\(\int \frac {(d x)^{7/2}}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [462]

Optimal result

Integrand size = 28, antiderivative size = 259 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}-\frac {5 d^3 \sqrt {d x}}{48 b^2 \left (a+b x^2\right )^2}+\frac {5 d^3 \sqrt {d x}}{192 a b^2 \left (a+b x^2\right )}-\frac {5 d^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{7/4} b^{9/4}}+\frac {5 d^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{7/4} b^{9/4}}+\frac {5 d^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{128 \sqrt {2} a^{7/4} b^{9/4}} \] Output:

-1/6*d*(d*x)^(5/2)/b/(b*x^2+a)^3-5/48*d^3*(d*x)^(1/2)/b^2/(b*x^2+a)^2+5/19 
2*d^3*(d*x)^(1/2)/a/b^2/(b*x^2+a)-5/256*d^(7/2)*arctan(1-2^(1/2)*b^(1/4)*( 
d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(7/4)/b^(9/4)+5/256*d^(7/2)*arctan(1 
+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(7/4)/b^(9/4)+5/25 
6*d^(7/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(d*x)^(1/2)/d^(1/2)/(a^(1/2)+b^( 
1/2)*x))*2^(1/2)/a^(7/4)/b^(9/4)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.63 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {d^3 \sqrt {d x} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-15 a^2-42 a b x^2+5 b^2 x^4\right )}{\left (a+b x^2\right )^3}-15 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 a^{7/4} b^{9/4} \sqrt {x}} \] Input:

Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
 

Output:

(d^3*Sqrt[d*x]*((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-15*a^2 - 42*a*b*x^2 + 5*b^2*x 
^4))/(a + b*x^2)^3 - 15*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1 
/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x 
])/(Sqrt[a] + Sqrt[b]*x)]))/(768*a^(7/4)*b^(9/4)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.45, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {1380, 27, 252, 252, 253, 266, 755, 27, 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.

\(\displaystyle \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^4 \int \frac {(d x)^{7/2}}{b^4 \left (b x^2+a\right )^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{7/2}}{\left (a+b x^2\right )^4}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {5 d^2 \int \frac {(d x)^{3/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {5 d^2 \left (\frac {d^2 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 b}-\frac {d \sqrt {d x}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{5/2}}{6 b \left (a+b x^2\right )^3}\)

Input:

Int[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
 

Output:

-1/6*(d*(d*x)^(5/2))/(b*(a + b*x^2)^3) + (5*d^2*(-1/4*(d*Sqrt[d*x])/(b*(a 
+ b*x^2)^2) + (d^2*(Sqrt[d*x]/(2*a*d*(a + b*x^2)) + (3*((d*(-(ArcTan[1 - ( 
Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr 
t[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2 
]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b 
]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4 
)*Sqrt[d]) + Log[Sqrt[a]*d + Sqrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d] 
*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a])))/(2*a*d)))/ 
(8*b)))/(12*b)
 

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -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 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 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

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 = 2.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(2 d^{7} \left (\frac {\frac {5 \left (d x \right )^{\frac {9}{2}}}{384 a \,d^{2}}-\frac {7 \left (d x \right )^{\frac {5}{2}}}{64 b}-\frac {5 a \,d^{2} \sqrt {d x}}{128 b^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{2} d^{4} b^{2}}\right )\)	\(206\)
default	\(2 d^{7} \left (\frac {\frac {5 \left (d x \right )^{\frac {9}{2}}}{384 a \,d^{2}}-\frac {7 \left (d x \right )^{\frac {5}{2}}}{64 b}-\frac {5 a \,d^{2} \sqrt {d x}}{128 b^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{2} d^{4} b^{2}}\right )\)	\(206\)
pseudoelliptic	\(\frac {5 \left (\frac {\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{3}}{2}+\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )-4 \sqrt {d x}\, a \left (-\frac {1}{3} b^{2} x^{4}+\frac {14}{5} a b \,x^{2}+a^{2}\right )\right ) d^{3}}{256 b^{2} a^{2} \left (b \,x^{2}+a \right )^{3}}\)	\(234\)

