\(\int \frac {A+B x^2}{x^{5/2} (b x^2+c x^4)} \, dx\) [126]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=-\frac {2 A}{7 b x^{7/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}+\frac {c^{3/4} (b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{11/4}}-\frac {c^{3/4} (b B-A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{11/4}}-\frac {c^{3/4} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} b^{11/4}} \] Output:

-2/7*A/b/x^(7/2)-2/3*(-A*c+B*b)/b^2/x^(3/2)+1/2*c^(3/4)*(-A*c+B*b)*arctan( 
1-2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(11/4)-1/2*c^(3/4)*(-A*c+B*b) 
*arctan(1+2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(11/4)-1/2*c^(3/4)*(- 
A*c+B*b)*arctanh(2^(1/2)*b^(1/4)*c^(1/4)*x^(1/2)/(b^(1/2)+c^(1/2)*x))*2^(1 
/2)/b^(11/4)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=-\frac {2 \left (3 A b+7 b B x^2-7 A c x^2\right )}{21 b^2 x^{7/2}}+\frac {c^{3/4} (b B-A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} b^{11/4}}-\frac {c^{3/4} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} b^{11/4}} \] Input:

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

Output:

(-2*(3*A*b + 7*b*B*x^2 - 7*A*c*x^2))/(21*b^2*x^(7/2)) + (c^(3/4)*(b*B - A* 
c)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])])/(Sqrt[ 
2]*b^(11/4)) - (c^(3/4)*(b*B - A*c)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[ 
x])/(Sqrt[b] + Sqrt[c]*x)])/(Sqrt[2]*b^(11/4))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {9, 359, 264, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 9

\(\displaystyle \int \frac {A+B x^2}{x^{9/2} \left (b+c x^2\right )}dx\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {(b B-A c) \int \frac {1}{x^{5/2} \left (c x^2+b\right )}dx}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {(b B-A c) \left (-\frac {c \int \frac {1}{\sqrt {x} \left (c x^2+b\right )}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2 A}{7 b x^{7/2}}\)

Input:

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

Output:

(-2*A)/(7*b*x^(7/2)) + ((b*B - A*c)*(-2/(3*b*x^(3/2)) - (2*c*((-(ArcTan[1 
- (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4))) + ArcTan[1 
 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[b 
]) + (-1/2*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(Sqr 
t[2]*b^(1/4)*c^(1/4)) + Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sq 
rt[c]*x]/(2*Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[b])))/b))/b
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

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 217

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

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {c \left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3}}-\frac {2 A}{7 b \,x^{\frac {7}{2}}}-\frac {2 \left (-A c +B b \right )}{3 b^{2} x^{\frac {3}{2}}}\)	\(141\)
default	\(\frac {c \left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3}}-\frac {2 A}{7 b \,x^{\frac {7}{2}}}-\frac {2 \left (-A c +B b \right )}{3 b^{2} x^{\frac {3}{2}}}\)	\(141\)
risch	\(-\frac {2 \left (-7 A c \,x^{2}+7 B b \,x^{2}+3 A b \right )}{21 b^{2} x^{\frac {7}{2}}}+\frac {c \left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3}}\)	\(143\)

Input:

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

Output:

1/4*c*(A*c-B*b)/b^3*(b/c)^(1/4)*2^(1/2)*(ln((x+(b/c)^(1/4)*x^(1/2)*2^(1/2) 
+(b/c)^(1/2))/(x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))+2*arctan(2^(1/2 
)/(b/c)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1))-2/7*A/b/ 
x^(7/2)-2/3*(-A*c+B*b)/b^2/x^(3/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.31 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=\frac {21 \, b^{2} x^{4} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} \log \left (b^{3} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} - {\left (B b c - A c^{2}\right )} \sqrt {x}\right ) + 21 i \, b^{2} x^{4} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} \log \left (i \, b^{3} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} - {\left (B b c - A c^{2}\right )} \sqrt {x}\right ) - 21 i \, b^{2} x^{4} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} - {\left (B b c - A c^{2}\right )} \sqrt {x}\right ) - 21 \, b^{2} x^{4} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-b^{3} \left (-\frac {B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac {1}{4}} - {\left (B b c - A c^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (7 \, {\left (B b - A c\right )} x^{2} + 3 \, A b\right )} \sqrt {x}}{42 \, b^{2} x^{4}} \] Input:

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

Output:

1/42*(21*b^2*x^4*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4* 
A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4)*log(b^3*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^ 
4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4) - (B*b*c - A* 
c^2)*sqrt(x)) + 21*I*b^2*x^4*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2* 
b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4)*log(I*b^3*(-(B^4*b^4*c^3 - 
4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4) 
 - (B*b*c - A*c^2)*sqrt(x)) - 21*I*b^2*x^4*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^ 
4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4)*log(-I*b^3*(- 
(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c 
^7)/b^11)^(1/4) - (B*b*c - A*c^2)*sqrt(x)) - 21*b^2*x^4*(-(B^4*b^4*c^3 - 4 
*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4)* 
log(-b^3*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b* 
c^6 + A^4*c^7)/b^11)^(1/4) - (B*b*c - A*c^2)*sqrt(x)) - 4*(7*(B*b - A*c)*x 
^2 + 3*A*b)*sqrt(x))/(b^2*x^4)
 

Sympy [A] (verification not implemented)

Time = 27.26 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.56 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{c} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b} & \text {for}\: c = 0 \\- \frac {2 A}{7 b x^{\frac {7}{2}}} + \frac {2 A c}{3 b^{2} x^{\frac {3}{2}}} - \frac {A c^{2} \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{3}} + \frac {A c^{2} \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{3}} + \frac {A c^{2} \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b^{3}} - \frac {2 B}{3 b x^{\frac {3}{2}}} + \frac {B c \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{2}} - \frac {B c \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{2}} - \frac {B c \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b^{2}} & \text {otherwise} \end {cases} \] Input:

integrate((B*x**2+A)/x**(5/2)/(c*x**4+b*x**2),x)
 

Output:

Piecewise((zoo*(-2*A/(11*x**(11/2)) - 2*B/(7*x**(7/2))), Eq(b, 0) & Eq(c, 
0)), ((-2*A/(11*x**(11/2)) - 2*B/(7*x**(7/2)))/c, Eq(b, 0)), ((-2*A/(7*x** 
(7/2)) - 2*B/(3*x**(3/2)))/b, Eq(c, 0)), (-2*A/(7*b*x**(7/2)) + 2*A*c/(3*b 
**2*x**(3/2)) - A*c**2*(-b/c)**(1/4)*log(sqrt(x) - (-b/c)**(1/4))/(2*b**3) 
 + A*c**2*(-b/c)**(1/4)*log(sqrt(x) + (-b/c)**(1/4))/(2*b**3) + A*c**2*(-b 
/c)**(1/4)*atan(sqrt(x)/(-b/c)**(1/4))/b**3 - 2*B/(3*b*x**(3/2)) + B*c*(-b 
/c)**(1/4)*log(sqrt(x) - (-b/c)**(1/4))/(2*b**2) - B*c*(-b/c)**(1/4)*log(s 
qrt(x) + (-b/c)**(1/4))/(2*b**2) - B*c*(-b/c)**(1/4)*atan(sqrt(x)/(-b/c)** 
(1/4))/b**2, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=-\frac {\frac {2 \, \sqrt {2} {\left (B b c - A c^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (B b c - A c^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (B b c - A c^{2}\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B b c - A c^{2}\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}}{4 \, b^{2}} - \frac {2 \, {\left (7 \, {\left (B b - A c\right )} x^{2} + 3 \, A b\right )}}{21 \, b^{2} x^{\frac {7}{2}}} \] Input:

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

Output:

