\(\int \frac {(A+B x) (c^2-d^2 x^2)^{5/2}}{(c+d x)^6} \, dx\) [34]

Optimal result

Integrand size = 29, antiderivative size = 169 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\frac {B \sqrt {c^2-d^2 x^2}}{d^2}+\frac {2 (5 B c-A d) \sqrt {c^2-d^2 x^2}}{d^2 (c+d x)}-\frac {2 (3 B c-A d) \left (c^2-d^2 x^2\right )^{3/2}}{3 d^2 (c+d x)^3}+\frac {2 (B c-A d) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^2 (c+d x)^5}+\frac {(6 B c-A d) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (141 B c^3-26 A c^2 d+333 B c^2 d x-48 A c d^2 x+231 B c d^2 x^2-46 A d^3 x^2+15 B d^3 x^3\right )}{15 d^2 (c+d x)^3}-\frac {\sqrt {-d^2} (-6 B c+A d) \log \left (-\sqrt {-d^2} x+\sqrt {c^2-d^2 x^2}\right )}{d^3} \] Input:

Integrate[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^6,x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*(141*B*c^3 - 26*A*c^2*d + 333*B*c^2*d*x - 48*A*c*d^2* 
x + 231*B*c*d^2*x^2 - 46*A*d^3*x^2 + 15*B*d^3*x^3))/(15*d^2*(c + d*x)^3) - 
 (Sqrt[-d^2]*(-6*B*c + A*d)*Log[-(Sqrt[-d^2]*x) + Sqrt[c^2 - d^2*x^2]])/d^ 
3
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {671, 465, 463, 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.

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

\(\Big \downarrow \) 671

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

\(\Big \downarrow \) 465

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

\(\Big \downarrow \) 463

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

Int[((A + B*x)*(c^2 - d^2*x^2)^(5/2))/(c + d*x)^6,x]
 

Output:

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

Defintions of rubi rules used

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 463

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x 
))), x] - Simp[d^(2*n + 2)/b^(n + 1)   Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ 
(2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F 
reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, 
-3/2]
 

rule 465

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 + p + 1))), x] - Simp[b*(p/(d^2*(n + 
 p + 1)))   Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, 
b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LtQ[n, -2] || EqQ[n 
+ 2*p + 1, 0]) && NeQ[n + p + 1, 0] && IntegerQ[2*p]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(155)=310\).

Time = 0.66 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.20

method	result	size
risch	\(\frac {B \sqrt {-d^{2} x^{2}+c^{2}}}{d^{2}}-\frac {\left (A d -6 B c \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{d \sqrt {d^{2}}}-\frac {6 \left (A d -3 B c \right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{d^{3} \left (x +\frac {c}{d}\right )}-\frac {4 c^{2} \left (3 A d -5 B c \right ) \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{d^{3}}+\frac {8 c^{3} \left (A d -B c \right ) \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{5 c d \left (x +\frac {c}{d}\right )^{3}}+\frac {2 d \left (-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c d \left (x +\frac {c}{d}\right )^{2}}-\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{3 c^{2} \left (x +\frac {c}{d}\right )}\right )}{5 c}\right )}{d^{4}}\)	\(372\)
default	\(\frac {B \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{3 c d \left (x +\frac {c}{d}\right )^{5}}-\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{c d \left (x +\frac {c}{d}\right )^{4}}-\frac {3 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{c d \left (x +\frac {c}{d}\right )^{3}}+\frac {4 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{3 c d \left (x +\frac {c}{d}\right )^{2}}+\frac {5 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5}+c d \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{4 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )\right )}{3 c}\right )}{c}\right )}{c}\right )}{3 c}\right )}{d^{6}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{5 c d \left (x +\frac {c}{d}\right )^{6}}-\frac {d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{3 c d \left (x +\frac {c}{d}\right )^{5}}-\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{c d \left (x +\frac {c}{d}\right )^{4}}-\frac {3 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{c d \left (x +\frac {c}{d}\right )^{3}}+\frac {4 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {7}{2}}}{3 c d \left (x +\frac {c}{d}\right )^{2}}+\frac {5 d \left (\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {5}{2}}}{5}+c d \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {\left (-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d \right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{4 d^{2}}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )\right )}{3 c}\right )}{c}\right )}{c}\right )}{3 c}\right )}{5 c}\right )}{d^{7}}\)	\(859\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^6,x,method=_RETURNVERBOSE)
 

