\(\int \frac {(A+B x) (c^2-d^2 x^2)^{3/2}}{(c+d x)^{9/2}} \, dx\) [79]

Optimal result

Integrand size = 31, antiderivative size = 176 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=-\frac {(11 B c-3 A d) \sqrt {c^2-d^2 x^2}}{4 d^2 (c+d x)^{3/2}}-\frac {2 B \sqrt {c^2-d^2 x^2}}{d^2 \sqrt {c+d x}}+\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{3/2}}{2 d^2 (c+d x)^{7/2}}+\frac {3 (9 B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{4 \sqrt {2} \sqrt {c} d^2} \] Output:

-1/4*(-3*A*d+11*B*c)*(-d^2*x^2+c^2)^(1/2)/d^2/(d*x+c)^(3/2)-2*B*(-d^2*x^2+ 
c^2)^(1/2)/d^2/(d*x+c)^(1/2)+1/2*(-A*d+B*c)*(-d^2*x^2+c^2)^(3/2)/d^2/(d*x+ 
c)^(7/2)+3/8*(-A*d+9*B*c)*arctanh(2^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-d^2*x^2+ 
c^2)^(1/2))*2^(1/2)/c^(1/2)/d^2
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=\frac {-\frac {2 \sqrt {c^2-d^2 x^2} \left (-A d (c+5 d x)+B \left (17 c^2+29 c d x+8 d^2 x^2\right )\right )}{(c+d x)^{5/2}}+\frac {3 \sqrt {2} (9 B c-A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {c}}}{8 d^2} \] Input:

Integrate[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^(9/2),x]
 

Output:

((-2*Sqrt[c^2 - d^2*x^2]*(-(A*d*(c + 5*d*x)) + B*(17*c^2 + 29*c*d*x + 8*d^ 
2*x^2)))/(c + d*x)^(5/2) + (3*Sqrt[2]*(9*B*c - A*d)*ArcTanh[(Sqrt[2]*Sqrt[ 
c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/Sqrt[c])/(8*d^2)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {671, 465, 466, 471, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 671

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

\(\Big \downarrow \) 465

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

\(\Big \downarrow \) 466

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

Input:

Int[((A + B*x)*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^(9/2),x]
 

Output:

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

Defintions of rubi rules used

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 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 466

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

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {2 B \sqrt {-d x +c}\, \sqrt {\frac {-d^{2} x^{2}+c^{2}}{d x +c}}\, \sqrt {d x +c}}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {2 \left (\frac {\left (-\frac {5 A d}{8}+\frac {13 B c}{8}\right ) \left (-d x +c \right )^{\frac {3}{2}}+\left (-\frac {11}{4} B \,c^{2}+\frac {3}{4} A c d \right ) \sqrt {-d x +c}}{\left (-d x -c \right )^{2}}-\frac {3 \left (A d -9 B c \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}\right ) \sqrt {\frac {-d^{2} x^{2}+c^{2}}{d x +c}}\, \sqrt {d x +c}}{d^{2} \sqrt {-d^{2} x^{2}+c^{2}}}\)	\(192\)
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{3} x^{2}-27 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{2} x^{2}+6 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{2} x -54 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d x +16 B \,d^{2} x^{2} \sqrt {-d x +c}\, \sqrt {c}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d -10 A \,d^{2} x \sqrt {-d x +c}\, \sqrt {c}-27 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3}+58 B \,c^{\frac {3}{2}} d x \sqrt {-d x +c}-2 A \sqrt {-d x +c}\, c^{\frac {3}{2}} d +34 B \sqrt {-d x +c}\, c^{\frac {5}{2}}\right )}{8 \left (d x +c \right )^{\frac {5}{2}} \sqrt {-d x +c}\, d^{2} \sqrt {c}}\)	\(290\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-2*B*(-d*x+c)^(1/2)/d^2*((-d^2*x^2+c^2)/(d*x+c))^(1/2)/(-d^2*x^2+c^2)^(1/2 
)*(d*x+c)^(1/2)+2/d^2*(((-5/8*A*d+13/8*B*c)*(-d*x+c)^(3/2)+(-11/4*B*c^2+3/ 
4*A*c*d)*(-d*x+c)^(1/2))/(-d*x-c)^2-3/16*(A*d-9*B*c)*2^(1/2)/c^(1/2)*arcta 
nh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2)))*((-d^2*x^2+c^2)/(d*x+c))^(1/2)/(-d 
^2*x^2+c^2)^(1/2)*(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.77 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (9 \, B c^{4} - A c^{3} d + {\left (9 \, B c d^{3} - A d^{4}\right )} x^{3} + 3 \, {\left (9 \, B c^{2} d^{2} - A c d^{3}\right )} x^{2} + 3 \, {\left (9 \, B c^{3} d - A c^{2} d^{2}\right )} x\right )} \sqrt {c} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x + 2 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {c} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 4 \, {\left (8 \, B c d^{2} x^{2} + 17 \, B c^{3} - A c^{2} d + {\left (29 \, B c^{2} d - 5 \, A c d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{16 \, {\left (c d^{5} x^{3} + 3 \, c^{2} d^{4} x^{2} + 3 \, c^{3} d^{3} x + c^{4} d^{2}\right )}}, -\frac {3 \, \sqrt {2} {\left (9 \, B c^{4} - A c^{3} d + {\left (9 \, B c d^{3} - A d^{4}\right )} x^{3} + 3 \, {\left (9 \, B c^{2} d^{2} - A c d^{3}\right )} x^{2} + 3 \, {\left (9 \, B c^{3} d - A c^{2} d^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) + 2 \, {\left (8 \, B c d^{2} x^{2} + 17 \, B c^{3} - A c^{2} d + {\left (29 \, B c^{2} d - 5 \, A c d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{8 \, {\left (c d^{5} x^{3} + 3 \, c^{2} d^{4} x^{2} + 3 \, c^{3} d^{3} x + c^{4} d^{2}\right )}}\right ] \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(9/2),x, algorithm="fricas" 
)
 

