\(\int \frac {x^4 (a+b x^2)}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx\) [22]

Optimal result

Integrand size = 31, antiderivative size = 164 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {c^2 \left (5 b c^2+6 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^6}+\frac {\left (5 b c^2+6 a d^2\right ) x^3 \sqrt {-c+d x} \sqrt {c+d x}}{24 d^4}+\frac {b x^5 \sqrt {-c+d x} \sqrt {c+d x}}{6 d^2}+\frac {c^4 \left (5 b c^2+6 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^7} \] Output:

1/16*c^2*(6*a*d^2+5*b*c^2)*x*(d*x-c)^(1/2)*(d*x+c)^(1/2)/d^6+1/24*(6*a*d^2 
+5*b*c^2)*x^3*(d*x-c)^(1/2)*(d*x+c)^(1/2)/d^4+1/6*b*x^5*(d*x-c)^(1/2)*(d*x 
+c)^(1/2)/d^2+1/8*c^4*(6*a*d^2+5*b*c^2)*arctanh((d*x-c)^(1/2)/(d*x+c)^(1/2 
))/d^7
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {d x \sqrt {-c+d x} \sqrt {c+d x} \left (6 a d^2 \left (3 c^2+2 d^2 x^2\right )+b \left (15 c^4+10 c^2 d^2 x^2+8 d^4 x^4\right )\right )+6 c^4 \left (5 b c^2+6 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{48 d^7} \] Input:

Integrate[(x^4*(a + b*x^2))/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
 

Output:

(d*x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(6*a*d^2*(3*c^2 + 2*d^2*x^2) + b*(15*c^4 
 + 10*c^2*d^2*x^2 + 8*d^4*x^4)) + 6*c^4*(5*b*c^2 + 6*a*d^2)*ArcTanh[Sqrt[- 
c + d*x]/Sqrt[c + d*x]])/(48*d^7)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {960, 111, 27, 101, 27, 45, 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.

\(\displaystyle \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {d x-c} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 960

\(\displaystyle \frac {1}{6} \left (6 a+\frac {5 b c^2}{d^2}\right ) \int \frac {x^4}{\sqrt {d x-c} \sqrt {c+d x}}dx+\frac {b x^5 \sqrt {d x-c} \sqrt {c+d x}}{6 d^2}\)

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 45

\(\displaystyle \frac {1}{6} \left (6 a+\frac {5 b c^2}{d^2}\right ) \left (\frac {3 c^2 \left (\frac {c^2 \int \frac {1}{d-\frac {d (d x-c)}{c+d x}}d\frac {\sqrt {d x-c}}{\sqrt {c+d x}}}{d^2}+\frac {x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\right )}{4 d^2}+\frac {x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\right )+\frac {b x^5 \sqrt {d x-c} \sqrt {c+d x}}{6 d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{6} \left (6 a+\frac {5 b c^2}{d^2}\right ) \left (\frac {3 c^2 \left (\frac {c^2 \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}+\frac {x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\right )}{4 d^2}+\frac {x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\right )+\frac {b x^5 \sqrt {d x-c} \sqrt {c+d x}}{6 d^2}\)

Input:

Int[(x^4*(a + b*x^2))/(Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
 

Output:

(b*x^5*Sqrt[-c + d*x]*Sqrt[c + d*x])/(6*d^2) + ((6*a + (5*b*c^2)/d^2)*((x^ 
3*Sqrt[-c + d*x]*Sqrt[c + d*x])/(4*d^2) + (3*c^2*((x*Sqrt[-c + d*x]*Sqrt[c 
 + d*x])/(2*d^2) + (c^2*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/d^3))/(4*d^ 
2)))/6
 

Defintions of rubi rules used

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 45

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] &&  !GtQ[c, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

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 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {x \left (8 b \,x^{4} d^{4}+12 a \,d^{4} x^{2}+10 b \,c^{2} d^{2} x^{2}+18 a \,c^{2} d^{2}+15 b \,c^{4}\right ) \left (-d x +c \right ) \sqrt {d x +c}}{48 d^{6} \sqrt {d x -c}}+\frac {c^{4} \left (6 a \,d^{2}+5 b \,c^{2}\right ) \ln \left (\frac {d^{2} x}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{16 d^{6} \sqrt {d^{2}}\, \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(160\)
default	\(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (8 \,\operatorname {csgn}\left (d \right ) b \,d^{5} x^{5} \sqrt {d^{2} x^{2}-c^{2}}+12 \,\operatorname {csgn}\left (d \right ) a \,d^{5} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+10 \,\operatorname {csgn}\left (d \right ) b \,c^{2} d^{3} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+18 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {d^{2} x^{2}-c^{2}}\, a \,c^{2} x +15 \,\operatorname {csgn}\left (d \right ) d \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{4} x +18 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) a \,c^{4} d^{2}+15 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{6}\right ) \operatorname {csgn}\left (d \right )}{48 d^{7} \sqrt {d^{2} x^{2}-c^{2}}}\)	\(240\)

