\(\int \frac {(d x)^{21/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [651]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [C] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [A] (verification not implemented)
Mupad [F(-1)]
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 30, antiderivative size = 494 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} b^{23/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

-1045/1024*d^7*(d*x)^(7/2)/b^4/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(19/2)/b/(b 
*x^2+a)^3/((b*x^2+a)^2)^(1/2)-19/96*d^3*(d*x)^(15/2)/b^2/(b*x^2+a)^2/((b*x 
^2+a)^2)^(1/2)-95/256*d^5*(d*x)^(11/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)+7 
315/3072*d^9*(d*x)^(3/2)*(b*x^2+a)/b^5/((b*x^2+a)^2)^(1/2)+7315/4096*a^(3/ 
4)*d^(21/2)*(b*x^2+a)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2) 
)*2^(1/2)/b^(23/4)/((b*x^2+a)^2)^(1/2)-7315/4096*a^(3/4)*d^(21/2)*(b*x^2+a 
)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(23/4)/( 
(b*x^2+a)^2)^(1/2)+7315/4096*a^(3/4)*d^(21/2)*(b*x^2+a)*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^(23/4)/(( 
b*x^2+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.44 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^9 (d x)^{3/2} \left (4 b^{3/4} x^{3/2} \left (7315 a^4+26125 a^3 b x^2+33345 a^2 b^2 x^4+16967 a b^3 x^6+2048 b^4 x^8\right )-21945 \sqrt {2} a^{3/4} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21945 \sqrt {2} a^{3/4} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 b^{23/4} x^{3/2} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \] Input:

Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(d^9*(d*x)^(3/2)*(4*b^(3/4)*x^(3/2)*(7315*a^4 + 26125*a^3*b*x^2 + 33345*a^ 
2*b^2*x^4 + 16967*a*b^3*x^6 + 2048*b^4*x^8) - 21945*Sqrt[2]*a^(3/4)*(a + b 
*x^2)^4*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 
 21945*Sqrt[2]*a^(3/4)*(a + b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt 
[x])/(Sqrt[a] + Sqrt[b]*x)]))/(12288*b^(23/4)*x^(3/2)*(a + b*x^2)^3*Sqrt[( 
a + b*x^2)^2])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.93, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {1384, 27, 252, 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.

\(\displaystyle \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {(d x)^{21/2}}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{21/2}}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \int \frac {(d x)^{17/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \int \frac {(d x)^{13/2}}{\left (b x^2+a\right )^3}dx}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \int \frac {(d x)^{9/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \int \frac {(d x)^{5/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {a d^2 \int \frac {\sqrt {d x}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {19 d^2 \left (\frac {5 d^2 \left (\frac {11 d^2 \left (\frac {7 d^2 \left (\frac {2 d (d x)^{3/2}}{3 b}-\frac {2 a d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{b}\right )}{4 b}-\frac {d (d x)^{7/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{11/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{15/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{19/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

Input:

Int[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

((a + b*x^2)*(-1/8*(d*(d*x)^(19/2))/(b*(a + b*x^2)^4) + (19*d^2*(-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 + (Sqrt[2]*b^( 
1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*S 
qrt[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)))/(16*b)))/Sqrt[a^2 + 
 2*a*b*x^2 + b^2*x^4]
 

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 1384
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac 
Part[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 - 1/2] && NeQ[u, x^(n 
- 1)] && NeQ[u, x^(2*n - 1)] &&  !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
 

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 = 0.21 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.56

method result size
risch \(\frac {2 x^{2} d^{11} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b^{5} \sqrt {d x}\, \left (b \,x^{2}+a \right )}-\frac {a \left (\frac {-\frac {5267 a^{3} d^{6} \left (d x \right )^{\frac {3}{2}}}{3072}-\frac {17933 a^{2} d^{4} b \left (d x \right )^{\frac {7}{2}}}{3072}-\frac {7019 a \,d^{2} b^{2} \left (d x \right )^{\frac {11}{2}}}{1024}-\frac {2925 b^{3} \left (d x \right )^{\frac {15}{2}}}{1024}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {7315 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8192 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) d^{11} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{5} \left (b \,x^{2}+a \right )}\) \(276\)
default \(\text {Expression too large to display}\) \(1171\)

Input:

int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*x^2/b^5/(d*x)^(1/2)*d^11*((b*x^2+a)^2)^(1/2)/(b*x^2+a)-a/b^5*(2*(-5267 
/6144*a^3*d^6*(d*x)^(3/2)-17933/6144*a^2*d^4*b*(d*x)^(7/2)-7019/2048*a*d^2 
*b^2*(d*x)^(11/2)-2925/2048*b^3*(d*x)^(15/2))/(b*d^2*x^2+a*d^2)^4+7315/819 
2/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)))*d^11*((b*x^2+a)^2)^(1/2)/(b*x^2+a)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.01 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} + 391419980875 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) + 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (-i \, b^{9} x^{8} - 4 i \, a b^{8} x^{6} - 6 i \, a^{2} b^{7} x^{4} - 4 i \, a^{3} b^{6} x^{2} - i \, a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} + 391419980875 i \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) + 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (i \, b^{9} x^{8} + 4 i \, a b^{8} x^{6} + 6 i \, a^{2} b^{7} x^{4} + 4 i \, a^{3} b^{6} x^{2} + i \, a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} - 391419980875 i \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) - 21945 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt {d x} a^{2} d^{31} - 391419980875 \, \left (-\frac {a^{3} d^{42}}{b^{23}}\right )^{\frac {3}{4}} b^{17}\right ) - 4 \, {\left (2048 \, b^{4} d^{10} x^{9} + 16967 \, a b^{3} d^{10} x^{7} + 33345 \, a^{2} b^{2} d^{10} x^{5} + 26125 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt {d x}}{12288 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \] Input:

