\(\int \frac {(d x)^{15/2}}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [458]

Optimal result

Integrand size = 28, antiderivative size = 273 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {195 \sqrt [4]{a} d^{15/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{128 \sqrt {2} b^{17/4}} \] Output:

195/64*d^7*(d*x)^(1/2)/b^4-1/6*d*(d*x)^(13/2)/b/(b*x^2+a)^3-13/48*d^3*(d*x 
)^(9/2)/b^2/(b*x^2+a)^2-39/64*d^5*(d*x)^(5/2)/b^3/(b*x^2+a)+195/256*a^(1/4 
)*d^(15/2)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b 
^(17/4)-195/256*a^(1/4)*d^(15/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1 
/4)/d^(1/2))*2^(1/2)/b^(17/4)-195/256*a^(1/4)*d^(15/2)*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^(17/4)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.64 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {d^7 \sqrt {d x} \left (\frac {4 \sqrt [4]{b} \sqrt {x} \left (585 a^3+1638 a^2 b x^2+1469 a b^2 x^4+384 b^3 x^6\right )}{\left (a+b x^2\right )^3}+585 \sqrt {2} \sqrt [4]{a} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-585 \sqrt {2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 b^{17/4} \sqrt {x}} \] Input:

Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
 

Output:

(d^7*Sqrt[d*x]*((4*b^(1/4)*Sqrt[x]*(585*a^3 + 1638*a^2*b*x^2 + 1469*a*b^2* 
x^4 + 384*b^3*x^6))/(a + b*x^2)^3 + 585*Sqrt[2]*a^(1/4)*ArcTan[(Sqrt[a] - 
Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 585*Sqrt[2]*a^(1/4)*ArcTan 
h[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(768*b^(17/4) 
*Sqrt[x])
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.44, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1380, 27, 252, 252, 252, 262, 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.

\(\displaystyle \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^4 \int \frac {(d x)^{15/2}}{b^4 \left (b x^2+a\right )^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{15/2}}{\left (a+b x^2\right )^4}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \int \frac {(d x)^{7/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \int \frac {(d x)^{3/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {a d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \int \frac {1}{b x^2+a}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {13 d^2 \left (\frac {9 d^2 \left (\frac {5 d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{9/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}\)

Input:

Int[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
 

Output:

-1/6*(d*(d*x)^(13/2))/(b*(a + b*x^2)^3) + (13*d^2*(-1/4*(d*(d*x)^(9/2))/(b 
*(a + b*x^2)^2) + (9*d^2*(-1/2*(d*(d*x)^(5/2))/(b*(a + b*x^2)) + (5*d^2*(( 
2*d*Sqrt[d*x])/b - (2*a*d*((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])) + 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])))/b))/(4*b)))/(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 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 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 = 2.38 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(2 d^{7} \left (\frac {\sqrt {d x}}{b^{4}}-\frac {a \,d^{2} \left (\frac {-\frac {317 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {81 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {67 \sqrt {d x}\, a^{2} d^{4}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{4}}\right )\)	\(220\)
default	\(2 d^{7} \left (\frac {\sqrt {d x}}{b^{4}}-\frac {a \,d^{2} \left (\frac {-\frac {317 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {81 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {67 \sqrt {d x}\, a^{2} d^{4}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{4}}\right )\)	\(220\)
risch	\(\frac {2 x \,d^{8}}{b^{4} \sqrt {d x}}-\frac {2 a \,d^{9} \left (\frac {-\frac {317 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {81 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {67 \sqrt {d x}\, a^{2} d^{4}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{4}}\)	\(220\)
pseudoelliptic	\(-\frac {d^{7} \left (\left (-3072 b^{3} x^{6}-11752 b^{2} x^{4} a -13104 a^{2} b \,x^{2}-4680 a^{3}\right ) \sqrt {d x}+585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{1536 \left (b \,x^{2}+a \right )^{3} b^{4}}\)	\(223\)

Input:

int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2*d^7*((d*x)^(1/2)/b^4-1/b^4*a*d^2*((-317/384*b^2*(d*x)^(9/2)-81/64*a*b*d^ 
2*(d*x)^(5/2)-67/128*(d*x)^(1/2)*a^2*d^4)/(b*d^2*x^2+a*d^2)^3+195/1024*(a* 
d^2/b)^(1/4)/a/d^2*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 = 0.17 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.46 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} + 195 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (i \, b^{7} x^{6} + 3 i \, a b^{6} x^{4} + 3 i \, a^{2} b^{5} x^{2} + i \, a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} + 195 i \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (-i \, b^{7} x^{6} - 3 i \, a b^{6} x^{4} - 3 i \, a^{2} b^{5} x^{2} - i \, a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} - 195 i \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} - 195 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 4 \, {\left (384 \, b^{3} d^{7} x^{6} + 1469 \, a b^{2} d^{7} x^{4} + 1638 \, a^{2} b d^{7} x^{2} + 585 \, a^{3} d^{7}\right )} \sqrt {d x}}{768 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} \] Input:

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
 

Output:

-1/768*(585*(-a*d^30/b^17)^(1/4)*(b^7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + 
a^3*b^4)*log(195*sqrt(d*x)*d^7 + 195*(-a*d^30/b^17)^(1/4)*b^4) + 585*(-a*d 
^30/b^17)^(1/4)*(I*b^7*x^6 + 3*I*a*b^6*x^4 + 3*I*a^2*b^5*x^2 + I*a^3*b^4)* 
log(195*sqrt(d*x)*d^7 + 195*I*(-a*d^30/b^17)^(1/4)*b^4) + 585*(-a*d^30/b^1 
7)^(1/4)*(-I*b^7*x^6 - 3*I*a*b^6*x^4 - 3*I*a^2*b^5*x^2 - I*a^3*b^4)*log(19 
5*sqrt(d*x)*d^7 - 195*I*(-a*d^30/b^17)^(1/4)*b^4) - 585*(-a*d^30/b^17)^(1/ 
4)*(b^7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + a^3*b^4)*log(195*sqrt(d*x)*d^7 
 - 195*(-a*d^30/b^17)^(1/4)*b^4) - 4*(384*b^3*d^7*x^6 + 1469*a*b^2*d^7*x^4 
 + 1638*a^2*b*d^7*x^2 + 585*a^3*d^7)*sqrt(d*x))/(b^7*x^6 + 3*a*b^6*x^4 + 3 
*a^2*b^5*x^2 + a^3*b^4)
 

Sympy [F]

\[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {15}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \] Input:

integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
 

Output:

Integral((d*x)**(15/2)/(a + b*x**2)**4, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.26 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {3072 \, \sqrt {d x} d^{8}}{b^{4}} + \frac {8 \, {\left (317 \, \left (d x\right )^{\frac {9}{2}} a b^{2} d^{10} + 486 \, \left (d x\right )^{\frac {5}{2}} a^{2} b d^{12} + 201 \, \sqrt {d x} a^{3} d^{14}\right )}}{b^{7} d^{6} x^{6} + 3 \, a b^{6} d^{6} x^{4} + 3 \, a^{2} b^{5} d^{6} x^{2} + a^{3} b^{4} d^{6}} - \frac {585 \, {\left (\frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b^{4}}}{1536 \, d} \] Input:

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
 

Output:

1/1536*(3072*sqrt(d*x)*d^8/b^4 + 8*(317*(d*x)^(9/2)*a*b^2*d^10 + 486*(d*x) 
^(5/2)*a^2*b*d^12 + 201*sqrt(d*x)*a^3*d^14)/(b^7*d^6*x^6 + 3*a*b^6*d^6*x^4 
 + 3*a^2*b^5*d^6*x^2 + a^3*b^4*d^6) - 585*(sqrt(2)*d^10*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^10*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^9*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^9*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)))*a/b^4)/d
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.16 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5}} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5}} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{8} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5}} - \frac {3072 \, \sqrt {d x} d^{8}}{b^{4}} - \frac {8 \, {\left (317 \, \sqrt {d x} a b^{2} d^{14} x^{4} + 486 \, \sqrt {d x} a^{2} b d^{14} x^{2} + 201 \, \sqrt {d x} a^{3} d^{14}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{4}}}{1536 \, d} \] Input:

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
 

Output:

-1/1536*(1170*sqrt(2)*(a*b^3*d^2)^(1/4)*d^8*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 + 1170*sqrt(2)*(a*b^3*d^ 
2)^(1/4)*d^8*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 + 585*sqrt(2)*(a*b^3*d^2)^(1/4)*d^8*log(d*x + sqrt(2)* 
(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/b^5 - 585*sqrt(2)*(a*b^3*d^2)^( 
1/4)*d^8*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/b^5 
- 3072*sqrt(d*x)*d^8/b^4 - 8*(317*sqrt(d*x)*a*b^2*d^14*x^4 + 486*sqrt(d*x) 
*a^2*b*d^14*x^2 + 201*sqrt(d*x)*a^3*d^14)/((b*d^2*x^2 + a*d^2)^3*b^4))/d
 

Mupad [B] (verification not implemented)

Time = 17.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.63 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {67\,a^3\,d^{13}\,\sqrt {d\,x}}{64}+\frac {81\,a^2\,b\,d^{11}\,{\left (d\,x\right )}^{5/2}}{32}+\frac {317\,a\,b^2\,d^9\,{\left (d\,x\right )}^{9/2}}{192}}{a^3\,b^4\,d^6+3\,a^2\,b^5\,d^6\,x^2+3\,a\,b^6\,d^6\,x^4+b^7\,d^6\,x^6}+\frac {2\,d^7\,\sqrt {d\,x}}{b^4}-\frac {195\,{\left (-a\right )}^{1/4}\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,b^{17/4}}+\frac {{\left (-a\right )}^{1/4}\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,195{}\mathrm {i}}{128\,b^{17/4}} \] Input:

int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
 

Output:

((67*a^3*d^13*(d*x)^(1/2))/64 + (81*a^2*b*d^11*(d*x)^(5/2))/32 + (317*a*b^ 
2*d^9*(d*x)^(9/2))/192)/(a^3*b^4*d^6 + b^7*d^6*x^6 + 3*a*b^6*d^6*x^4 + 3*a 
^2*b^5*d^6*x^2) + (2*d^7*(d*x)^(1/2))/b^4 - (195*(-a)^(1/4)*d^(15/2)*atan( 
(b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(128*b^(17/4)) + ((-a)^(1/4)* 
d^(15/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*195i)/(128*b^ 
(17/4))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.42 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
 

Output:

(sqrt(d)*d**7*(1170*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 + 3510*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 + 3510*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)))*a*b**2*x**4 + 1170*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 - 1170 
*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqr 
t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3 - 3510*b**(3/4)*a**(1/4)*sqrt(2)*a 
tan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr 
t(2)))*a**2*b*x**2 - 3510*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 - 
1170*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 + 585*b**(3/4)*a**(1/4)*s 
qrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a** 
3 + 1755*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**2*b*x**2 + 1755*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 + 
585*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + s 
qrt(a) + sqrt(b)*x)*b**3*x**6 - 585*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(...