Output:

[-1/16*(3*sqrt(2)*(9*B*c^4 - A*c^3*d + (9*B*c*d^3 - A*d^4)*x^3 + 3*(9*B*c^ 
2*d^2 - A*c*d^3)*x^2 + 3*(9*B*c^3*d - A*c^2*d^2)*x)*sqrt(c)*log(-(d^2*x^2 
- 2*c*d*x + 2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(c) - 3*c^2)/ 
(d^2*x^2 + 2*c*d*x + c^2)) + 4*(8*B*c*d^2*x^2 + 17*B*c^3 - A*c^2*d + (29*B 
*c^2*d - 5*A*c*d^2)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c*d^5*x^3 + 3* 
c^2*d^4*x^2 + 3*c^3*d^3*x + c^4*d^2), -1/8*(3*sqrt(2)*(9*B*c^4 - A*c^3*d + 
 (9*B*c*d^3 - A*d^4)*x^3 + 3*(9*B*c^2*d^2 - A*c*d^3)*x^2 + 3*(9*B*c^3*d - 
A*c^2*d^2)*x)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + 
c)*sqrt(-c)/(c*d*x + c^2)) + 2*(8*B*c*d^2*x^2 + 17*B*c^3 - A*c^2*d + (29*B 
*c^2*d - 5*A*c*d^2)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c*d^5*x^3 + 3* 
c^2*d^4*x^2 + 3*c^3*d^3*x + c^4*d^2)]
 

Sympy [F]

\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (c + d x\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(3/2)/(d*x+c)**(9/2),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(3/2)*(A + B*x)/(c + d*x)**(9/2), x)
 

Maxima [F]

\[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (d x + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(9/2),x, algorithm="maxima" 
)
 

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*(B*x + A)/(d*x + c)^(9/2), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} {\left (9 \, B c - A d\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + 16 \, \sqrt {-d x + c} B - \frac {2 \, {\left (13 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c - 22 \, \sqrt {-d x + c} B c^{2} - 5 \, {\left (-d x + c\right )}^{\frac {3}{2}} A d + 6 \, \sqrt {-d x + c} A c d\right )}}{{\left (d x + c\right )}^{2}}}{8 \, d^{2}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(9/2),x, algorithm="giac")
 

Output:

-1/8*(3*sqrt(2)*(9*B*c - A*d)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c))/ 
sqrt(-c) + 16*sqrt(-d*x + c)*B - 2*(13*(-d*x + c)^(3/2)*B*c - 22*sqrt(-d*x 
 + c)*B*c^2 - 5*(-d*x + c)^(3/2)*A*d + 6*sqrt(-d*x + c)*A*c*d)/(d*x + c)^2 
)/d^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (c+d\,x\right )}^{9/2}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x))/(c + d*x)^(9/2),x)
 

Output:

int(((c^2 - d^2*x^2)^(3/2)*(A + B*x))/(c + d*x)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) \left (c^2-d^2 x^2\right )^{3/2}}{(c+d x)^{9/2}} \, dx=\frac {2 \sqrt {-d x +c}\, a \,c^{2} d +10 \sqrt {-d x +c}\, a c \,d^{2} x -34 \sqrt {-d x +c}\, b \,c^{3}-58 \sqrt {-d x +c}\, b \,c^{2} d x -16 \sqrt {-d x +c}\, b c \,d^{2} x^{2}+3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) a \,c^{2} d +6 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) a c \,d^{2} x +3 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) a \,d^{3} x^{2}-27 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) b \,c^{3}-54 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) b \,c^{2} d x -27 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )\right ) b c \,d^{2} x^{2}-\sqrt {c}\, \sqrt {2}\, a \,c^{2} d -2 \sqrt {c}\, \sqrt {2}\, a c \,d^{2} x -\sqrt {c}\, \sqrt {2}\, a \,d^{3} x^{2}+19 \sqrt {c}\, \sqrt {2}\, b \,c^{3}+38 \sqrt {c}\, \sqrt {2}\, b \,c^{2} d x +19 \sqrt {c}\, \sqrt {2}\, b c \,d^{2} x^{2}}{8 c \,d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(3/2)/(d*x+c)^(9/2),x)
 

Output:

(2*sqrt(c - d*x)*a*c**2*d + 10*sqrt(c - d*x)*a*c*d**2*x - 34*sqrt(c - d*x) 
*b*c**3 - 58*sqrt(c - d*x)*b*c**2*d*x - 16*sqrt(c - d*x)*b*c*d**2*x**2 + 3 
*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*c**2* 
d + 6*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a* 
c*d**2*x + 3*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2))) 
/2))*a*d**3*x**2 - 27*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)* 
sqrt(2)))/2))*b*c**3 - 54*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt 
(c)*sqrt(2)))/2))*b*c**2*d*x - 27*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d* 
x)/(sqrt(c)*sqrt(2)))/2))*b*c*d**2*x**2 - sqrt(c)*sqrt(2)*a*c**2*d - 2*sqr 
t(c)*sqrt(2)*a*c*d**2*x - sqrt(c)*sqrt(2)*a*d**3*x**2 + 19*sqrt(c)*sqrt(2) 
*b*c**3 + 38*sqrt(c)*sqrt(2)*b*c**2*d*x + 19*sqrt(c)*sqrt(2)*b*c*d**2*x**2 
)/(8*c*d**2*(c**2 + 2*c*d*x + d**2*x**2))