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas" 
)
 

Output:

-1/12288*(21945*(-a^3*d^42/b^23)^(1/4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7* 
x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*log(391419980875*sqrt(d*x)*a^2*d^31 + 39141 
9980875*(-a^3*d^42/b^23)^(3/4)*b^17) + 21945*(-a^3*d^42/b^23)^(1/4)*(-I*b^ 
9*x^8 - 4*I*a*b^8*x^6 - 6*I*a^2*b^7*x^4 - 4*I*a^3*b^6*x^2 - I*a^4*b^5)*log 
(391419980875*sqrt(d*x)*a^2*d^31 + 391419980875*I*(-a^3*d^42/b^23)^(3/4)*b 
^17) + 21945*(-a^3*d^42/b^23)^(1/4)*(I*b^9*x^8 + 4*I*a*b^8*x^6 + 6*I*a^2*b 
^7*x^4 + 4*I*a^3*b^6*x^2 + I*a^4*b^5)*log(391419980875*sqrt(d*x)*a^2*d^31 
- 391419980875*I*(-a^3*d^42/b^23)^(3/4)*b^17) - 21945*(-a^3*d^42/b^23)^(1/ 
4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*log(3 
91419980875*sqrt(d*x)*a^2*d^31 - 391419980875*(-a^3*d^42/b^23)^(3/4)*b^17) 
 - 4*(2048*b^4*d^10*x^9 + 16967*a*b^3*d^10*x^7 + 33345*a^2*b^2*d^10*x^5 + 
26125*a^3*b*d^10*x^3 + 7315*a^4*d^10*x)*sqrt(d*x))/(b^9*x^8 + 4*a*b^8*x^6 
+ 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x)**(21/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {21}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima" 
)
 

Output:

-4*a*d^(21/2)*integrate(sqrt(x)/(b^6*x^2 + a*b^5), x) + d^(21/2)*integrate 
(x^(5/2)/(b^5*x^2 + a*b^4), x) + 2925/8192*a*d^(21/2)*(2*sqrt(2)*arctan(1/ 
2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt( 
b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt 
(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt 
(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt 
(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)* 
sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/b^5 + 1/3072*(8775*a*b^3 
*d^(21/2)*x^(15/2) + 29649*a^2*b^2*d^(21/2)*x^(11/2) + 34285*a^3*b*d^(21/2 
)*x^(7/2) + 13795*a^4*d^(21/2)*x^(3/2))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7 
*x^4 + 4*a^3*b^6*x^2 + a^4*b^5) - 1/192*((537*a^2*b^4*d^(21/2)*x^5 + 1210* 
a^3*b^3*d^(21/2)*x^3 + 705*a^4*b^2*d^(21/2)*x)*x^(9/2) + 2*(443*a^3*b^3*d^ 
(21/2)*x^5 + 1014*a^4*b^2*d^(21/2)*x^3 + 603*a^5*b*d^(21/2)*x)*x^(5/2) + ( 
381*a^4*b^2*d^(21/2)*x^5 + 882*a^5*b*d^(21/2)*x^3 + 533*a^6*d^(21/2)*x)*sq 
rt(x))/(a^3*b^8*x^6 + 3*a^4*b^7*x^4 + 3*a^5*b^6*x^2 + a^6*b^5 + (b^11*x^6 
+ 3*a*b^10*x^4 + 3*a^2*b^9*x^2 + a^3*b^8)*x^6 + 3*(a*b^10*x^6 + 3*a^2*b^9* 
x^4 + 3*a^3*b^8*x^2 + a^4*b^7)*x^4 + 3*(a^2*b^9*x^6 + 3*a^3*b^8*x^4 + 3*a^ 
4*b^7*x^2 + a^5*b^6)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.80 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{24576} \, d^{10} {\left (\frac {16384 \, \sqrt {d x} x}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {43890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{8} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {43890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{8} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {21945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{8} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {21945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{8} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (8775 \, \sqrt {d x} a b^{3} d^{8} x^{7} + 21057 \, \sqrt {d x} a^{2} b^{2} d^{8} x^{5} + 17933 \, \sqrt {d x} a^{3} b d^{8} x^{3} + 5267 \, \sqrt {d x} a^{4} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \] Input:

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

1/24576*d^10*(16384*sqrt(d*x)*x/(b^5*sgn(b*x^2 + a)) - 43890*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^8*d*sgn(b*x^2 + a)) - 43890*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^8*d*sgn(b*x^2 + a)) + 21945*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^8*d*sgn(b*x^2 + a)) - 2194 
5*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^8*d*sgn(b*x^2 + a)) + 8*(8775*sqrt(d*x)*a*b^3*d^8*x^7 + 
21057*sqrt(d*x)*a^2*b^2*d^8*x^5 + 17933*sqrt(d*x)*a^3*b*d^8*x^3 + 5267*sqr 
t(d*x)*a^4*d^8*x)/((b*d^2*x^2 + a*d^2)^4*b^5*sgn(b*x^2 + a)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{21/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d*x)^(21/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
 

Output:

int((d*x)^(21/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.68 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(sqrt(d)*d**10*(43890*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq 
rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4 + 175560*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*b*x**2 + 263340*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**2*x**4 + 175560*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** 
3*x**6 + 43890*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**4*x**8 - 43890*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**4 - 175560*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*b*x**2 - 263340*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq 
rt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**2*x**4 - 1 
75560*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**3*x**6 - 43890*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**4*x**8 - 21945*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)*a**4 - 87780*b**(1/4 
)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) +...