\(\int \frac {(d x)^{23/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [650]

Optimal result

Integrand size = 30, antiderivative size = 541 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

-1547/1024*d^7*(d*x)^(9/2)/b^4/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(21/2)/b/(b 
*x^2+a)^3/((b*x^2+a)^2)^(1/2)-7/32*d^3*(d*x)^(17/2)/b^2/(b*x^2+a)^2/((b*x^ 
2+a)^2)^(1/2)-119/256*d^5*(d*x)^(13/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-1 
3923/1024*a*d^11*(d*x)^(1/2)*(b*x^2+a)/b^6/((b*x^2+a)^2)^(1/2)+13923/5120* 
d^9*(d*x)^(5/2)*(b*x^2+a)/b^5/((b*x^2+a)^2)^(1/2)-13923/4096*a^(5/4)*d^(23 
/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^(25/4)/((b*x^2+a)^2)^(1/2)+13923/4096*a^(5/4)*d^(23/2)*(b*x^2+a)*arcta 
n(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(25/4)/((b*x^2+ 
a)^2)^(1/2)+13923/4096*a^(5/4)*d^(23/2)*(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^(25/4)/((b*x^2+ 
a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.42 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^{11} \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-69615 a^5-264537 a^4 b x^2-369733 a^3 b^2 x^4-220507 a^2 b^3 x^6-43008 a b^4 x^8+2048 b^5 x^{10}\right )+69615 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+69615 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{20480 b^{25/4} \sqrt {x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \] Input:

Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(d^11*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(-69615*a^5 - 264537*a^4*b*x^2 - 369733 
*a^3*b^2*x^4 - 220507*a^2*b^3*x^6 - 43008*a*b^4*x^8 + 2048*b^5*x^10) + 696 
15*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^ 
(1/4)*b^(1/4)*Sqrt[x])] + 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTanh[(Sqr 
t[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(20480*b^(25/4)*Sqr 
t[x]*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.89, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1384, 27, 252, 252, 252, 252, 262, 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.

\(\displaystyle \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {(d x)^{23/2}}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{23/2}}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \int \frac {(d x)^{19/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \int \frac {(d x)^{15/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \int \frac {(d x)^{7/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \int \frac {(d x)^{3/2}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {a d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \int \frac {1}{b x^2+a}d\sqrt {d x}}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 d^2 \left (\frac {17 d^2 \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

Input:

Int[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

((a + b*x^2)*(-1/8*(d*(d*x)^(21/2))/(b*(a + b*x^2)^4) + (21*d^2*(-1/6*(d*( 
d*x)^(17/2))/(b*(a + b*x^2)^3) + (17*d^2*(-1/4*(d*(d*x)^(13/2))/(b*(a + b* 
x^2)^2) + (13*d^2*(-1/2*(d*(d*x)^(9/2))/(b*(a + b*x^2)) + (9*d^2*((2*d*(d* 
x)^(5/2))/(5*b) - (a*d^2*((2*d*Sqrt[d*x])/b - (2*a*d*((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])) + 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]*S 
qrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a])))/b))/b))/(4*b 
)))/(8*b)))/(12*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 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 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 = 288, normalized size of antiderivative = 0.53

method	result	size
risch	\(-\frac {2 \left (-b \,x^{2}+25 a \right ) x \,d^{12} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 b^{6} \sqrt {d x}\, \left (b \,x^{2}+a \right )}+\frac {2 a^{2} d^{13} \left (\frac {-\frac {3683 a^{3} d^{6} \sqrt {d x}}{2048}-\frac {12357 b \,d^{4} a^{2} \left (d x \right )^{\frac {5}{2}}}{2048}-\frac {14145 a \,d^{2} b^{2} \left (d x \right )^{\frac {9}{2}}}{2048}-\frac {5599 b^{3} \left (d x \right )^{\frac {13}{2}}}{2048}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16384 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{6} \left (b \,x^{2}+a \right )}\)	\(288\)
default	\(\text {Expression too large to display}\)	\(1287\)

Input:

int((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/5*(-b*x^2+25*a)*x/b^6/(d*x)^(1/2)*d^12*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+2* 
a^2/b^6*d^13*((-3683/2048*a^3*d^6*(d*x)^(1/2)-12357/2048*b*d^4*a^2*(d*x)^( 
5/2)-14145/2048*a*d^2*b^2*(d*x)^(9/2)-5599/2048*b^3*(d*x)^(13/2))/(b*d^2*x 
^2+a*d^2)^4+13923/16384*(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)))*((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 = 502, normalized size of antiderivative = 0.93 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} + 13923 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (-i \, b^{10} x^{8} - 4 i \, a b^{9} x^{6} - 6 i \, a^{2} b^{8} x^{4} - 4 i \, a^{3} b^{7} x^{2} - i \, a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} + 13923 i \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (i \, b^{10} x^{8} + 4 i \, a b^{9} x^{6} + 6 i \, a^{2} b^{8} x^{4} + 4 i \, a^{3} b^{7} x^{2} + i \, a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} - 13923 i \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} - 13923 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 4 \, {\left (2048 \, b^{5} d^{11} x^{10} - 43008 \, a b^{4} d^{11} x^{8} - 220507 \, a^{2} b^{3} d^{11} x^{6} - 369733 \, a^{3} b^{2} d^{11} x^{4} - 264537 \, a^{4} b d^{11} x^{2} - 69615 \, a^{5} d^{11}\right )} \sqrt {d x}}{20480 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} \] Input:

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas" 
)
 

Output:

1/20480*(69615*(-a^5*d^46/b^25)^(1/4)*(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8* 
x^4 + 4*a^3*b^7*x^2 + a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 + 13923*(-a^5*d^ 
46/b^25)^(1/4)*b^6) - 69615*(-a^5*d^46/b^25)^(1/4)*(-I*b^10*x^8 - 4*I*a*b^ 
9*x^6 - 6*I*a^2*b^8*x^4 - 4*I*a^3*b^7*x^2 - I*a^4*b^6)*log(13923*sqrt(d*x) 
*a*d^11 + 13923*I*(-a^5*d^46/b^25)^(1/4)*b^6) - 69615*(-a^5*d^46/b^25)^(1/ 
4)*(I*b^10*x^8 + 4*I*a*b^9*x^6 + 6*I*a^2*b^8*x^4 + 4*I*a^3*b^7*x^2 + I*a^4 
*b^6)*log(13923*sqrt(d*x)*a*d^11 - 13923*I*(-a^5*d^46/b^25)^(1/4)*b^6) - 6 
9615*(-a^5*d^46/b^25)^(1/4)*(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^ 
3*b^7*x^2 + a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 - 13923*(-a^5*d^46/b^25)^( 
1/4)*b^6) + 4*(2048*b^5*d^11*x^10 - 43008*a*b^4*d^11*x^8 - 220507*a^2*b^3* 
d^11*x^6 - 369733*a^3*b^2*d^11*x^4 - 264537*a^4*b*d^11*x^2 - 69615*a^5*d^1 
1)*sqrt(d*x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^ 
4*b^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x)**(23/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {23}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima" 
)
 

Output:

-4*a*d^(23/2)*integrate(x^(3/2)/(b^6*x^2 + a*b^5), x) + d^(23/2)*integrate 
(x^(7/2)/(b^5*x^2 + a*b^4), x) + 3683/8192*(2*sqrt(2)*a^(3/2)*arctan(1/2*s 
qrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)) 
)/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(3/2)*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)*sq 
rt(b)) + sqrt(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + 
 sqrt(a))/b^(1/4) - sqrt(2)*a^(5/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + 
 sqrt(b)*x + sqrt(a))/b^(1/4))*d^(23/2)/b^6 - 1/3072*(6925*a^2*b^3*d^(23/2 
)*x^(13/2) + 23395*a^3*b^2*d^(23/2)*x^(9/2) + 27135*a^4*b*d^(23/2)*x^(5/2) 
 + 11049*a^5*d^(23/2)*sqrt(x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4 
*a^3*b^7*x^2 + a^4*b^6) - 1/192*((617*a^2*b^4*d^(23/2)*x^5 + 1386*a^3*b^3* 
d^(23/2)*x^3 + 801*a^4*b^2*d^(23/2)*x)*x^(11/2) + 2*(519*a^3*b^3*d^(23/2)* 
x^5 + 1182*a^4*b^2*d^(23/2)*x^3 + 695*a^5*b*d^(23/2)*x)*x^(7/2) + (453*a^4 
*b^2*d^(23/2)*x^5 + 1042*a^5*b*d^(23/2)*x^3 + 621*a^6*d^(23/2)*x)*x^(3/2)) 
/(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 = 0.16 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.78 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {\frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{12} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{12} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{12} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{12} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {40 \, {\left (5599 \, \sqrt {d x} a^{2} b^{3} d^{20} x^{6} + 14145 \, \sqrt {d x} a^{3} b^{2} d^{20} x^{4} + 12357 \, \sqrt {d x} a^{4} b d^{20} x^{2} + 3683 \, \sqrt {d x} a^{5} d^{20}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {16384 \, {\left (\sqrt {d x} b^{20} d^{12} x^{2} - 25 \, \sqrt {d x} a b^{19} d^{12}\right )}}{b^{25} \mathrm {sgn}\left (b x^{2} + a\right )}}{40960 \, d} \] Input:

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

1/40960*(139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^12*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*sgn(b*x^2 + a)) + 
139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^12*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*sgn(b*x^2 + a)) + 69615*sq 
rt(2)*(a*b^3*d^2)^(1/4)*a*d^12*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) 
 + sqrt(a*d^2/b))/(b^7*sgn(b*x^2 + a)) - 69615*sqrt(2)*(a*b^3*d^2)^(1/4)*a 
*d^12*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sg 
n(b*x^2 + a)) - 40*(5599*sqrt(d*x)*a^2*b^3*d^20*x^6 + 14145*sqrt(d*x)*a^3* 
b^2*d^20*x^4 + 12357*sqrt(d*x)*a^4*b*d^20*x^2 + 3683*sqrt(d*x)*a^5*d^20)/( 
(b*d^2*x^2 + a*d^2)^4*b^6*sgn(b*x^2 + a)) + 16384*(sqrt(d*x)*b^20*d^12*x^2 
 - 25*sqrt(d*x)*a*b^19*d^12)/(b^25*sgn(b*x^2 + a)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{23/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d*x)^(23/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
 

Output:

int((d*x)^(23/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.56 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(sqrt(d)*d**11*( - 139230*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**5 - 556920* 
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**4*b*x**2 - 835380*b**(3/4)*a**(1/4)*s 
qrt(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**2*x**4 - 556920*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**3*x**6 - 139230*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s 
qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**4*x**8 + 139 
230*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**5 + 556920*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**4*b*x**2 + 835380*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*b* 
*2*x**4 + 556920*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**3*x**6 + 139230 
*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*b**4*x**8 - 69615*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** 
5 - 278460*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*s...