Output:

B*(-d^2*x^2+c^2)^(1/2)/d^2-(A*d-6*B*c)/d/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/ 
(-d^2*x^2+c^2)^(1/2))-6/d^3*(A*d-3*B*c)/(x+c/d)*(-d^2*(x+c/d)^2+2*c*d*(x+c 
/d))^(1/2)-4*c^2/d^3*(3*A*d-5*B*c)*(-1/3/c/d/(x+c/d)^2*(-d^2*(x+c/d)^2+2*c 
*d*(x+c/d))^(1/2)-1/3/c^2/(x+c/d)*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2))+8* 
c^3*(A*d-B*c)/d^4*(-1/5/c/d/(x+c/d)^3*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2) 
+2/5*d/c*(-1/3/c/d/(x+c/d)^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)-1/3/c^2/ 
(x+c/d)*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\frac {141 \, B c^{4} - 26 \, A c^{3} d + {\left (141 \, B c d^{3} - 26 \, A d^{4}\right )} x^{3} + 3 \, {\left (141 \, B c^{2} d^{2} - 26 \, A c d^{3}\right )} x^{2} + 3 \, {\left (141 \, B c^{3} d - 26 \, A c^{2} d^{2}\right )} x - 30 \, {\left (6 \, B c^{4} - A c^{3} d + {\left (6 \, B c d^{3} - A d^{4}\right )} x^{3} + 3 \, {\left (6 \, B c^{2} d^{2} - A c d^{3}\right )} x^{2} + 3 \, {\left (6 \, B c^{3} d - A c^{2} d^{2}\right )} x\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (15 \, B d^{3} x^{3} + 141 \, B c^{3} - 26 \, A c^{2} d + {\left (231 \, B c d^{2} - 46 \, A d^{3}\right )} x^{2} + 3 \, {\left (111 \, B c^{2} d - 16 \, A c d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{15 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^6,x, algorithm="fricas")
 

Output:

1/15*(141*B*c^4 - 26*A*c^3*d + (141*B*c*d^3 - 26*A*d^4)*x^3 + 3*(141*B*c^2 
*d^2 - 26*A*c*d^3)*x^2 + 3*(141*B*c^3*d - 26*A*c^2*d^2)*x - 30*(6*B*c^4 - 
A*c^3*d + (6*B*c*d^3 - A*d^4)*x^3 + 3*(6*B*c^2*d^2 - A*c*d^3)*x^2 + 3*(6*B 
*c^3*d - A*c^2*d^2)*x)*arctan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (15*B*d 
^3*x^3 + 141*B*c^3 - 26*A*c^2*d + (231*B*c*d^2 - 46*A*d^3)*x^2 + 3*(111*B* 
c^2*d - 16*A*c*d^2)*x)*sqrt(-d^2*x^2 + c^2))/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^ 
2*d^3*x + c^3*d^2)
 

Sympy [F]

\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (c + d x\right )^{6}}\, dx \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(5/2)/(d*x+c)**6,x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(5/2)*(A + B*x)/(c + d*x)**6, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (155) = 310\).