Input:

int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2*d^7*((5/384/a/d^2*(d*x)^(9/2)-7/64/b*(d*x)^(5/2)-5/128*a*d^2/b^2*(d*x)^( 
1/2))/(b*d^2*x^2+a*d^2)^3+5/1024/a^2/d^4/b^2*(a*d^2/b)^(1/4)*2^(1/2)*(ln(( 
d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^(1 
/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4) 
*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.65 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {15 \, {\left (a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{4} + 3 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (5 \, a^{2} b^{2} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {d x} d^{3}\right ) - 15 \, {\left (-i \, a b^{5} x^{6} - 3 i \, a^{2} b^{4} x^{4} - 3 i \, a^{3} b^{3} x^{2} - i \, a^{4} b^{2}\right )} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (5 i \, a^{2} b^{2} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {d x} d^{3}\right ) - 15 \, {\left (i \, a b^{5} x^{6} + 3 i \, a^{2} b^{4} x^{4} + 3 i \, a^{3} b^{3} x^{2} + i \, a^{4} b^{2}\right )} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-5 i \, a^{2} b^{2} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {d x} d^{3}\right ) - 15 \, {\left (a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{4} + 3 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-5 \, a^{2} b^{2} \left (-\frac {d^{14}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {d x} d^{3}\right ) + 4 \, {\left (5 \, b^{2} d^{3} x^{4} - 42 \, a b d^{3} x^{2} - 15 \, a^{2} d^{3}\right )} \sqrt {d x}}{768 \, {\left (a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{4} + 3 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \] Input:

integrate((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
 

Output:

1/768*(15*(a*b^5*x^6 + 3*a^2*b^4*x^4 + 3*a^3*b^3*x^2 + a^4*b^2)*(-d^14/(a^ 
7*b^9))^(1/4)*log(5*a^2*b^2*(-d^14/(a^7*b^9))^(1/4) + 5*sqrt(d*x)*d^3) - 1 
5*(-I*a*b^5*x^6 - 3*I*a^2*b^4*x^4 - 3*I*a^3*b^3*x^2 - I*a^4*b^2)*(-d^14/(a 
^7*b^9))^(1/4)*log(5*I*a^2*b^2*(-d^14/(a^7*b^9))^(1/4) + 5*sqrt(d*x)*d^3) 
- 15*(I*a*b^5*x^6 + 3*I*a^2*b^4*x^4 + 3*I*a^3*b^3*x^2 + I*a^4*b^2)*(-d^14/ 
(a^7*b^9))^(1/4)*log(-5*I*a^2*b^2*(-d^14/(a^7*b^9))^(1/4) + 5*sqrt(d*x)*d^ 
3) - 15*(a*b^5*x^6 + 3*a^2*b^4*x^4 + 3*a^3*b^3*x^2 + a^4*b^2)*(-d^14/(a^7* 
b^9))^(1/4)*log(-5*a^2*b^2*(-d^14/(a^7*b^9))^(1/4) + 5*sqrt(d*x)*d^3) + 4* 
(5*b^2*d^3*x^4 - 42*a*b*d^3*x^2 - 15*a^2*d^3)*sqrt(d*x))/(a*b^5*x^6 + 3*a^ 
2*b^4*x^4 + 3*a^3*b^3*x^2 + a^4*b^2)
 

Sympy [F]

\[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {7}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \] Input:

integrate((d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
 

Output:

Integral((d*x)**(7/2)/(a + b*x**2)**4, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.28 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {8 \, {\left (5 \, \left (d x\right )^{\frac {9}{2}} b^{2} d^{6} - 42 \, \left (d x\right )^{\frac {5}{2}} a b d^{8} - 15 \, \sqrt {d x} a^{2} d^{10}\right )}}{a b^{5} d^{6} x^{6} + 3 \, a^{2} b^{4} d^{6} x^{4} + 3 \, a^{3} b^{3} d^{6} x^{2} + a^{4} b^{2} d^{6}} + \frac {15 \, {\left (\frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a b^{2}}}{1536 \, d} \] Input:

integrate((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
 

Output:

1/1536*(8*(5*(d*x)^(9/2)*b^2*d^6 - 42*(d*x)^(5/2)*a*b*d^8 - 15*sqrt(d*x)*a 
^2*d^10)/(a*b^5*d^6*x^6 + 3*a^2*b^4*d^6*x^4 + 3*a^3*b^3*d^6*x^2 + a^4*b^2* 
d^6) + 15*(sqrt(2)*d^6*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b 
^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^6*log(sqrt(b)*d*x 
- sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1 
/4)) + 2*sqrt(2)*d^5*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2 
*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt 
(a)) + 2*sqrt(2)*d^5*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 
2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqr 
t(a)))/(a*b^2))/d
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.22 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {30 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{3}} + \frac {30 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{3}} + \frac {15 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{3}} - \frac {15 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{3}} + \frac {8 \, {\left (5 \, \sqrt {d x} b^{2} d^{10} x^{4} - 42 \, \sqrt {d x} a b d^{10} x^{2} - 15 \, \sqrt {d x} a^{2} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a b^{2}}}{1536 \, d} \] Input:

integrate((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
 

Output:

1/1536*(30*sqrt(2)*(a*b^3*d^2)^(1/4)*d^4*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^ 
2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^2*b^3) + 30*sqrt(2)*(a*b^3*d 
^2)^(1/4)*d^4*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/ 
(a*d^2/b)^(1/4))/(a^2*b^3) + 15*sqrt(2)*(a*b^3*d^2)^(1/4)*d^4*log(d*x + sq 
rt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^2*b^3) - 15*sqrt(2)*(a 
*b^3*d^2)^(1/4)*d^4*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d 
^2/b))/(a^2*b^3) + 8*(5*sqrt(d*x)*b^2*d^10*x^4 - 42*sqrt(d*x)*a*b*d^10*x^2 
 - 15*sqrt(d*x)*a^2*d^10)/((b*d^2*x^2 + a*d^2)^3*a*b^2))/d
 

Mupad [B] (verification not implemented)

Time = 18.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.58 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {5\,d^{7/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\frac {7\,d^7\,{\left (d\,x\right )}^{5/2}}{32\,b}-\frac {5\,d^5\,{\left (d\,x\right )}^{9/2}}{192\,a}+\frac {5\,a\,d^9\,\sqrt {d\,x}}{64\,b^2}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}+\frac {5\,d^{7/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{7/4}\,b^{9/4}} \] Input:

int((d*x)^(7/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
 

Output:

(5*d^(7/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(128*(-a)^(7/ 
4)*b^(9/4)) - ((7*d^7*(d*x)^(5/2))/(32*b) - (5*d^5*(d*x)^(9/2))/(192*a) + 
(5*a*d^9*(d*x)^(1/2))/(64*b^2))/(a^3*d^6 + b^3*d^6*x^6 + 3*a^2*b*d^6*x^2 + 
 3*a*b^2*d^6*x^4) + (5*d^(7/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^( 
1/2))))/(128*(-a)^(7/4)*b^(9/4))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.52 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
 

Output:

(sqrt(d)*d**3*( - 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3 - 90*b**(3/4)* 
a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b** 
(1/4)*a**(1/4)*sqrt(2)))*a**2*b*x**2 - 90*b**(3/4)*a**(1/4)*sqrt(2)*atan(( 
b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)) 
)*a*b**2*x**4 - 30*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt( 
2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**3*x**6 + 30*b**(3/ 
4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/( 
b**(1/4)*a**(1/4)*sqrt(2)))*a**3 + 90*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**( 
1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a* 
*2*b*x**2 + 90*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 
 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*x**4 + 30*b**(3/4) 
*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b* 
*(1/4)*a**(1/4)*sqrt(2)))*b**3*x**6 - 15*b**(3/4)*a**(1/4)*sqrt(2)*log( - 
sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3 - 45*b**(3/4 
)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq 
rt(b)*x)*a**2*b*x**2 - 45*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4 
)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b**2*x**4 - 15*b**(3/4)*a**(1/ 
4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x) 
*b**3*x**6 + 15*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)...