\(\int \frac {x^{7/2} (A+B x^2)}{(a+b x^2)^3} \, dx\) [177]

Optimal result

Integrand size = 22, antiderivative size = 239 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {2 B \sqrt {x}}{b^3}+\frac {a (A b-a B) \sqrt {x}}{4 b^3 \left (a+b x^2\right )^2}-\frac {(9 A b-17 a B) \sqrt {x}}{16 b^3 \left (a+b x^2\right )}-\frac {5 (A b-9 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}}+\frac {5 (A b-9 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{3/4} b^{13/4}} \] Output:

2*B*x^(1/2)/b^3+1/4*a*(A*b-B*a)*x^(1/2)/b^3/(b*x^2+a)^2-1/16*(9*A*b-17*B*a 
)*x^(1/2)/b^3/(b*x^2+a)-5/64*(A*b-9*B*a)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/ 
a^(1/4))*2^(1/2)/a^(3/4)/b^(13/4)+5/64*(A*b-9*B*a)*arctan(1+2^(1/2)*b^(1/4 
)*x^(1/2)/a^(1/4))*2^(1/2)/a^(3/4)/b^(13/4)+5/64*(A*b-9*B*a)*arctanh(2^(1/ 
2)*a^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(3/4)/b^(13/4)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.77 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (-5 a A b+45 a^2 B-9 A b^2 x^2+81 a b B x^2+32 b^2 B x^4\right )}{\left (a+b x^2\right )^2}+\frac {5 \sqrt {2} (-A b+9 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {5 \sqrt {2} (A b-9 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{64 b^{13/4}} \] Input:

Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x]
 

Output:

((4*b^(1/4)*Sqrt[x]*(-5*a*A*b + 45*a^2*B - 9*A*b^2*x^2 + 81*a*b*B*x^2 + 32 
*b^2*B*x^4))/(a + b*x^2)^2 + (5*Sqrt[2]*(-(A*b) + 9*a*B)*ArcTan[(Sqrt[a] - 
 Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(3/4) + (5*Sqrt[2]*(A*b 
- 9*a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]) 
/a^(3/4))/(64*b^(13/4))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.29, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {362, 252, 262, 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.

\(\displaystyle \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \int \frac {x^{7/2}}{\left (b x^2+a\right )^2}dx}{8 a b}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \int \frac {x^{3/2}}{b x^2+a}dx}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{b x^2+a}d\sqrt {x}}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}-\frac {(A b-9 a B) \left (\frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}\)

Input:

Int[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x]
 

Output:

((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x^2)^2) - ((A*b - 9*a*B)*(-1/2*x^(5/2) 
/(b*(a + b*x^2)) + (5*((2*Sqrt[x])/b - (2*a*((-(ArcTan[1 - (Sqrt[2]*b^(1/4 
)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/ 
4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sq 
rt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1 
/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[ 
2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/b))/(4*b)))/(8*a*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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(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 = 0.46 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{32}\right )}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\)	\(172\)
default	\(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{32}\right )}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\)	\(172\)
risch	\(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (-\frac {9}{32} b^{2} A +\frac {17}{32} a b B \right ) x^{\frac {5}{2}}-\frac {a \left (5 A b -13 B a \right ) \sqrt {x}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (A b -9 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{b^{3}}\)	\(172\)

Input:

int(x^(7/2)*(B*x^2+A)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*B*x^(1/2)/b^3+2/b^3*(((-9/32*b^2*A+17/32*a*b*B)*x^(5/2)-1/32*a*(5*A*b-13 
*B*a)*x^(1/2))/(b*x^2+a)^2+5/256*(A*b-9*B*a)*(a/b)^(1/4)/a*2^(1/2)*(ln((x+ 
(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a 
/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b) 
^(1/4)*x^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.13 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

1/64*(5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3 
*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log 
(5*a*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3 
*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) - 5*(-I*b 
^5*x^4 - 2*I*a*b^4*x^2 - I*a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 4 
86*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(5*I*a 
*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a 
*b^3 + A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) - 5*(I*b^5*x^ 
4 + 2*I*a*b^4*x^2 + I*a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^ 
2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(-5*I*a*b^3 
*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 
 + A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) - 5*(b^5*x^4 + 2* 
a*b^4*x^2 + a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2* 
b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(-5*a*b^3*(-(6561*B^4 
*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/ 
(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) + 4*(32*B*b^2*x^4 + 45*B*a^2 
- 5*A*a*b + 9*(9*B*a*b - A*b^2)*x^2)*sqrt(x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2 
*b^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(7/2)*(B*x**2+A)/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.18 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (17 \, B a b - 9 \, A b^{2}\right )} x^{\frac {5}{2}} + {\left (13 \, B a^{2} - 5 \, A a b\right )} \sqrt {x}}{16 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, B \sqrt {x}}{b^{3}} - \frac {5 \, {\left (\frac {2 \, \sqrt {2} {\left (9 \, B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (9 \, B a - A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (9 \, B a - A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (9 \, B a - A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, b^{3}} \] Input:

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

1/16*((17*B*a*b - 9*A*b^2)*x^(5/2) + (13*B*a^2 - 5*A*a*b)*sqrt(x))/(b^5*x^ 
4 + 2*a*b^4*x^2 + a^2*b^3) + 2*B*sqrt(x)/b^3 - 5/128*(2*sqrt(2)*(9*B*a - A 
*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt( 
sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(9*B*a - A*b 
)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(s 
qrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(9*B*a - A*b)*l 
og(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4) 
) - sqrt(2)*(9*B*a - A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x 
 + sqrt(a))/(a^(3/4)*b^(1/4)))/b^3
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.27 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {2 \, B \sqrt {x}}{b^{3}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a b^{4}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a b^{4}} - \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a b^{4}} + \frac {5 \, \sqrt {2} {\left (9 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a b^{4}} + \frac {17 \, B a b x^{\frac {5}{2}} - 9 \, A b^{2} x^{\frac {5}{2}} + 13 \, B a^{2} \sqrt {x} - 5 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} b^{3}} \] Input:

integrate(x^(7/2)*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

2*B*sqrt(x)/b^3 - 5/64*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*a 
rctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) - 
 5/64*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2 
)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) - 5/128*sqrt(2)*( 
9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + 
 x + sqrt(a/b))/(a*b^4) + 5/128*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/ 
4)*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^4) + 1/16*( 
17*B*a*b*x^(5/2) - 9*A*b^2*x^(5/2) + 13*B*a^2*sqrt(x) - 5*A*a*b*sqrt(x))/( 
(b*x^2 + a)^2*b^3)
 

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 760, normalized size of antiderivative = 3.18 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x)
 

Output:

(x^(1/2)*((13*B*a^2)/16 - (5*A*a*b)/16) - x^(5/2)*((9*A*b^2)/16 - (17*B*a* 
b)/16))/(a^2*b^3 + b^5*x^4 + 2*a*b^4*x^2) + (2*B*x^(1/2))/b^3 - (atan((((A 
*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) - ( 
5*(45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a))/(64*(-a)^(3/4)*b^(13/4)))*5i)/(64*(- 
a)^(3/4)*b^(13/4)) + ((A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 1 
8*A*B*a*b))/(64*b^3) + (5*(45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a))/(64*(-a)^(3/ 
4)*b^(13/4)))*5i)/(64*(-a)^(3/4)*b^(13/4)))/((5*(A*b - 9*B*a)*((25*x^(1/2) 
*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) - (5*(45*B*a^2 - 5*A*a*b)*( 
A*b - 9*B*a))/(64*(-a)^(3/4)*b^(13/4))))/(64*(-a)^(3/4)*b^(13/4)) - (5*(A* 
b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) + (5 
*(45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a))/(64*(-a)^(3/4)*b^(13/4))))/(64*(-a)^( 
3/4)*b^(13/4))))*(A*b - 9*B*a)*5i)/(32*(-a)^(3/4)*b^(13/4)) - (5*atan(((5* 
(A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) - 
 ((45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a)*5i)/(64*(-a)^(3/4)*b^(13/4))))/(64*(- 
a)^(3/4)*b^(13/4)) + (5*(A*b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 
 18*A*B*a*b))/(64*b^3) + ((45*B*a^2 - 5*A*a*b)*(A*b - 9*B*a)*5i)/(64*(-a)^ 
(3/4)*b^(13/4))))/(64*(-a)^(3/4)*b^(13/4)))/(((A*b - 9*B*a)*((25*x^(1/2)*( 
A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) - ((45*B*a^2 - 5*A*a*b)*(A*b 
- 9*B*a)*5i)/(64*(-a)^(3/4)*b^(13/4)))*5i)/(64*(-a)^(3/4)*b^(13/4)) - ((A* 
b - 9*B*a)*((25*x^(1/2)*(A^2*b^2 + 81*B^2*a^2 - 18*A*B*a*b))/(64*b^3) +...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.31 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+10 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-10 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+5 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-5 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+40 \sqrt {x}\, a b +32 \sqrt {x}\, b^{2} x^{2}}{16 b^{3} \left (b \,x^{2}+a \right )} \] Input:

int(x^(7/2)*(B*x^2+A)/(b*x^2+a)^3,x)
 

Output:

(10*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 + 10*b**(3/4)*a**(1/4)*sqrt(2)*ata 
n((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 
2)))*b*x**2 - 10*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 - 10*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*x**2 + 5*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 + 5*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*x**2 
- 5*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 - 5*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*x**2 + 40*sqrt(x)*a*b + 32*sqrt(x)*b 
**2*x**2)/(16*b**3*(a + b*x**2))