\(\int \frac {(a+b x^2)^2}{x^{5/2} (c+d x^2)} \, dx\) [743]

Optimal result

Integrand size = 24, antiderivative size = 195 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx=-\frac {2 a^2}{3 c x^{3/2}}+\frac {2 b^2 \sqrt {x}}{d}+\frac {(b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{7/4} d^{5/4}}-\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{7/4} d^{5/4}} \] Output:

-2/3*a^2/c/x^(3/2)+2*b^2*x^(1/2)/d+1/2*(-a*d+b*c)^2*arctan(1-2^(1/2)*d^(1/ 
4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(7/4)/d^(5/4)-1/2*(-a*d+b*c)^2*arctan(1+2^(1 
/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(7/4)/d^(5/4)-1/2*(-a*d+b*c)^2*arct 
anh(2^(1/2)*c^(1/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(7/4)/d 
^(5/4)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx=\frac {\frac {4 c^{3/4} \sqrt [4]{d} \left (-a^2 d+3 b^2 c x^2\right )}{x^{3/2}}+3 \sqrt {2} (b c-a d)^2 \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-3 \sqrt {2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{6 c^{7/4} d^{5/4}} \] Input:

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

Output:

((4*c^(3/4)*d^(1/4)*(-(a^2*d) + 3*b^2*c*x^2))/x^(3/2) + 3*Sqrt[2]*(b*c - a 
*d)^2*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - 3* 
Sqrt[2]*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + 
 Sqrt[d]*x)])/(6*c^(7/4)*d^(5/4))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {365, 27, 363, 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.

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

\(\Big \downarrow \) 365

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-2*a^2)/(3*c*x^(3/2)) + ((2*b^2*c*Sqrt[x])/d - (2*(b*c - a*d)^2*((-(ArcTa 
n[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4))) + ArcT 
an[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sq 
rt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/ 
(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] 
+ Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c])))/d)/c
 

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 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 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 363

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

rule 365

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x 
_Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] 
- Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p 
+ 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -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 = 0.48 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {2 b^{2} \sqrt {x}}{d}-\frac {2 a^{2}}{3 c \,x^{\frac {3}{2}}}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} d}\)	\(155\)
default	\(\frac {2 b^{2} \sqrt {x}}{d}-\frac {2 a^{2}}{3 c \,x^{\frac {3}{2}}}+\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} d}\)	\(155\)
risch	\(-\frac {2 \left (-3 b^{2} c \,x^{2}+d \,a^{2}\right )}{3 d \,x^{\frac {3}{2}} c}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \,c^{2}}\)	\(157\)

Input:

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

Output:

2*b^2*x^(1/2)/d-2/3*a^2/c/x^(3/2)+1/4/c^2/d*(-a^2*d^2+2*a*b*c*d-b^2*c^2)*( 
c/d)^(1/4)*2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d 
)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2) 
+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 1112, normalized size of antiderivative = 5.70 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(3*c*d*x^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b 
^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 
- 8*a^7*b*c*d^7 + a^8*d^8)/(c^7*d^5))^(1/4)*log(c^2*d*(-(b^8*c^8 - 8*a*b^7 
*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56 
*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^7*d^5) 
)^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 3*I*c*d*x^2*(-(b^8*c^ 
8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c 
^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8 
)/(c^7*d^5))^(1/4)*log(I*c^2*d*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6 
*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a 
^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^7*d^5))^(1/4) + (b^2*c^2 - 2* 
a*b*c*d + a^2*d^2)*sqrt(x)) - 3*I*c*d*x^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28* 
a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3 
*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^7*d^5))^(1/4)*log( 
-I*c^2*d*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5* 
d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7 
*b*c*d^7 + a^8*d^8)/(c^7*d^5))^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqr 
t(x)) - 3*c*d*x^2*(-(b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3 
*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^ 
6 - 8*a^7*b*c*d^7 + a^8*d^8)/(c^7*d^5))^(1/4)*log(-c^2*d*(-(b^8*c^8 - 8...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (182) = 364\).

