\(\int \frac {(d x)^{17/2}}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [457]

Optimal result

Integrand size = 28, antiderivative size = 273 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {385 d^7 (d x)^{3/2}}{192 b^4}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}-\frac {5 d^3 (d x)^{11/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac {55 d^5 (d x)^{7/2}}{64 b^3 \left (a+b x^2\right )}+\frac {385 a^{3/4} d^{17/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{19/4}}-\frac {385 a^{3/4} d^{17/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{19/4}}+\frac {385 a^{3/4} d^{17/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{128 \sqrt {2} b^{19/4}} \] Output:

385/192*d^7*(d*x)^(3/2)/b^4-1/6*d*(d*x)^(15/2)/b/(b*x^2+a)^3-5/16*d^3*(d*x 
)^(11/2)/b^2/(b*x^2+a)^2-55/64*d^5*(d*x)^(7/2)/b^3/(b*x^2+a)+385/256*a^(3/ 
4)*d^(17/2)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/ 
b^(19/4)-385/256*a^(3/4)*d^(17/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^( 
1/4)/d^(1/2))*2^(1/2)/b^(19/4)+385/256*a^(3/4)*d^(17/2)*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^(19/4)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.71 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {d^8 \sqrt {d x} \left (4 b^{3/4} x^{3/2} \left (385 a^3+990 a^2 b x^2+765 a b^2 x^4+128 b^3 x^6\right )-1155 \sqrt {2} a^{3/4} \left (a+b x^2\right )^3 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+1155 \sqrt {2} a^{3/4} \left (a+b x^2\right )^3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 b^{19/4} \sqrt {x} \left (a+b x^2\right )^3} \] Input:

Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
 

Output:

(d^8*Sqrt[d*x]*(4*b^(3/4)*x^(3/2)*(385*a^3 + 990*a^2*b*x^2 + 765*a*b^2*x^4 
 + 128*b^3*x^6) - 1155*Sqrt[2]*a^(3/4)*(a + b*x^2)^3*ArcTan[(-Sqrt[a] + Sq 
rt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 1155*Sqrt[2]*a^(3/4)*(a + b* 
x^2)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/ 
(768*b^(19/4)*Sqrt[x]*(a + b*x^2)^3)
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.45, 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, 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.

\(\displaystyle \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^4 \int \frac {(d x)^{17/2}}{b^4 \left (b x^2+a\right )^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{17/2}}{\left (a+b x^2\right )^4}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {5 d^2 \int \frac {(d x)^{13/2}}{\left (b x^2+a\right )^3}dx}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \int \frac {(d x)^{9/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \int \frac {(d x)^{5/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {a d^2 \int \frac {\sqrt {d x}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\)

Input:

Int[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
 

Output:

-1/6*(d*(d*x)^(15/2))/(b*(a + b*x^2)^3) + (5*d^2*(-1/4*(d*(d*x)^(11/2))/(b 
*(a + b*x^2)^2) + (11*d^2*(-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])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sq 
rt[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*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[b])))/b))/(4*b)))/(8*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 = 2.50 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(2 d^{7} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{3 b^{4}}-\frac {a \,d^{2} \left (\frac {-\frac {127 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}-\frac {101 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}-\frac {257 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\right )\)	\(218\)
default	\(2 d^{7} \left (\frac {\left (d x \right )^{\frac {3}{2}}}{3 b^{4}}-\frac {a \,d^{2} \left (\frac {-\frac {127 b^{2} \left (d x \right )^{\frac {11}{2}}}{128}-\frac {101 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{64}-\frac {257 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{384}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{4}}\right )\)	\(218\)
risch	\(\frac {2 x^{2} d^{9}}{3 b^{4} \sqrt {d x}}-\frac {a \left (\frac {-\frac {127 b^{2} \left (d x \right )^{\frac {11}{2}}}{64}-\frac {101 a b \,d^{2} \left (d x \right )^{\frac {7}{2}}}{32}-\frac {257 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}{192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{512 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) d^{9}}{b^{4}}\)	\(220\)
pseudoelliptic	\(-\frac {385 d^{8} \left (-8 \sqrt {d x}\, \left (\frac {128}{385} b^{3} x^{6}+\frac {153}{77} b^{2} x^{4} a +\frac {18}{7} a^{2} b \,x^{2}+a^{3}\right ) b x \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+3 \sqrt {2}\, a d \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{1536 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{5} \left (b \,x^{2}+a \right )^{3}}\)	\(236\)

