\(\int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx\) [99]

Optimal result

Integrand size = 31, antiderivative size = 195 \[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\frac {(B c-A d) \sqrt {c^2-d^2 x^2}}{6 c d^2 (c+d x)^{7/2}}-\frac {(7 B c+5 A d) \sqrt {c^2-d^2 x^2}}{48 c^2 d^2 (c+d x)^{5/2}}-\frac {(7 B c+5 A d) \sqrt {c^2-d^2 x^2}}{64 c^3 d^2 (c+d x)^{3/2}}-\frac {(7 B c+5 A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{64 \sqrt {2} c^{7/2} d^2} \] Output:

1/6*(-A*d+B*c)*(-d^2*x^2+c^2)^(1/2)/c/d^2/(d*x+c)^(7/2)-1/48*(5*A*d+7*B*c) 
*(-d^2*x^2+c^2)^(1/2)/c^2/d^2/(d*x+c)^(5/2)-1/64*(5*A*d+7*B*c)*(-d^2*x^2+c 
^2)^(1/2)/c^3/d^2/(d*x+c)^(3/2)-1/128*(5*A*d+7*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^(7/2)/d^2
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (A d \left (67 c^2+50 c d x+15 d^2 x^2\right )+B c \left (17 c^2+70 c d x+21 d^2 x^2\right )\right )}{(c+d x)^{7/2}}-3 \sqrt {2} (7 B c+5 A d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{384 c^{7/2} d^2} \] Input:

Integrate[(A + B*x)/((c + d*x)^(7/2)*Sqrt[c^2 - d^2*x^2]),x]
 

Output:

((-2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(A*d*(67*c^2 + 50*c*d*x + 15*d^2*x^2) + B 
*c*(17*c^2 + 70*c*d*x + 21*d^2*x^2)))/(c + d*x)^(7/2) - 3*Sqrt[2]*(7*B*c + 
 5*A*d)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d^2*x^2]])/(384 
*c^(7/2)*d^2)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {671, 470, 470, 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.

\(\displaystyle \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {(5 A d+7 B c) \int \frac {1}{(c+d x)^{5/2} \sqrt {c^2-d^2 x^2}}dx}{12 c d}+\frac {\sqrt {c^2-d^2 x^2} (B c-A d)}{6 c d^2 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {(5 A d+7 B c) \left (\frac {3 \int \frac {1}{(c+d x)^{3/2} \sqrt {c^2-d^2 x^2}}dx}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c d}+\frac {\sqrt {c^2-d^2 x^2} (B c-A d)}{6 c d^2 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {(5 A d+7 B c) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c d}+\frac {\sqrt {c^2-d^2 x^2} (B c-A d)}{6 c d^2 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 471

\(\displaystyle \frac {(5 A d+7 B c) \left (\frac {3 \left (\frac {d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}}{2 c}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c d}+\frac {\sqrt {c^2-d^2 x^2} (B c-A d)}{6 c d^2 (c+d x)^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(5 A d+7 B c) \left (\frac {3 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{2 \sqrt {2} c^{3/2} d}-\frac {\sqrt {c^2-d^2 x^2}}{2 c d (c+d x)^{3/2}}\right )}{8 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c d (c+d x)^{5/2}}\right )}{12 c d}+\frac {\sqrt {c^2-d^2 x^2} (B c-A d)}{6 c d^2 (c+d x)^{7/2}}\)

Input:

Int[(A + B*x)/((c + d*x)^(7/2)*Sqrt[c^2 - d^2*x^2]),x]
 

Output:

((B*c - A*d)*Sqrt[c^2 - d^2*x^2])/(6*c*d^2*(c + d*x)^(7/2)) + ((7*B*c + 5* 
A*d)*(-1/4*Sqrt[c^2 - d^2*x^2]/(c*d*(c + d*x)^(5/2)) + (3*(-1/2*Sqrt[c^2 - 
 d^2*x^2]/(c*d*(c + d*x)^(3/2)) - ArcTanh[Sqrt[c^2 - d^2*x^2]/(Sqrt[2]*Sqr 
t[c]*Sqrt[c + d*x])]/(2*Sqrt[2]*c^(3/2)*d)))/(8*c)))/(12*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 470

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

Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(163)=326\).

Time = 0.36 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.92

method	result	size
default	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (15 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{4} x^{3}+21 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{3} x^{3}+45 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{3} x^{2}+63 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{2} x^{2}+45 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{2} x +30 A \,d^{3} x^{2} \sqrt {-d x +c}\, \sqrt {c}+63 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d x +42 B \,c^{\frac {3}{2}} d^{2} x^{2} \sqrt {-d x +c}+15 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d +100 A \,c^{\frac {3}{2}} d^{2} x \sqrt {-d x +c}+21 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}+140 B \,c^{\frac {5}{2}} d x \sqrt {-d x +c}+134 A \sqrt {-d x +c}\, c^{\frac {5}{2}} d +34 B \sqrt {-d x +c}\, c^{\frac {7}{2}}\right )}{384 c^{\frac {7}{2}} \left (d x +c \right )^{\frac {7}{2}} \sqrt {-d x +c}\, d^{2}}\)	\(374\)