Input:

int(x^4*(b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/48*x*(8*b*d^4*x^4+12*a*d^4*x^2+10*b*c^2*d^2*x^2+18*a*c^2*d^2+15*b*c^4)* 
(-d*x+c)*(d*x+c)^(1/2)/d^6/(d*x-c)^(1/2)+1/16*c^4*(6*a*d^2+5*b*c^2)/d^6*ln 
(d^2*x/(d^2)^(1/2)+(d^2*x^2-c^2)^(1/2))/(d^2)^(1/2)*((d*x-c)*(d*x+c))^(1/2 
)/(d*x-c)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.70 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {{\left (8 \, b d^{5} x^{5} + 2 \, {\left (5 \, b c^{2} d^{3} + 6 \, a d^{5}\right )} x^{3} + 3 \, {\left (5 \, b c^{4} d + 6 \, a c^{2} d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} - 3 \, {\left (5 \, b c^{6} + 6 \, a c^{4} d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{48 \, d^{7}} \] Input:

integrate(x^4*(b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

1/48*((8*b*d^5*x^5 + 2*(5*b*c^2*d^3 + 6*a*d^5)*x^3 + 3*(5*b*c^4*d + 6*a*c^ 
2*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) - 3*(5*b*c^6 + 6*a*c^4*d^2)*log(-d*x 
 + sqrt(d*x + c)*sqrt(d*x - c)))/d^7
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

integrate(x**4*(b*x**2+a)/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d^{2} x^{2} - c^{2}} b x^{5}}{6 \, d^{2}} + \frac {5 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{2} x^{3}}{24 \, d^{4}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a x^{3}}{4 \, d^{2}} + \frac {5 \, b c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{16 \, d^{7}} + \frac {3 \, a c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{8 \, d^{5}} + \frac {5 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{4} x}{16 \, d^{6}} + \frac {3 \, \sqrt {d^{2} x^{2} - c^{2}} a c^{2} x}{8 \, d^{4}} \] Input:

integrate(x^4*(b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

1/6*sqrt(d^2*x^2 - c^2)*b*x^5/d^2 + 5/24*sqrt(d^2*x^2 - c^2)*b*c^2*x^3/d^4 
 + 1/4*sqrt(d^2*x^2 - c^2)*a*x^3/d^2 + 5/16*b*c^6*log(2*d^2*x + 2*sqrt(d^2 
*x^2 - c^2)*d)/d^7 + 3/8*a*c^4*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^5 
+ 5/16*sqrt(d^2*x^2 - c^2)*b*c^4*x/d^6 + 3/8*sqrt(d^2*x^2 - c^2)*a*c^2*x/d 
^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.24 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {{\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{6}} - \frac {5 \, b c}{d^{6}}\right )} + \frac {3 \, {\left (15 \, b c^{2} d^{36} + 2 \, a d^{38}\right )}}{d^{42}}\right )} - \frac {55 \, b c^{3} d^{36} + 18 \, a c d^{38}}{d^{42}}\right )} {\left (d x + c\right )} + \frac {85 \, b c^{4} d^{36} + 54 \, a c^{2} d^{38}}{d^{42}}\right )} {\left (d x + c\right )} - \frac {3 \, {\left (11 \, b c^{5} d^{36} + 10 \, a c^{3} d^{38}\right )}}{d^{42}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {6 \, {\left (5 \, b c^{6} + 6 \, a c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{6}}}{48 \, d} \] Input:

integrate(x^4*(b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

1/48*(((2*((d*x + c)*(4*(d*x + c)*((d*x + c)*b/d^6 - 5*b*c/d^6) + 3*(15*b* 
c^2*d^36 + 2*a*d^38)/d^42) - (55*b*c^3*d^36 + 18*a*c*d^38)/d^42)*(d*x + c) 
 + (85*b*c^4*d^36 + 54*a*c^2*d^38)/d^42)*(d*x + c) - 3*(11*b*c^5*d^36 + 10 
*a*c^3*d^38)/d^42)*sqrt(d*x + c)*sqrt(d*x - c) - 6*(5*b*c^6 + 6*a*c^4*d^2) 
*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^6)/d
 

Mupad [B] (verification not implemented)

Time = 48.11 (sec) , antiderivative size = 1682, normalized size of antiderivative = 10.26 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\text {Too large to display} \] Input:

int((x^4*(a + b*x^2))/((c + d*x)^(1/2)*(d*x - c)^(1/2)),x)
 