Time = 8.06 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}}{d} & \text {for}\: c = 0 \\\frac {- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {5}{2}}}{5}}{c} & \text {for}\: d = 0 \\- \frac {2 a^{2}}{3 c x^{\frac {3}{2}}} + \frac {a^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 c^{2}} - \frac {a^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 c^{2}} - \frac {a^{2} d \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c^{2}} - \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{c} + \frac {a b \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{c} + \frac {2 a b \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{c} + \frac {2 b^{2} \sqrt {x}}{d} + \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{2 d} - \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{2 d} - \frac {b^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{d} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((zoo*(-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2*b**2*sqrt(x) 
), Eq(c, 0) & Eq(d, 0)), ((-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2*b 
**2*sqrt(x))/d, Eq(c, 0)), ((-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2 
*x**(5/2)/5)/c, Eq(d, 0)), (-2*a**2/(3*c*x**(3/2)) + a**2*d*(-c/d)**(1/4)* 
log(sqrt(x) - (-c/d)**(1/4))/(2*c**2) - a**2*d*(-c/d)**(1/4)*log(sqrt(x) + 
 (-c/d)**(1/4))/(2*c**2) - a**2*d*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4) 
)/c**2 - a*b*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/c + a*b*(-c/d)**(1 
/4)*log(sqrt(x) + (-c/d)**(1/4))/c + 2*a*b*(-c/d)**(1/4)*atan(sqrt(x)/(-c/ 
d)**(1/4))/c + 2*b**2*sqrt(x)/d + b**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)* 
*(1/4))/(2*d) - b**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(2*d) - b* 
*2*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/d, True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx=\frac {2 \, b^{2} \sqrt {x}}{d} - \frac {2 \, a^{2}}{3 \, c x^{\frac {3}{2}}} - \frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{4 \, c d} \] Input:

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

Output:

2*b^2*sqrt(x)/d - 2/3*a^2/(c*x^(3/2)) - 1/4*(2*sqrt(2)*(b^2*c^2 - 2*a*b*c* 
d + a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt( 
x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(b^ 
2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) 
- 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d)) 
) + sqrt(2)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sq 
rt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(b^2*c^2 - 2*a*b* 
c*d + a^2*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c)) 
/(c^(3/4)*d^(1/4)))/(c*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (144) = 288\).

Time = 0.14 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx=\frac {2 \, b^{2} \sqrt {x}}{d} - \frac {2 \, a^{2}}{3 \, c x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c^{2} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c^{2} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c^{2} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, c^{2} d^{2}} \] Input:

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

Output:

2*b^2*sqrt(x)/d - 2/3*a^2/(c*x^(3/2)) - 1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 
 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sq 
rt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^2) - 1/2*sqrt(2)*((c*d^ 
3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*arctan 
(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^2) - 1 
/4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4 
)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^2) + 1/ 
4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4) 
*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^2)
 