Input:

int((B*x+A)/(d*x+c)^(7/2)/(-d^2*x^2+c^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/384*(-d^2*x^2+c^2)^(1/2)/c^(7/2)*(15*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/ 
2)*2^(1/2)/c^(1/2))*d^4*x^3+21*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2 
)/c^(1/2))*c*d^3*x^3+45*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/ 
2))*c*d^3*x^2+63*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2 
*d^2*x^2+45*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2*d^2* 
x+30*A*d^3*x^2*(-d*x+c)^(1/2)*c^(1/2)+63*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1 
/2)*2^(1/2)/c^(1/2))*c^3*d*x+42*B*c^(3/2)*d^2*x^2*(-d*x+c)^(1/2)+15*A*2^(1 
/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^3*d+100*A*c^(3/2)*d^2*x* 
(-d*x+c)^(1/2)+21*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^ 
4+140*B*c^(5/2)*d*x*(-d*x+c)^(1/2)+134*A*(-d*x+c)^(1/2)*c^(5/2)*d+34*B*(-d 
*x+c)^(1/2)*c^(7/2))/(d*x+c)^(7/2)/(-d*x+c)^(1/2)/d^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 591, normalized size of antiderivative = 3.03 \[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (7 \, B c^{5} + 5 \, A c^{4} d + {\left (7 \, B c d^{4} + 5 \, A d^{5}\right )} x^{4} + 4 \, {\left (7 \, B c^{2} d^{3} + 5 \, A c d^{4}\right )} x^{3} + 6 \, {\left (7 \, B c^{3} d^{2} + 5 \, A c^{2} d^{3}\right )} x^{2} + 4 \, {\left (7 \, B c^{4} d + 5 \, A c^{3} d^{2}\right )} x\right )} \sqrt {c} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x + 2 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {c} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 4 \, {\left (17 \, B c^{4} + 67 \, A c^{3} d + 3 \, {\left (7 \, B c^{2} d^{2} + 5 \, A c d^{3}\right )} x^{2} + 10 \, {\left (7 \, B c^{3} d + 5 \, A c^{2} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{768 \, {\left (c^{4} d^{6} x^{4} + 4 \, c^{5} d^{5} x^{3} + 6 \, c^{6} d^{4} x^{2} + 4 \, c^{7} d^{3} x + c^{8} d^{2}\right )}}, \frac {3 \, \sqrt {2} {\left (7 \, B c^{5} + 5 \, A c^{4} d + {\left (7 \, B c d^{4} + 5 \, A d^{5}\right )} x^{4} + 4 \, {\left (7 \, B c^{2} d^{3} + 5 \, A c d^{4}\right )} x^{3} + 6 \, {\left (7 \, B c^{3} d^{2} + 5 \, A c^{2} d^{3}\right )} x^{2} + 4 \, {\left (7 \, B c^{4} d + 5 \, A c^{3} d^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) - 2 \, {\left (17 \, B c^{4} + 67 \, A c^{3} d + 3 \, {\left (7 \, B c^{2} d^{2} + 5 \, A c d^{3}\right )} x^{2} + 10 \, {\left (7 \, B c^{3} d + 5 \, A c^{2} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{384 \, {\left (c^{4} d^{6} x^{4} + 4 \, c^{5} d^{5} x^{3} + 6 \, c^{6} d^{4} x^{2} + 4 \, c^{7} d^{3} x + c^{8} d^{2}\right )}}\right ] \] Input:

integrate((B*x+A)/(d*x+c)^(7/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="fricas" 
)
 

Output:

[1/768*(3*sqrt(2)*(7*B*c^5 + 5*A*c^4*d + (7*B*c*d^4 + 5*A*d^5)*x^4 + 4*(7* 
B*c^2*d^3 + 5*A*c*d^4)*x^3 + 6*(7*B*c^3*d^2 + 5*A*c^2*d^3)*x^2 + 4*(7*B*c^ 
4*d + 5*A*c^3*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 
*(17*B*c^4 + 67*A*c^3*d + 3*(7*B*c^2*d^2 + 5*A*c*d^3)*x^2 + 10*(7*B*c^3*d 
+ 5*A*c^2*d^2)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^4*d^6*x^4 + 4*c^5 
*d^5*x^3 + 6*c^6*d^4*x^2 + 4*c^7*d^3*x + c^8*d^2), 1/384*(3*sqrt(2)*(7*B*c 
^5 + 5*A*c^4*d + (7*B*c*d^4 + 5*A*d^5)*x^4 + 4*(7*B*c^2*d^3 + 5*A*c*d^4)*x 
^3 + 6*(7*B*c^3*d^2 + 5*A*c^2*d^3)*x^2 + 4*(7*B*c^4*d + 5*A*c^3*d^2)*x)*sq 
rt(-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*(17*B*c^4 + 67*A*c^3*d + 3*(7*B*c^2*d^2 + 5*A*c*d^3)*x^2 + 
10*(7*B*c^3*d + 5*A*c^2*d^2)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^4*d 
^6*x^4 + 4*c^5*d^5*x^3 + 6*c^6*d^4*x^2 + 4*c^7*d^3*x + c^8*d^2)]
 

