\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2) (e+f x^2)} \, dx\) [308]

Optimal result

Integrand size = 30, antiderivative size = 157 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d f}+\frac {(b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d (d e-c f)}-\frac {(b e-a f)^{3/2} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f (d e-c f)} \] Output:

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {(-b c+a d)^{3/2} \sqrt {e} f \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )+\sqrt {c} \left (-d (-b e+a f)^{3/2} \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )+b^{3/2} \sqrt {e} (d e-c f) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{\sqrt {c} d \sqrt {e} f (-d e+c f)} \] Input:

Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)*(e + f*x^2)),x]
 

Output:

((-(b*c) + a*d)^(3/2)*Sqrt[e]*f*ArcTan[(-(d*x*Sqrt[a + b*x^2]) + Sqrt[b]*( 
c + d*x^2))/(Sqrt[c]*Sqrt[-(b*c) + a*d])] + Sqrt[c]*(-(d*(-(b*e) + a*f)^(3 
/2)*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sqrt[-( 
b*e) + a*f])]) + b^(3/2)*Sqrt[e]*(d*e - c*f)*Log[-(Sqrt[b]*x) + Sqrt[a + b 
*x^2]]))/(Sqrt[c]*d*Sqrt[e]*f*(-(d*e) + c*f))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {420, 301, 224, 219, 291, 221, 422, 301, 224, 219, 291, 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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{d}\)

\(\Big \downarrow \) 422

\(\displaystyle \frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{d}\)

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d}-\frac {(b c-a d) \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{d}\)

Input:

Int[(a + b*x^2)^(3/2)/((c + d*x^2)*(e + f*x^2)),x]
 

Output:

(b*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - (Sqrt[b*e - a*f]*Ar 
cTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*f)))/d - (( 
b*c - a*d)*((d*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - (Sqrt[b 
*c - a*d]*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c] 
*d)))/(d*e - c*f) - (f*((Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/f - 
 (Sqrt[b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/ 
(Sqrt[e]*f)))/(d*e - c*f)))/d
 

Defintions of rubi rules used

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 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 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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 420

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

rule 422

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

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {\left (a d -b c \right )^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{d \left (c f -d e \right ) \sqrt {\left (a d -b c \right ) c}}+\frac {b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{f d}-\frac {\left (a f -b e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{f \left (c f -d e \right ) \sqrt {\left (a f -b e \right ) e}}\)	\(156\)
default	\(\text {Expression too large to display}\)	\(2624\)

Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right ) \left (e + f x^{2}\right )}\, dx \] Input:

integrate((b*x**2+a)**(3/2)/(d*x**2+c)/(f*x**2+e),x)
 

Output:

Integral((a + b*x**2)**(3/2)/((c + d*x**2)*(e + f*x**2)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)*(f*x^2 + e)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),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. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{\left (d\,x^2+c\right )\,\left (f\,x^2+e\right )} \,d x \] Input:

int((a + b*x^2)^(3/2)/((c + d*x^2)*(e + f*x^2)),x)
 

Output:

int((a + b*x^2)^(3/2)/((c + d*x^2)*(e + f*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.29 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) a d e f -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) b c e f +\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) a d e f -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) b c e f -\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) a c d f +\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b c d e -\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) a c d f +\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) b c d e +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} e f -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b c d \,e^{2}}{c d e f \left (c f -d e \right )} \] Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)/(f*x^2+e),x)
 

Output:

(sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) 
- sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d*e*f - sqrt(c)*sqrt(a*d - b*c)* 
atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) - sqrt(d)*sqrt(b)*x)/(sqr 
t(c)*sqrt(b)))*b*c*e*f + sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + s 
qrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*a*d*e*f - 
sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + 
 sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*b*c*e*f - sqrt(e)*sqrt(a*f - b*e)*a 
tan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt 
(e)*sqrt(b)))*a*c*d*f + sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sq 
rt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*b*c*d*e - s 
qrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + 
sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*c*d*f + sqrt(e)*sqrt(a*f - b*e)*at 
an((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt( 
e)*sqrt(b)))*b*c*d*e + sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a)) 
*b*c**2*e*f - sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b*c*d*e* 
*2)/(c*d*e*f*(c*f - d*e))