\(\int \frac {(c+d x^2)^{5/2}}{x^4 (a+b x^2)} \, dx\) [947]

Optimal result

Integrand size = 24, antiderivative size = 130 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\frac {c (b c-2 a d) \sqrt {c+d x^2}}{a^2 x}-\frac {c \left (c+d x^2\right )^{3/2}}{3 a x^3}+\frac {(b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} b}+\frac {d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=-\frac {c \sqrt {c+d x^2} \left (-3 b c x^2+a \left (c+7 d x^2\right )\right )}{3 a^2 x^3}-\frac {(b c-a d)^{5/2} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2} b}-\frac {d^{5/2} \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{b} \] Input:

Integrate[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x]
 

Output:

-1/3*(c*Sqrt[c + d*x^2]*(-3*b*c*x^2 + a*(c + 7*d*x^2)))/(a^2*x^3) - ((b*c 
- a*d)^(5/2)*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[ 
a]*Sqrt[b*c - a*d])])/(a^(5/2)*b) - (d^(5/2)*Log[-(Sqrt[d]*x) + Sqrt[c + d 
*x^2]])/b
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {376, 27, 442, 25, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 376

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 442

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[(c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

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 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 442

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

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {-x^{3} \left (a d -b c \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )+\left (d^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right ) a^{2} x^{3}-\frac {\left (\left (7 x^{2} d +c \right ) a -3 x^{2} b c \right ) c \sqrt {x^{2} d +c}\, b}{3}\right ) \sqrt {\left (a d -b c \right ) a}}{\sqrt {\left (a d -b c \right ) a}\, a^{2} x^{3} b}\)	\(138\)
risch	\(-\frac {\sqrt {x^{2} d +c}\, c \left (7 a d \,x^{2}-3 x^{2} b c +a c \right )}{3 a^{2} x^{3}}+\frac {\frac {a^{2} d^{\frac {5}{2}} \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{a^{2}}\)	\(451\)
default	\(\text {Expression too large to display}\)	\(2302\)

Input:

int((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

1/((a*d-b*c)*a)^(1/2)*(-x^3*(a*d-b*c)^3*arctanh((d*x^2+c)^(1/2)/x*a/((a*d- 
b*c)*a)^(1/2))+(d^(5/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))*a^2*x^3-1/3*((7 
*d*x^2+c)*a-3*x^2*b*c)*c*(d*x^2+c)^(1/2)*b)*((a*d-b*c)*a)^(1/2))/a^2/x^3/b
 

Fricas [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 901, normalized size of antiderivative = 6.93 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x, algorithm="fricas")
 

Output:

[1/12*(6*a^2*d^(5/2)*x^3*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 
 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^3*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)) - 4*(a*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sq 
rt(d*x^2 + c))/(a^2*b*x^3), -1/12*(12*a^2*sqrt(-d)*d^2*x^3*arctan(sqrt(-d) 
*x/sqrt(d*x^2 + c)) - 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^3*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)*sq 
rt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(a*b*c^2 - (3*b^2*c^2 
 - 7*a*b*c*d)*x^2)*sqrt(d*x^2 + c))/(a^2*b*x^3), 1/6*(3*a^2*d^(5/2)*x^3*lo 
g(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 3*(b^2*c^2 - 2*a*b*c*d + a 
^2*d^2)*x^3*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)) 
- 2*(a*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sqrt(d*x^2 + c))/(a^2*b*x^3), 
-1/6*(6*a^2*sqrt(-d)*d^2*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - 3*(b^2*c 
^2 - 2*a*b*c*d + a^2*d^2)*x^3*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)) + 2*(a*b*c^2 - (3*b^2*c^2 - 7*a*b*c*d)*x^2)*sqrt(d*x^2 + 
 c))/(a^2*b*x^3)]
 

Sympy [F]

\[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{x^{4} \left (a + b x^{2}\right )}\, dx \] Input:

integrate((d*x**2+c)**(5/2)/x**4/(b*x**2+a),x)
 

Output:

Integral((c + d*x**2)**(5/2)/(x**4*(a + b*x**2)), x)
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} x^{4}} \,d x } \] Input:

integrate((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^(5/2)/((b*x^2 + a)*x^4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x^2+c)^(5/2)/x^4/(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. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{x^4\,\left (b\,x^2+a\right )} \,d x \] Input:

int((c + d*x^2)^(5/2)/(x^4*(a + b*x^2)),x)
 

Output:

int((c + d*x^2)^(5/2)/(x^4*(a + b*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 709, normalized size of antiderivative = 5.45 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )} \, dx=\frac {-3 \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^{3}+6 \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^{3}-3 \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^{3}-3 \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^{3}+6 \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^{3}-3 \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^{3}+3 \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^{3}-6 \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^{3}+3 \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^{3}-2 \sqrt {d \,x^{2}+c}\, a^{2} b \,c^{2}-14 \sqrt {d \,x^{2}+c}\, a^{2} b c d \,x^{2}+6 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{2} x^{2}+6 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{3} d^{2} x^{3}+2 \sqrt {d}\, a^{2} b c d \,x^{3}-2 \sqrt {d}\, a \,b^{2} c^{2} x^{3}}{6 a^{3} b \,x^{3}} \] Input:

int((d*x^2+c)^(5/2)/x^4/(b*x^2+a),x)
 

Output:

( - 3*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**3 + 6*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**3 - 3*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**3 - 3*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**3 + 6*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**3 - 3*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**3 + 3*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**3 
 - 6*sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqr 
t(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*a*b*c*d*x**3 + 3*sqrt(a)*s 
qrt(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**3 - 2*sqrt(c + d*x**2)*a**2 
*b*c**2 - 14*sqrt(c + d*x**2)*a**2*b*c*d*x**2 + 6*sqrt(c + d*x**2)*a*b**2* 
c**2*x**2 + 6*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt(c))*a**3*...