Sympy [F]

\[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {A + B x}{\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((B*x+A)/(d*x+c)**(7/2)/(-d**2*x**2+c**2)**(1/2),x)
 

Output:

Integral((A + B*x)/(sqrt(-(-c + d*x)*(c + d*x))*(c + d*x)**(7/2)), x)
 

Maxima [F]

\[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {B x + A}{\sqrt {-d^{2} x^{2} + c^{2}} {\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x+A)/(d*x+c)^(7/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((B*x + A)/(sqrt(-d^2*x^2 + c^2)*(d*x + c)^(7/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left (7 \, B c + 5 \, A d\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c^{3}} - \frac {2 \, {\left (21 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} B c - 112 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c^{2} + 108 \, \sqrt {-d x + c} B c^{3} + 15 \, {\left (d x - c\right )}^{2} \sqrt {-d x + c} A d - 80 \, {\left (-d x + c\right )}^{\frac {3}{2}} A c d + 132 \, \sqrt {-d x + c} A c^{2} d\right )}}{{\left (d x + c\right )}^{3} c^{3}}}{384 \, d^{2}} \] Input:

integrate((B*x+A)/(d*x+c)^(7/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="giac")
 

Output:

1/384*(3*sqrt(2)*(7*B*c + 5*A*d)*arctan(1/2*sqrt(2)*sqrt(-d*x + c)/sqrt(-c 
))/(sqrt(-c)*c^3) - 2*(21*(d*x - c)^2*sqrt(-d*x + c)*B*c - 112*(-d*x + c)^ 
(3/2)*B*c^2 + 108*sqrt(-d*x + c)*B*c^3 + 15*(d*x - c)^2*sqrt(-d*x + c)*A*d 
 - 80*(-d*x + c)^(3/2)*A*c*d + 132*sqrt(-d*x + c)*A*c^2*d)/((d*x + c)^3*c^ 
3))/d^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c^2-d^2\,x^2}\,{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:

int((A + B*x)/((c^2 - d^2*x^2)^(1/2)*(c + d*x)^(7/2)),x)
 

Output:

int((A + B*x)/((c^2 - d^2*x^2)^(1/2)*(c + d*x)^(7/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.94 \[ \int \frac {A+B x}{(c+d x)^{7/2} \sqrt {c^2-d^2 x^2}} \, dx=\frac {-268 \sqrt {-d x +c}\, a \,c^{3} d -200 \sqrt {-d x +c}\, a \,c^{2} d^{2} x -60 \sqrt {-d x +c}\, a c \,d^{3} x^{2}-68 \sqrt {-d x +c}\, b \,c^{4}-280 \sqrt {-d x +c}\, b \,c^{3} d x -84 \sqrt {-d x +c}\, b \,c^{2} d^{2} x^{2}+15 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a \,c^{3} d +45 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a \,c^{2} d^{2} x +45 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a c \,d^{3} x^{2}+15 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a \,d^{4} x^{3}+21 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b \,c^{4}+63 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b \,c^{3} d x +63 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d^{2} x^{2}+21 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b c \,d^{3} x^{3}-15 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a \,c^{3} d -45 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a \,c^{2} d^{2} x -45 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a c \,d^{3} x^{2}-15 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a \,d^{4} x^{3}-21 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b \,c^{4}-63 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b \,c^{3} d x -63 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d^{2} x^{2}-21 \sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b c \,d^{3} x^{3}}{768 c^{4} 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*x+c)^(7/2)/(-d^2*x^2+c^2)^(1/2),x)
 

Output:

( - 268*sqrt(c - d*x)*a*c**3*d - 200*sqrt(c - d*x)*a*c**2*d**2*x - 60*sqrt 
(c - d*x)*a*c*d**3*x**2 - 68*sqrt(c - d*x)*b*c**4 - 280*sqrt(c - d*x)*b*c* 
*3*d*x - 84*sqrt(c - d*x)*b*c**2*d**2*x**2 + 15*sqrt(c)*sqrt(2)*log(sqrt(c 
 - d*x) - sqrt(c)*sqrt(2))*a*c**3*d + 45*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) 
 - sqrt(c)*sqrt(2))*a*c**2*d**2*x + 45*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - 
 sqrt(c)*sqrt(2))*a*c*d**3*x**2 + 15*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - s 
qrt(c)*sqrt(2))*a*d**4*x**3 + 21*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt( 
c)*sqrt(2))*b*c**4 + 63*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2 
))*b*c**3*d*x + 63*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b* 
c**2*d**2*x**2 + 21*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b 
*c*d**3*x**3 - 15*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c 
**3*d - 45*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c**2*d** 
2*x - 45*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*c*d**3*x** 
2 - 15*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*a*d**4*x**3 - 
21*sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**4 - 63*sqrt(c 
)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**3*d*x - 63*sqrt(c)*sqr 
t(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c**2*d**2*x**2 - 21*sqrt(c)*sq 
rt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c*d**3*x**3)/(768*c**4*d**2*( 
c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))