Input:

int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2*d^7*(1/3*(d*x)^(3/2)/b^4-a*d^2/b^4*((-127/128*b^2*(d*x)^(11/2)-101/64*a* 
b*d^2*(d*x)^(7/2)-257/384*a^2*d^4*(d*x)^(3/2))/(b*d^2*x^2+a*d^2)^3+385/102 
4/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 = 0.14 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.57 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1155 \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (57066625 \, \sqrt {d x} a^{2} d^{25} + 57066625 \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {3}{4}} b^{14}\right ) + 1155 \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {1}{4}} {\left (-i \, b^{7} x^{6} - 3 i \, a b^{6} x^{4} - 3 i \, a^{2} b^{5} x^{2} - i \, a^{3} b^{4}\right )} \log \left (57066625 \, \sqrt {d x} a^{2} d^{25} + 57066625 i \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {3}{4}} b^{14}\right ) + 1155 \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {1}{4}} {\left (i \, b^{7} x^{6} + 3 i \, a b^{6} x^{4} + 3 i \, a^{2} b^{5} x^{2} + i \, a^{3} b^{4}\right )} \log \left (57066625 \, \sqrt {d x} a^{2} d^{25} - 57066625 i \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {3}{4}} b^{14}\right ) - 1155 \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (57066625 \, \sqrt {d x} a^{2} d^{25} - 57066625 \, \left (-\frac {a^{3} d^{34}}{b^{19}}\right )^{\frac {3}{4}} b^{14}\right ) - 4 \, {\left (128 \, b^{3} d^{8} x^{7} + 765 \, a b^{2} d^{8} x^{5} + 990 \, a^{2} b d^{8} x^{3} + 385 \, a^{3} d^{8} x\right )} \sqrt {d x}}{768 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} \] Input:

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
 

Output:

-1/768*(1155*(-a^3*d^34/b^19)^(1/4)*(b^7*x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 
 + a^3*b^4)*log(57066625*sqrt(d*x)*a^2*d^25 + 57066625*(-a^3*d^34/b^19)^(3 
/4)*b^14) + 1155*(-a^3*d^34/b^19)^(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(57066625*sqrt(d*x)*a^2*d^25 + 57066625*I*(-a^ 
3*d^34/b^19)^(3/4)*b^14) + 1155*(-a^3*d^34/b^19)^(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(57066625*sqrt(d*x)*a^2*d^25 - 5 
7066625*I*(-a^3*d^34/b^19)^(3/4)*b^14) - 1155*(-a^3*d^34/b^19)^(1/4)*(b^7* 
x^6 + 3*a*b^6*x^4 + 3*a^2*b^5*x^2 + a^3*b^4)*log(57066625*sqrt(d*x)*a^2*d^ 
25 - 57066625*(-a^3*d^34/b^19)^(3/4)*b^14) - 4*(128*b^3*d^8*x^7 + 765*a*b^ 
2*d^8*x^5 + 990*a^2*b*d^8*x^3 + 385*a^3*d^8*x)*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]

\[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {17}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \] Input:

integrate((d*x)**(17/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
 

Output:

