\(\int \frac {(c+d x^2)^{5/2}}{x^2 (a+b x^2)} \, dx\) [946]

Optimal result

Integrand size = 24, antiderivative size = 145 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {d (2 b c+a d) x \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{a x}-\frac {(b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} b^2}+\frac {d^{3/2} (5 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^2} \] Output:

1/2*d*(a*d+2*b*c)*x*(d*x^2+c)^(1/2)/a/b-c*(d*x^2+c)^(3/2)/a/x-(-a*d+b*c)^( 
5/2)*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2))/a^(3/2)/b^2+1/2*d^ 
(3/2)*(-2*a*d+5*b*c)*arctanh(d^(1/2)*x/(d*x^2+c)^(1/2))/b^2
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {\frac {b \sqrt {c+d x^2} \left (-2 b c^2+a d^2 x^2\right )}{a x}+\frac {2 (b c-a d)^{5/2} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{3/2}}+d^{3/2} (-5 b c+2 a d) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{2 b^2} \] Input:

Integrate[(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)),x]
 

Output:

((b*Sqrt[c + d*x^2]*(-2*b*c^2 + a*d^2*x^2))/(a*x) + (2*(b*c - a*d)^(5/2)*A 
rcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - 
a*d])])/a^(3/2) + d^(3/2)*(-5*b*c + 2*a*d)*Log[-(Sqrt[d]*x) + Sqrt[c + d*x 
^2]])/(2*b^2)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {376, 25, 403, 398, 224, 219, 291, 218}

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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 376

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[(c + d*x^2)^(5/2)/(x^2*(a + b*x^2)),x]
 

Output:

-((c*(c + d*x^2)^(3/2))/(a*x)) - (-1/2*(d*(2*b*c + a*d)*x*Sqrt[c + d*x^2]) 
/b + ((2*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d* 
x^2])])/(Sqrt[a]*b) - (a*d^(3/2)*(5*b*c - 2*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[ 
c + d*x^2]])/b)/(2*b))/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 218

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[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 291

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

rule 376

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 
)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^ 
2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* 
d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre 
eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & 
& IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 403

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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {\left (a d -b c \right )^{3} \sqrt {d}\, x \,\operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )+\frac {\left (x \left (-2 a^{2} d^{3}+5 a c \,d^{2} b \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )+\sqrt {d}\, \left (a \,d^{2} x^{2}-2 b \,c^{2}\right ) b \sqrt {x^{2} d +c}\right ) \sqrt {\left (a d -b c \right ) a}}{2}}{\sqrt {\left (a d -b c \right ) a}\, \sqrt {d}\, a x \,b^{2}}\)	\(149\)
risch	\(\frac {\sqrt {x^{2} d +c}\, \left (a \,d^{2} x^{2}-2 b \,c^{2}\right )}{2 b a x}-\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {a \,d^{\frac {3}{2}} \left (2 a d -5 b c \right ) \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b}}{2 a b}\)	\(460\)
default	\(\text {Expression too large to display}\)	\(2177\)

Input:

int((d*x^2+c)^(5/2)/x^2/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

((a*d-b*c)^3*d^(1/2)*x*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2))+1/ 
2*(x*(-2*a^2*d^3+5*a*b*c*d^2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))+d^(1/2)*( 
a*d^2*x^2-2*b*c^2)*b*(d*x^2+c)^(1/2))*((a*d-b*c)*a)^(1/2))/((a*d-b*c)*a)^( 
1/2)/d^(1/2)/a/x/b^2
 

Fricas [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 887, normalized size of antiderivative = 6.12 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((d*x^2+c)^(5/2)/x^2/(b*x^2+a),x, algorithm="fricas")
 

Output:

[-1/4*((5*a*b*c*d - 2*a^2*d^2)*sqrt(d)*x*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)* 
sqrt(d)*x - c) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt(-(b*c - a*d)/a)*lo 
g(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2* 
c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c 
- a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(a*b*d^2*x^2 - 2*b^2*c^2)*sqrt 
(d*x^2 + c))/(a*b^2*x), -1/4*(2*(5*a*b*c*d - 2*a^2*d^2)*sqrt(-d)*x*arctan( 
sqrt(-d)*x/sqrt(d*x^2 + c)) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt(-(b*c 
 - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b 
*c^2 - 4*a^2*c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c 
)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(a*b*d^2*x^2 - 2* 
b^2*c^2)*sqrt(d*x^2 + c))/(a*b^2*x), -1/4*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^ 
2)*x*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + 
 c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 - a*c*d)*x)) + (5*a* 
b*c*d - 2*a^2*d^2)*sqrt(d)*x*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - 
c) - 2*(a*b*d^2*x^2 - 2*b^2*c^2)*sqrt(d*x^2 + c))/(a*b^2*x), -1/2*((5*a*b* 
c*d - 2*a^2*d^2)*sqrt(-d)*x*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (b^2*c^2 
- 2*a*b*c*d + a^2*d^2)*x*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 
 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 
- a*c*d)*x)) - (a*b*d^2*x^2 - 2*b^2*c^2)*sqrt(d*x^2 + c))/(a*b^2*x)]
 

Sympy [F]

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

integrate((d*x**2+c)**(5/2)/x**2/(b*x**2+a),x)
 

Output:

Integral((c + d*x**2)**(5/2)/(x**2*(a + b*x**2)), x)
 

Maxima [F]

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

integrate((d*x^2+c)^(5/2)/x^2/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*x^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x^2+c)^(5/2)/x^2/(b*x^2+a),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

int((c + d*x^2)^(5/2)/(x^2*(a + b*x^2)),x)
                                                                                    
                                                                                    
 

Output:

int((c + d*x^2)^(5/2)/(x^2*(a + b*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 696, normalized size of antiderivative = 4.80 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )} \, dx=\frac {4 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a^{2} d^{2} x -8 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a b c d x +4 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b^{2} c^{2} x +4 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a^{2} d^{2} x -8 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a b c d x +4 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b^{2} c^{2} x -4 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) a^{2} d^{2} x +8 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) a b c d x -4 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b^{2} c^{2} x +4 \sqrt {d \,x^{2}+c}\, a^{2} b \,d^{2} x^{2}-8 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{2}-8 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{3} d^{2} x +20 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} b c d x -\sqrt {d}\, a^{2} b c d x -8 \sqrt {d}\, a \,b^{2} c^{2} x}{8 a^{2} b^{2} x} \] Input:

int((d*x^2+c)^(5/2)/x^2/(b*x^2+a),x)
 

Output:

(4*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 
 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*d**2*x 
- 8*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
- 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*c*d*x + 
 4*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 
 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b**2*c**2*x 
+ 4*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2 
*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*d**2*x - 
8*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a 
*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*c*d*x + 4*sq 
rt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + 
 b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b**2*c**2*x - 4*sqrt 
(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt 
(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*a**2*d**2*x + 8*sqrt(a)*sqrt(a*d - 
b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b* 
x + 2*a*d + 2*b*d*x**2)*a*b*c*d*x - 4*sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d 
)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d 
*x**2)*b**2*c**2*x + 4*sqrt(c + d*x**2)*a**2*b*d**2*x**2 - 8*sqrt(c + d*x* 
*2)*a*b**2*c**2 - 8*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt(c))*a* 
*3*d**2*x + 20*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt(c))*a**2...