Time = 0.14 (sec) , antiderivative size = 718, normalized size of antiderivative = 4.25 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B c}{5 \, {\left (d^{7} x^{5} + 5 \, c d^{6} x^{4} + 10 \, c^{2} d^{5} x^{3} + 10 \, c^{3} d^{4} x^{2} + 5 \, c^{4} d^{3} x + c^{5} d^{2}\right )}} + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} B c^{2}}{d^{6} x^{4} + 4 \, c d^{5} x^{3} + 6 \, c^{2} d^{4} x^{2} + 4 \, c^{3} d^{3} x + c^{4} d^{2}} - \frac {6 \, \sqrt {-d^{2} x^{2} + c^{2}} B c^{3}}{5 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} A}{5 \, {\left (d^{6} x^{5} + 5 \, c d^{5} x^{4} + 10 \, c^{2} d^{4} x^{3} + 10 \, c^{3} d^{3} x^{2} + 5 \, c^{4} d^{2} x + c^{5} d\right )}} + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} B}{d^{6} x^{4} + 4 \, c d^{5} x^{3} + 6 \, c^{2} d^{4} x^{2} + 4 \, c^{3} d^{3} x + c^{4} d^{2}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A c}{d^{5} x^{4} + 4 \, c d^{4} x^{3} + 6 \, c^{2} d^{3} x^{2} + 4 \, c^{3} d^{2} x + c^{4} d} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} B c}{d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}} + \frac {6 \, \sqrt {-d^{2} x^{2} + c^{2}} A c^{2}}{5 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} - \frac {19 \, \sqrt {-d^{2} x^{2} + c^{2}} B c^{2}}{5 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} + \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} A}{3 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} + \frac {7 \, \sqrt {-d^{2} x^{2} + c^{2}} A c}{15 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} + \frac {71 \, \sqrt {-d^{2} x^{2} + c^{2}} B c}{5 \, {\left (d^{3} x + c d^{2}\right )}} + \frac {6 \, B c \arcsin \left (\frac {d x}{c}\right )}{d^{2}} - \frac {A \arcsin \left (\frac {d x}{c}\right )}{d} - \frac {38 \, \sqrt {-d^{2} x^{2} + c^{2}} A}{15 \, {\left (d^{2} x + c d\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^6,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/5*(-d^2*x^2 + c^2)^(5/2)*B*c/(d^7*x^5 + 5*c*d^6*x^4 + 10*c^2*d^5*x^3 + 1 
0*c^3*d^4*x^2 + 5*c^4*d^3*x + c^5*d^2) + (-d^2*x^2 + c^2)^(3/2)*B*c^2/(d^6 
*x^4 + 4*c*d^5*x^3 + 6*c^2*d^4*x^2 + 4*c^3*d^3*x + c^4*d^2) - 6/5*sqrt(-d^ 
2*x^2 + c^2)*B*c^3/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2) - 1/5*( 
-d^2*x^2 + c^2)^(5/2)*A/(d^6*x^5 + 5*c*d^5*x^4 + 10*c^2*d^4*x^3 + 10*c^3*d 
^3*x^2 + 5*c^4*d^2*x + c^5*d) + (-d^2*x^2 + c^2)^(5/2)*B/(d^6*x^4 + 4*c*d^ 
5*x^3 + 6*c^2*d^4*x^2 + 4*c^3*d^3*x + c^4*d^2) - (-d^2*x^2 + c^2)^(3/2)*A* 
c/(d^5*x^4 + 4*c*d^4*x^3 + 6*c^2*d^3*x^2 + 4*c^3*d^2*x + c^4*d) - 2*(-d^2* 
x^2 + c^2)^(3/2)*B*c/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2) + 6/5 
*sqrt(-d^2*x^2 + c^2)*A*c^2/(d^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d) 
- 19/5*sqrt(-d^2*x^2 + c^2)*B*c^2/(d^4*x^2 + 2*c*d^3*x + c^2*d^2) + 1/3*(- 
d^2*x^2 + c^2)^(3/2)*A/(d^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d) + 7/1 
5*sqrt(-d^2*x^2 + c^2)*A*c/(d^3*x^2 + 2*c*d^2*x + c^2*d) + 71/5*sqrt(-d^2* 
x^2 + c^2)*B*c/(d^3*x + c*d^2) + 6*B*c*arcsin(d*x/c)/d^2 - A*arcsin(d*x/c) 
/d - 38/15*sqrt(-d^2*x^2 + c^2)*A/(d^2*x + c*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (155) = 310\).

