\(\int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx\) [447]

Optimal result

Integrand size = 28, antiderivative size = 221 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {7 d^3 (d x)^{3/2}}{6 b^2}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}+\frac {7 a^{3/4} d^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{11/4}}-\frac {7 a^{3/4} d^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{11/4}}+\frac {7 a^{3/4} d^{9/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} b^{11/4}} \] Output:

7/6*d^3*(d*x)^(3/2)/b^2-1/2*d*(d*x)^(7/2)/b/(b*x^2+a)+7/8*a^(3/4)*d^(9/2)* 
arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(11/4)-7/8 
*a^(3/4)*d^(9/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^( 
1/2)/b^(11/4)+7/8*a^(3/4)*d^(9/2)*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)/b^(11/4)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {d^4 \sqrt {d x} \left (4 b^{3/4} x^{3/2} \left (7 a+4 b x^2\right )+21 \sqrt {2} a^{3/4} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} a^{3/4} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{24 b^{11/4} \sqrt {x} \left (a+b x^2\right )} \] Input:

Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]
 

Output:

(d^4*Sqrt[d*x]*(4*b^(3/4)*x^(3/2)*(7*a + 4*b*x^2) + 21*Sqrt[2]*a^(3/4)*(a 
+ b*x^2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 
 21*Sqrt[2]*a^(3/4)*(a + b*x^2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/ 
(Sqrt[a] + Sqrt[b]*x)]))/(24*b^(11/4)*Sqrt[x]*(a + b*x^2))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.48, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1380, 27, 252, 262, 266, 27, 826, 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.

\(\displaystyle \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^2 \int \frac {(d x)^{9/2}}{b^2 \left (b x^2+a\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{9/2}}{\left (a+b x^2\right )^2}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {7 d^2 \int \frac {(d x)^{5/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {a d^2 \int \frac {\sqrt {d x}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]
 

Output:

-1/2*(d*(d*x)^(7/2))/(b*(a + b*x^2)) + (7*d^2*((2*d*(d*x)^(3/2))/(3*b) - ( 
2*a*d^3*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqr 
t[2]*a^(1/4)*b^(1/4)*Sqrt[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[b]) - (-1/2*Lo 
g[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sq 
rt[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*Sq 
rt[b])))/b))/(4*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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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 = 0.96 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(2 d^{3} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \,d^{2} \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{4 \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}\right )\)	\(188\)
default	\(2 d^{3} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \,d^{2} \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{4 \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{2}}\right )\)	\(188\)
risch	\(\frac {2 x^{2} d^{5}}{3 b^{2} \sqrt {d x}}-\frac {a \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{2 \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) d^{5}}{b^{2}}\)	\(189\)
pseudoelliptic	\(-\frac {7 \left (\frac {\sqrt {2}\, a \,d^{2} \left (b \,x^{2}+a \right ) \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{2}+\sqrt {2}\, a \,d^{2} \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\sqrt {2}\, a \,d^{2} \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )-\frac {4 \left (\frac {4 b \,x^{2}}{7}+a \right ) b \left (d x \right )^{\frac {3}{2}} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{3}\right ) d^{3}}{8 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right ) b^{3}}\)	\(216\)

Input:

int((d*x)^(9/2)/(b^2*x^4+2*a*b*x^2+a^2),x,method=_RETURNVERBOSE)
 

Output:

2*d^3*(1/3*(d*x)^(3/2)/b^2-a*d^2/b^2*(-1/4*(d*x)^(3/2)/(b*d^2*x^2+a*d^2)+7 
/32/b/(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 = 0.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.31 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {21 \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {1}{4}} {\left (b^{3} x^{2} + a b^{2}\right )} \log \left (343 \, \sqrt {d x} a^{2} d^{13} + 343 \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {3}{4}} b^{8}\right ) + 21 \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {1}{4}} {\left (-i \, b^{3} x^{2} - i \, a b^{2}\right )} \log \left (343 \, \sqrt {d x} a^{2} d^{13} + 343 i \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {3}{4}} b^{8}\right ) + 21 \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {1}{4}} {\left (i \, b^{3} x^{2} + i \, a b^{2}\right )} \log \left (343 \, \sqrt {d x} a^{2} d^{13} - 343 i \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {3}{4}} b^{8}\right ) - 21 \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {1}{4}} {\left (b^{3} x^{2} + a b^{2}\right )} \log \left (343 \, \sqrt {d x} a^{2} d^{13} - 343 \, \left (-\frac {a^{3} d^{18}}{b^{11}}\right )^{\frac {3}{4}} b^{8}\right ) - 4 \, {\left (4 \, b d^{4} x^{3} + 7 \, a d^{4} x\right )} \sqrt {d x}}{24 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \] Input:

integrate((d*x)^(9/2)/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")
 

Output:

-1/24*(21*(-a^3*d^18/b^11)^(1/4)*(b^3*x^2 + a*b^2)*log(343*sqrt(d*x)*a^2*d 
^13 + 343*(-a^3*d^18/b^11)^(3/4)*b^8) + 21*(-a^3*d^18/b^11)^(1/4)*(-I*b^3* 
x^2 - I*a*b^2)*log(343*sqrt(d*x)*a^2*d^13 + 343*I*(-a^3*d^18/b^11)^(3/4)*b 
^8) + 21*(-a^3*d^18/b^11)^(1/4)*(I*b^3*x^2 + I*a*b^2)*log(343*sqrt(d*x)*a^ 
2*d^13 - 343*I*(-a^3*d^18/b^11)^(3/4)*b^8) - 21*(-a^3*d^18/b^11)^(1/4)*(b^ 
3*x^2 + a*b^2)*log(343*sqrt(d*x)*a^2*d^13 - 343*(-a^3*d^18/b^11)^(3/4)*b^8 
) - 4*(4*b*d^4*x^3 + 7*a*d^4*x)*sqrt(d*x))/(b^3*x^2 + a*b^2)
 

Sympy [F]

\[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\int \frac {\left (d x\right )^{\frac {9}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \] Input:

integrate((d*x)**(9/2)/(b**2*x**4+2*a*b*x**2+a**2),x)
 

Output:

Integral((d*x)**(9/2)/(a + b*x**2)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.24 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {\frac {24 \, \left (d x\right )^{\frac {3}{2}} a d^{6}}{b^{3} d^{2} x^{2} + a b^{2} d^{2}} - \frac {21 \, a d^{6} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b^{2}} + \frac {32 \, \left (d x\right )^{\frac {3}{2}} d^{4}}{b^{2}}}{48 \, d} \] Input:

integrate((d*x)^(9/2)/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")
 

Output:

1/48*(24*(d*x)^(3/2)*a*d^6/(b^3*d^2*x^2 + a*b^2*d^2) - 21*a*d^6*(2*sqrt(2) 
*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(b)) + 2*sqrt(2)*arc 
tan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqr 
t(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt 
(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1 
/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b 
^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/b^2 + 32*(d*x)^(3/2)*d^4/b^2) 
/d
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.25 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {1}{48} \, {\left (\frac {24 \, \sqrt {d x} a d^{2} x}{{\left (b d^{2} x^{2} + a d^{2}\right )} b^{2}} + \frac {32 \, \sqrt {d x} x}{b^{2}} - \frac {42 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5} d} - \frac {42 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5} d} + \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5} d} - \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5} d}\right )} d^{4} \] Input:

integrate((d*x)^(9/2)/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")
 

Output:

1/48*(24*sqrt(d*x)*a*d^2*x/((b*d^2*x^2 + a*d^2)*b^2) + 32*sqrt(d*x)*x/b^2 
- 42*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) 
 + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^5*d) - 42*sqrt(2)*(a*b^3*d^2)^(3/4)*ar 
ctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4)) 
/(b^5*d) + 21*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)* 
sqrt(d*x) + sqrt(a*d^2/b))/(b^5*d) - 21*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x 
- sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^5*d))*d^4
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.51 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {2\,d^3\,{\left (d\,x\right )}^{3/2}}{3\,b^2}+\frac {7\,{\left (-a\right )}^{3/4}\,d^{9/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,b^{11/4}}+\frac {a\,d^5\,{\left (d\,x\right )}^{3/2}}{2\,\left (b^3\,d^2\,x^2+a\,b^2\,d^2\right )}+\frac {{\left (-a\right )}^{3/4}\,d^{9/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,7{}\mathrm {i}}{4\,b^{11/4}} \] Input:

int((d*x)^(9/2)/(a^2 + b^2*x^4 + 2*a*b*x^2),x)
 

Output:

(2*d^3*(d*x)^(3/2))/(3*b^2) + (7*(-a)^(3/4)*d^(9/2)*atan((b^(1/4)*(d*x)^(1 
/2))/((-a)^(1/4)*d^(1/2))))/(4*b^(11/4)) + ((-a)^(3/4)*d^(9/2)*atan((b^(1/ 
4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*7i)/(4*b^(11/4)) + (a*d^5*(d*x)^( 
3/2))/(2*(a*b^2*d^2 + b^3*d^2*x^2))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.44 \[ \int \frac {(d x)^{9/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {\sqrt {d}\, d^{4} \left (42 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+42 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-42 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-42 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-21 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-21 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+21 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+21 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+56 \sqrt {x}\, a b x +32 \sqrt {x}\, b^{2} x^{3}\right )}{48 b^{3} \left (b \,x^{2}+a \right )} \] Input:

int((d*x)^(9/2)/(b^2*x^4+2*a*b*x^2+a^2),x)
 

Output:

(sqrt(d)*d**4*(42*b**(1/4)*a**(3/4)*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 + 42*b**(1/4)*a**(3/ 
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 - 42*b**(1/4)*a**(3/4)*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 - 42*b* 
*(1/4)*a**(3/4)*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 - 21*b**(1/4)*a**(3/4)*sqrt(2)*log( 
 - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a - 21*b**(1/4 
)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq 
rt(b)*x)*b*x**2 + 21*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/ 
4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a + 21*b**(1/4)*a**(3/4)*sqrt(2)*log(sqr 
t(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 + 56*sqrt(x)* 
a*b*x + 32*sqrt(x)*b**2*x**3))/(48*b**3*(a + b*x**2))