-1/4*(2*sqrt(2)*(B*b*c - A*c^2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4 
) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c 
))) + 2*sqrt(2)*(B*b*c - A*c^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/ 
4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt( 
c))) + sqrt(2)*(B*b*c - A*c^2)*log(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt( 
c)*x + sqrt(b))/(b^(3/4)*c^(1/4)) - sqrt(2)*(B*b*c - A*c^2)*log(-sqrt(2)*b 
^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)))/b^2 - 2/2 
1*(7*(B*b - A*c)*x^2 + 3*A*b)/(b^2*x^(7/2))
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=-\frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{3}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{3}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{3}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{3}} - \frac {2 \, {\left (7 \, B b x^{2} - 7 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{2} x^{\frac {7}{2}}} \] Input:

integrate((B*x^2+A)/x^(5/2)/(c*x^4+b*x^2),x, algorithm="giac")
 

Output:

-1/2*sqrt(2)*((b*c^3)^(1/4)*B*b - (b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2)*(s 
qrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/b^3 - 1/2*sqrt(2)*((b*c^3)^(1 
/4)*B*b - (b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2* 
sqrt(x))/(b/c)^(1/4))/b^3 - 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b - (b*c^3)^(1/4) 
*A*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^3 + 1/4*sqrt(2)*( 
(b*c^3)^(1/4)*B*b - (b*c^3)^(1/4)*A*c)*log(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + 
x + sqrt(b/c))/b^3 - 2/21*(7*B*b*x^2 - 7*A*c*x^2 + 3*A*b)/(b^2*x^(7/2))
 

Mupad [B] (verification not implemented)

Time = 8.99 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.86 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=-\frac {\frac {2\,A}{7\,b}-\frac {2\,x^2\,\left (A\,c-B\,b\right )}{3\,b^2}}{x^{7/2}}+\frac {{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^6\,c^7-32\,A\,B\,b^7\,c^6+16\,B^2\,b^8\,c^5\right )-\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (32\,A\,b^9\,c^5-32\,B\,b^{10}\,c^4\right )\,1{}\mathrm {i}}{2\,b^{11/4}}\right )}{2\,b^{11/4}}+\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^6\,c^7-32\,A\,B\,b^7\,c^6+16\,B^2\,b^8\,c^5\right )+\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (32\,A\,b^9\,c^5-32\,B\,b^{10}\,c^4\right )\,1{}\mathrm {i}}{2\,b^{11/4}}\right )}{2\,b^{11/4}}}{\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^6\,c^7-32\,A\,B\,b^7\,c^6+16\,B^2\,b^8\,c^5\right )-\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (32\,A\,b^9\,c^5-32\,B\,b^{10}\,c^4\right )\,1{}\mathrm {i}}{2\,b^{11/4}}\right )\,1{}\mathrm {i}}{2\,b^{11/4}}-\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^6\,c^7-32\,A\,B\,b^7\,c^6+16\,B^2\,b^8\,c^5\right )+\frac {{\left (-c\right )}^{3/4}\,\left (A\,c-B\,b\right )\,\left (32\,A\,b^9\,c^5-32\,B\,b^{10}\,c^4\right )\,1{}\mathrm {i}}{2\,b^{11/4}}\right )\,1{}\mathrm {i}}{2\,b^{11/4}}}\right )\,\left (A\,c-B\,b\right )}{b^{11/4}}-\frac {{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {A^3\,c^8\,\sqrt {x}\,1{}\mathrm {i}-B^3\,b^3\,c^5\,\sqrt {x}\,1{}\mathrm {i}-A^2\,B\,b\,c^7\,\sqrt {x}\,3{}\mathrm {i}+A\,B^2\,b^2\,c^6\,\sqrt {x}\,3{}\mathrm {i}}{b^{1/4}\,{\left (-c\right )}^{19/4}\,\left (c\,\left (c\,\left (A^3\,c-3\,A^2\,B\,b\right )+3\,A\,B^2\,b^2\right )-B^3\,b^3\right )}\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{b^{11/4}} \] Input:

int((A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)),x)
 

Output:

((-c)^(3/4)*atan((((-c)^(3/4)*(A*c - B*b)*(x^(1/2)*(16*A^2*b^6*c^7 + 16*B^ 
2*b^8*c^5 - 32*A*B*b^7*c^6) - ((-c)^(3/4)*(A*c - B*b)*(32*A*b^9*c^5 - 32*B 
*b^10*c^4)*1i)/(2*b^(11/4))))/(2*b^(11/4)) + ((-c)^(3/4)*(A*c - B*b)*(x^(1 
/2)*(16*A^2*b^6*c^7 + 16*B^2*b^8*c^5 - 32*A*B*b^7*c^6) + ((-c)^(3/4)*(A*c 
- B*b)*(32*A*b^9*c^5 - 32*B*b^10*c^4)*1i)/(2*b^(11/4))))/(2*b^(11/4)))/((( 
-c)^(3/4)*(A*c - B*b)*(x^(1/2)*(16*A^2*b^6*c^7 + 16*B^2*b^8*c^5 - 32*A*B*b 
^7*c^6) - ((-c)^(3/4)*(A*c - B*b)*(32*A*b^9*c^5 - 32*B*b^10*c^4)*1i)/(2*b^ 
(11/4)))*1i)/(2*b^(11/4)) - ((-c)^(3/4)*(A*c - B*b)*(x^(1/2)*(16*A^2*b^6*c 
^7 + 16*B^2*b^8*c^5 - 32*A*B*b^7*c^6) + ((-c)^(3/4)*(A*c - B*b)*(32*A*b^9* 
c^5 - 32*B*b^10*c^4)*1i)/(2*b^(11/4)))*1i)/(2*b^(11/4))))*(A*c - B*b))/b^( 
11/4) - ((2*A)/(7*b) - (2*x^2*(A*c - B*b))/(3*b^2))/x^(7/2) - ((-c)^(3/4)* 
atan((A^3*c^8*x^(1/2)*1i - B^3*b^3*c^5*x^(1/2)*1i - A^2*B*b*c^7*x^(1/2)*3i 
 + A*B^2*b^2*c^6*x^(1/2)*3i)/(b^(1/4)*(-c)^(19/4)*(c*(c*(A^3*c - 3*A^2*B*b 
) + 3*A*B^2*b^2) - B^3*b^3)))*(A*c - B*b)*1i)/b^(11/4)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx=\frac {-42 \sqrt {x}\, c^{\frac {7}{4}} b^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) a \,x^{3}+42 \sqrt {x}\, c^{\frac {3}{4}} b^{\frac {9}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}+42 \sqrt {x}\, c^{\frac {7}{4}} b^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) a \,x^{3}-42 \sqrt {x}\, c^{\frac {3}{4}} b^{\frac {9}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}-21 \sqrt {x}\, c^{\frac {7}{4}} b^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) a \,x^{3}+21 \sqrt {x}\, c^{\frac {3}{4}} b^{\frac {9}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) x^{3}+21 \sqrt {x}\, c^{\frac {7}{4}} b^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) a \,x^{3}-21 \sqrt {x}\, c^{\frac {3}{4}} b^{\frac {9}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) x^{3}-24 a \,b^{2}+56 a b c \,x^{2}-56 b^{3} x^{2}}{84 \sqrt {x}\, b^{3} x^{3}} \] Input:

int((B*x^2+A)/x^(5/2)/(c*x^4+b*x^2),x)
 

Output:

( - 42*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 
 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*c*x**3 + 42*sqrt(x)*c** 
(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c) 
)/(c**(1/4)*b**(1/4)*sqrt(2)))*b**2*x**3 + 42*sqrt(x)*c**(3/4)*b**(1/4)*sq 
rt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1 
/4)*sqrt(2)))*a*c*x**3 - 42*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/ 
4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*b**2 
*x**3 - 21*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*log( - sqrt(x)*c**(1/4)*b**(1 
/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*a*c*x**3 + 21*sqrt(x)*c**(3/4)*b**(1/4) 
*sqrt(2)*log( - sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*b 
**2*x**3 + 21*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*log(sqrt(x)*c**(1/4)*b**(1 
/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*a*c*x**3 - 21*sqrt(x)*c**(3/4)*b**(1/4) 
*sqrt(2)*log(sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*b**2 
*x**3 - 24*a*b**2 + 56*a*b*c*x**2 - 56*b**3*x**2)/(84*sqrt(x)*b**3*x**3)