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 1201, normalized size of antiderivative = 6.16 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*b^2*x^(1/2))/d - (2*a^2)/(3*c*x^(3/2)) - (atan(((((x^(1/2)*(16*a^4*c^3* 
d^10 + 16*b^4*c^7*d^6 - 64*a*b^3*c^6*d^7 - 64*a^3*b*c^4*d^9 + 96*a^2*b^2*c 
^5*d^8))/2 - ((a*d - b*c)^2*(16*a^2*c^5*d^9 + 16*b^2*c^7*d^7 - 32*a*b*c^6* 
d^8))/(2*(-c)^(7/4)*d^(5/4)))*(a*d - b*c)^2*1i)/((-c)^(7/4)*d^(5/4)) + ((( 
x^(1/2)*(16*a^4*c^3*d^10 + 16*b^4*c^7*d^6 - 64*a*b^3*c^6*d^7 - 64*a^3*b*c^ 
4*d^9 + 96*a^2*b^2*c^5*d^8))/2 + ((a*d - b*c)^2*(16*a^2*c^5*d^9 + 16*b^2*c 
^7*d^7 - 32*a*b*c^6*d^8))/(2*(-c)^(7/4)*d^(5/4)))*(a*d - b*c)^2*1i)/((-c)^ 
(7/4)*d^(5/4)))/((((x^(1/2)*(16*a^4*c^3*d^10 + 16*b^4*c^7*d^6 - 64*a*b^3*c 
^6*d^7 - 64*a^3*b*c^4*d^9 + 96*a^2*b^2*c^5*d^8))/2 - ((a*d - b*c)^2*(16*a^ 
2*c^5*d^9 + 16*b^2*c^7*d^7 - 32*a*b*c^6*d^8))/(2*(-c)^(7/4)*d^(5/4)))*(a*d 
 - b*c)^2)/((-c)^(7/4)*d^(5/4)) - (((x^(1/2)*(16*a^4*c^3*d^10 + 16*b^4*c^7 
*d^6 - 64*a*b^3*c^6*d^7 - 64*a^3*b*c^4*d^9 + 96*a^2*b^2*c^5*d^8))/2 + ((a* 
d - b*c)^2*(16*a^2*c^5*d^9 + 16*b^2*c^7*d^7 - 32*a*b*c^6*d^8))/(2*(-c)^(7/ 
4)*d^(5/4)))*(a*d - b*c)^2)/((-c)^(7/4)*d^(5/4))))*(a*d - b*c)^2*1i)/((-c) 
^(7/4)*d^(5/4)) - (atan(((((x^(1/2)*(16*a^4*c^3*d^10 + 16*b^4*c^7*d^6 - 64 
*a*b^3*c^6*d^7 - 64*a^3*b*c^4*d^9 + 96*a^2*b^2*c^5*d^8))/2 - ((a*d - b*c)^ 
2*(16*a^2*c^5*d^9 + 16*b^2*c^7*d^7 - 32*a*b*c^6*d^8)*1i)/(2*(-c)^(7/4)*d^( 
5/4)))*(a*d - b*c)^2)/((-c)^(7/4)*d^(5/4)) + (((x^(1/2)*(16*a^4*c^3*d^10 + 
 16*b^4*c^7*d^6 - 64*a*b^3*c^6*d^7 - 64*a^3*b*c^4*d^9 + 96*a^2*b^2*c^5*d^8 
))/2 + ((a*d - b*c)^2*(16*a^2*c^5*d^9 + 16*b^2*c^7*d^7 - 32*a*b*c^6*d^8...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(x)*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*s 
qrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*d**2*x - 12*sqrt(x)*d**( 
3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d)) 
/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d*x + 6*sqrt(x)*d**(3/4)*c**(1/4)*sqrt 
(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4 
)*sqrt(2)))*b**2*c**2*x - 6*sqrt(x)*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/ 
4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2 
*d**2*x + 12*sqrt(x)*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqr 
t(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d*x - 6*sqrt( 
x)*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*s 
qrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**2*x + 3*sqrt(x)*d**(3/4)*c**( 
1/4)*sqrt(2)*log( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)* 
x)*a**2*d**2*x - 6*sqrt(x)*d**(3/4)*c**(1/4)*sqrt(2)*log( - sqrt(x)*d**(1/ 
4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*a*b*c*d*x + 3*sqrt(x)*d**(3/4)* 
c**(1/4)*sqrt(2)*log( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt 
(d)*x)*b**2*c**2*x - 3*sqrt(x)*d**(3/4)*c**(1/4)*sqrt(2)*log(sqrt(x)*d**(1 
/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*a**2*d**2*x + 6*sqrt(x)*d**(3/ 
4)*c**(1/4)*sqrt(2)*log(sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt 
(d)*x)*a*b*c*d*x - 3*sqrt(x)*d**(3/4)*c**(1/4)*sqrt(2)*log(sqrt(x)*d**(1/4 
)*c**(1/4)*sqrt(2) + sqrt(c) + sqrt(d)*x)*b**2*c**2*x - 8*a**2*c*d**2 +...