Time = 0.16 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.09 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\frac {{\left (6 \, B c - A d\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{d {\left | d \right |}} + \frac {\sqrt {-d^{2} x^{2} + c^{2}} B}{d^{2}} - \frac {4 \, {\left (63 \, B c - 13 \, A d + \frac {270 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} B c}{d^{2} x} - \frac {50 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} A}{d x} + \frac {420 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} B c}{d^{4} x^{2}} - \frac {100 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} A}{d^{3} x^{2}} + \frac {210 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} B c}{d^{6} x^{3}} - \frac {30 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} A}{d^{5} x^{3}} + \frac {45 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} B c}{d^{8} x^{4}} - \frac {15 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} A}{d^{7} x^{4}}\right )}}{15 \, d {\left (\frac {c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}}{d^{2} x} + 1\right )}^{5} {\left | d \right |}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^6,x, algorithm="giac")
 

Output:

(6*B*c - A*d)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d*abs(d)) + sqrt(-d^2*x^2 + c^2 
)*B/d^2 - 4/15*(63*B*c - 13*A*d + 270*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))* 
B*c/(d^2*x) - 50*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) + 420*(c*d + 
sqrt(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^4*x^2) - 100*(c*d + sqrt(-d^2*x^2 + 
c^2)*abs(d))^2*A/(d^3*x^2) + 210*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c 
/(d^6*x^3) - 30*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^3) + 45*(c* 
d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^8*x^4) - 15*(c*d + sqrt(-d^2*x^2 
 + c^2)*abs(d))^4*A/(d^7*x^4))/(d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d))/(d^ 
2*x) + 1)^5*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (c+d\,x\right )}^6} \,d x \] Input:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x))/(c + d*x)^6,x)
 

Output:

int(((c^2 - d^2*x^2)^(5/2)*(A + B*x))/(c + d*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 727, normalized size of antiderivative = 4.30 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{5/2}}{(c+d x)^6} \, dx=\frac {15 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) a \,c^{3} d +45 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) a \,c^{2} d^{2} x +45 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) a c \,d^{3} x^{2}+15 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) a \,d^{4} x^{3}-90 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b \,c^{4}-270 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b \,c^{3} d x -270 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b \,c^{2} d^{2} x^{2}-90 \mathit {atan} \left (\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, c^{2}-2 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} x^{2}}{-2 d^{3} x^{3}+2 c^{2} d x}\right ) b c \,d^{3} x^{3}-52 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d -96 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{2} x -92 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{3} x^{2}+282 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3}+666 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d x +462 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{2} x^{2}+30 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{3} x^{3}}{30 d^{2} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(5/2)/(d*x+c)^6,x)
 

Output:

(15*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2 
)/(2*c**2*d*x - 2*d**3*x**3))*a*c**3*d + 45*atan((sqrt(c**2 - d**2*x**2)*c 
**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*a*c* 
*2*d**2*x + 45*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2 
)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*a*c*d**3*x**2 + 15*atan((sqrt(c** 
2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2* 
d**3*x**3))*a*d**4*x**3 - 90*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c* 
*2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*b*c**4 - 270*atan(( 
sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2* 
d*x - 2*d**3*x**3))*b*c**3*d*x - 270*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2 
*sqrt(c**2 - d**2*x**2)*d**2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*b*c**2*d**2 
*x**2 - 90*atan((sqrt(c**2 - d**2*x**2)*c**2 - 2*sqrt(c**2 - d**2*x**2)*d* 
*2*x**2)/(2*c**2*d*x - 2*d**3*x**3))*b*c*d**3*x**3 - 52*sqrt(c**2 - d**2*x 
**2)*a*c**2*d - 96*sqrt(c**2 - d**2*x**2)*a*c*d**2*x - 92*sqrt(c**2 - d**2 
*x**2)*a*d**3*x**2 + 282*sqrt(c**2 - d**2*x**2)*b*c**3 + 666*sqrt(c**2 - d 
**2*x**2)*b*c**2*d*x + 462*sqrt(c**2 - d**2*x**2)*b*c*d**2*x**2 + 30*sqrt( 
c**2 - d**2*x**2)*b*d**3*x**3)/(30*d**2*(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 
 + d**3*x**3))