)/ 
d^12)*x + 7*(5*B*c^2*d^15 + 8*A*c*d^16)/d^12)*x - 16*(17*B*c^3*d^14 - 13*A 
*c^2*d^15)/d^12)*x - 175*(11*B*c^4*d^13 + 8*A*c^3*d^14)/d^12)*x - 64*(11*B 
*c^5*d^12 + 41*A*c^4*d^13)/d^12)*x + 35*(27*B*c^6*d^11 - 56*A*c^5*d^12)/d^ 
12)*x + 128*(13*B*c^7*d^10 + 23*A*c^6*d^11)/d^12)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.77 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \, dx=\frac {2520 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{7} d +945 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{8}-2944 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d +1960 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{2} x +5248 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{3} x^{2}+2800 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{4} x^{3}-1664 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{5} x^{4}-2240 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{6} x^{5}-640 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{7} x^{6}-1664 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{7}-945 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d x +1408 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{2} x^{2}+3850 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{3} x^{3}+2176 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{4} x^{4}-1400 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{5} x^{5}-1920 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{6} x^{6}-560 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{7} x^{7}+2944 a \,c^{7} d +1664 b \,c^{8}}{4480 d^{2}} \] Input:

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

Output:

(2520*asin((d*x)/c)*a*c**7*d + 945*asin((d*x)/c)*b*c**8 - 2944*sqrt(c**2 - 
 d**2*x**2)*a*c**6*d + 1960*sqrt(c**2 - d**2*x**2)*a*c**5*d**2*x + 5248*sq 
rt(c**2 - d**2*x**2)*a*c**4*d**3*x**2 + 2800*sqrt(c**2 - d**2*x**2)*a*c**3 
*d**4*x**3 - 1664*sqrt(c**2 - d**2*x**2)*a*c**2*d**5*x**4 - 2240*sqrt(c**2 
 - d**2*x**2)*a*c*d**6*x**5 - 640*sqrt(c**2 - d**2*x**2)*a*d**7*x**6 - 166 
4*sqrt(c**2 - d**2*x**2)*b*c**7 - 945*sqrt(c**2 - d**2*x**2)*b*c**6*d*x + 
1408*sqrt(c**2 - d**2*x**2)*b*c**5*d**2*x**2 + 3850*sqrt(c**2 - d**2*x**2) 
*b*c**4*d**3*x**3 + 2176*sqrt(c**2 - d**2*x**2)*b*c**3*d**4*x**4 - 1400*sq 
rt(c**2 - d**2*x**2)*b*c**2*d**5*x**5 - 1920*sqrt(c**2 - d**2*x**2)*b*c*d* 
*6*x**6 - 560*sqrt(c**2 - d**2*x**2)*b*d**7*x**7 + 2944*a*c**7*d + 1664*b* 
c**8)/(4480*d**2)