Output:

((5*b*c^6*((c + d*x)^(1/2) - c^(1/2)))/(4*((-c)^(1/2) - (d*x - c)^(1/2))) 
- (175*b*c^6*((c + d*x)^(1/2) - c^(1/2))^3)/(12*((-c)^(1/2) - (d*x - c)^(1 
/2))^3) + (311*b*c^6*((c + d*x)^(1/2) - c^(1/2))^5)/(4*((-c)^(1/2) - (d*x 
- c)^(1/2))^5) + (8361*b*c^6*((c + d*x)^(1/2) - c^(1/2))^7)/(4*((-c)^(1/2) 
 - (d*x - c)^(1/2))^7) + (42259*b*c^6*((c + d*x)^(1/2) - c^(1/2))^9)/(6*(( 
-c)^(1/2) - (d*x - c)^(1/2))^9) + (25295*b*c^6*((c + d*x)^(1/2) - c^(1/2)) 
^11)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^11) + (25295*b*c^6*((c + d*x)^(1/2) 
 - c^(1/2))^13)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^13) + (42259*b*c^6*((c + 
 d*x)^(1/2) - c^(1/2))^15)/(6*((-c)^(1/2) - (d*x - c)^(1/2))^15) + (8361*b 
*c^6*((c + d*x)^(1/2) - c^(1/2))^17)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^17) 
 + (311*b*c^6*((c + d*x)^(1/2) - c^(1/2))^19)/(4*((-c)^(1/2) - (d*x - c)^( 
1/2))^19) - (175*b*c^6*((c + d*x)^(1/2) - c^(1/2))^21)/(12*((-c)^(1/2) - ( 
d*x - c)^(1/2))^21) + (5*b*c^6*((c + d*x)^(1/2) - c^(1/2))^23)/(4*((-c)^(1 
/2) - (d*x - c)^(1/2))^23))/(d^7 - (12*d^7*((c + d*x)^(1/2) - c^(1/2))^2)/ 
((-c)^(1/2) - (d*x - c)^(1/2))^2 + (66*d^7*((c + d*x)^(1/2) - c^(1/2))^4)/ 
((-c)^(1/2) - (d*x - c)^(1/2))^4 - (220*d^7*((c + d*x)^(1/2) - c^(1/2))^6) 
/((-c)^(1/2) - (d*x - c)^(1/2))^6 + (495*d^7*((c + d*x)^(1/2) - c^(1/2))^8 
)/((-c)^(1/2) - (d*x - c)^(1/2))^8 - (792*d^7*((c + d*x)^(1/2) - c^(1/2))^ 
10)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + (924*d^7*((c + d*x)^(1/2) - c^(1/2 
))^12)/((-c)^(1/2) - (d*x - c)^(1/2))^12 - (792*d^7*((c + d*x)^(1/2) - ...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15 \[ \int \frac {x^4 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {18 \sqrt {d x +c}\, \sqrt {d x -c}\, a \,c^{2} d^{3} x +12 \sqrt {d x +c}\, \sqrt {d x -c}\, a \,d^{5} x^{3}+15 \sqrt {d x +c}\, \sqrt {d x -c}\, b \,c^{4} d x +10 \sqrt {d x +c}\, \sqrt {d x -c}\, b \,c^{2} d^{3} x^{3}+8 \sqrt {d x +c}\, \sqrt {d x -c}\, b \,d^{5} x^{5}+36 \,\mathrm {log}\left (\frac {\sqrt {d x -c}+\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right ) a \,c^{4} d^{2}+30 \,\mathrm {log}\left (\frac {\sqrt {d x -c}+\sqrt {d x +c}}{\sqrt {c}\, \sqrt {2}}\right ) b \,c^{6}}{48 d^{7}} \] Input:

int(x^4*(b*x^2+a)/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)
 

Output:

(18*sqrt(c + d*x)*sqrt( - c + d*x)*a*c**2*d**3*x + 12*sqrt(c + d*x)*sqrt( 
- c + d*x)*a*d**5*x**3 + 15*sqrt(c + d*x)*sqrt( - c + d*x)*b*c**4*d*x + 10 
*sqrt(c + d*x)*sqrt( - c + d*x)*b*c**2*d**3*x**3 + 8*sqrt(c + d*x)*sqrt( - 
 c + d*x)*b*d**5*x**5 + 36*log((sqrt( - c + d*x) + sqrt(c + d*x))/(sqrt(c) 
*sqrt(2)))*a*c**4*d**2 + 30*log((sqrt( - c + d*x) + sqrt(c + d*x))/(sqrt(c 
)*sqrt(2)))*b*c**6)/(48*d**7)