Integral((d*x)**(17/2)/(a + b*x**2)**4, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.22 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {1155 \, a d^{10} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b^{4}} - \frac {1024 \, \left (d x\right )^{\frac {3}{2}} d^{8}}{b^{4}} - \frac {8 \, {\left (381 \, \left (d x\right )^{\frac {11}{2}} a b^{2} d^{10} + 606 \, \left (d x\right )^{\frac {7}{2}} a^{2} b d^{12} + 257 \, \left (d x\right )^{\frac {3}{2}} a^{3} d^{14}\right )}}{b^{7} d^{6} x^{6} + 3 \, a b^{6} d^{6} x^{4} + 3 \, a^{2} b^{5} d^{6} x^{2} + a^{3} b^{4} d^{6}}}{1536 \, d} \] Input:

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
 

Output:

-1/1536*(1155*a*d^10*(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)*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)) - 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 - sq 
rt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) 
)/b^4 - 1024*(d*x)^(3/2)*d^8/b^4 - 8*(381*(d*x)^(11/2)*a*b^2*d^10 + 606*(d 
*x)^(7/2)*a^2*b*d^12 + 257*(d*x)^(3/2)*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))/d
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.16 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {1}{1536} \, d^{8} {\left (\frac {1024 \, \sqrt {d x} x}{b^{4}} - \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7} d} - \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7} d} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7} d} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7} d} + \frac {8 \, {\left (381 \, \sqrt {d x} a b^{2} d^{6} x^{5} + 606 \, \sqrt {d x} a^{2} b d^{6} x^{3} + 257 \, \sqrt {d x} a^{3} d^{6} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{4}}\right )} \] Input:

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
 

Output:

1/1536*d^8*(1024*sqrt(d*x)*x/b^4 - 2310*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^7*d 
) - 2310*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^7*d) + 1155*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^7*d) 
- 1155*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^7*d) + 8*(381*sqrt(d*x)*a*b^2*d^6*x^5 + 606*sqrt(d* 
x)*a^2*b*d^6*x^3 + 257*sqrt(d*x)*a^3*d^6*x)/((b*d^2*x^2 + a*d^2)^3*b^4))
 

Mupad [B] (verification not implemented)

Time = 18.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.63 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {257\,a^3\,d^{13}\,{\left (d\,x\right )}^{3/2}}{192}+\frac {101\,a^2\,b\,d^{11}\,{\left (d\,x\right )}^{7/2}}{32}+\frac {127\,a\,b^2\,d^9\,{\left (d\,x\right )}^{11/2}}{64}}{a^3\,b^4\,d^6+3\,a^2\,b^5\,d^6\,x^2+3\,a\,b^6\,d^6\,x^4+b^7\,d^6\,x^6}+\frac {2\,d^7\,{\left (d\,x\right )}^{3/2}}{3\,b^4}+\frac {385\,{\left (-a\right )}^{3/4}\,d^{17/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,b^{19/4}}+\frac {{\left (-a\right )}^{3/4}\,d^{17/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,385{}\mathrm {i}}{128\,b^{19/4}} \] Input:

int((d*x)^(17/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
 

Output:

((257*a^3*d^13*(d*x)^(3/2))/192 + (101*a^2*b*d^11*(d*x)^(7/2))/32 + (127*a 
*b^2*d^9*(d*x)^(11/2))/64)/(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)^(3/2))/(3*b^4) + (385*(-a)^(3/4)*d^(17/2 
)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(128*b^(19/4)) + ((-a) 
^(3/4)*d^(17/2)*atan((b^(1/4)*(d*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*385i)/ 
(128*b^(19/4))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.42 \[ \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
 

Output:

(sqrt(d)*d**8*(2310*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**3 + 6930*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**2*b*x**2 + 6930*b**(1/4)*a**(3/4)*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 + 2310*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**3*x**6 - 2310 
*b**(1/4)*a**(3/4)*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 - 6930*b**(1/4)*a**(3/4)*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 - 6930*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*b**2*x**4 - 
2310*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**3*x**6 - 1155*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* 
*3 - 3465*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**2*b*x**2 - 3465*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*b**2*x**4 - 
 1155*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + 
 sqrt(a) + sqrt(b)*x)*b**3*x**6 + 1155*b**(1/4)*a**(3/4)*sqrt(2)*log(sq...