\(\int \frac {(d x)^{19/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [474]

Optimal result

Integrand size = 28, antiderivative size = 308 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^{19/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}} \] Output:

-1/10*d*(d*x)^(17/2)/b/(b*x^2+a)^5-17/160*d^3*(d*x)^(13/2)/b^2/(b*x^2+a)^4 
-221/1920*d^5*(d*x)^(9/2)/b^3/(b*x^2+a)^3-663/5120*d^7*(d*x)^(5/2)/b^4/(b* 
x^2+a)^2-663/4096*d^9*(d*x)^(1/2)/b^5/(b*x^2+a)-663/16384*d^(19/2)*arctan( 
1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(3/4)/b^(21/4)+66 
3/16384*d^(19/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^( 
1/2)/a^(3/4)/b^(21/4)+663/16384*d^(19/2)*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)/a^(3/4)/b^(21/4)
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.67 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^9 \sqrt {d x} \left (-4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (9945 a^4+47736 a^3 b x^2+90610 a^2 b^2 x^4+84320 a b^3 x^6+37645 b^4 x^8\right )+9945 \sqrt {2} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+9945 \sqrt {2} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{245760 a^{3/4} b^{21/4} \sqrt {x} \left (a+b x^2\right )^5} \] Input:

Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

(d^9*Sqrt[d*x]*(-4*a^(3/4)*b^(1/4)*Sqrt[x]*(9945*a^4 + 47736*a^3*b*x^2 + 9 
0610*a^2*b^2*x^4 + 84320*a*b^3*x^6 + 37645*b^4*x^8) + 9945*Sqrt[2]*(a + b* 
x^2)^5*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 
9945*Sqrt[2]*(a + b*x^2)^5*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt 
[a] + Sqrt[b]*x)]))/(245760*a^(3/4)*b^(21/4)*Sqrt[x]*(a + b*x^2)^5)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.43, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {1380, 27, 252, 252, 252, 252, 252, 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.

\(\displaystyle \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^6 \int \frac {(d x)^{19/2}}{b^6 \left (b x^2+a\right )^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{19/2}}{\left (a+b x^2\right )^6}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {17 d^2 \int \frac {(d x)^{15/2}}{\left (b x^2+a\right )^5}dx}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^4}dx}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \int \frac {(d x)^{7/2}}{\left (b x^2+a\right )^3}dx}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \int \frac {(d x)^{3/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {17 d^2 \left (\frac {13 d^2 \left (\frac {3 d^2 \left (\frac {5 d^2 \left (\frac {d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 b}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right )^2}\right )}{4 b}-\frac {d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{13/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}\)

Input:

Int[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/10*(d*(d*x)^(17/2))/(b*(a + b*x^2)^5) + (17*d^2*(-1/8*(d*(d*x)^(13/2))/ 
(b*(a + b*x^2)^4) + (13*d^2*(-1/6*(d*(d*x)^(9/2))/(b*(a + b*x^2)^3) + (3*d 
^2*(-1/4*(d*(d*x)^(5/2))/(b*(a + b*x^2)^2) + (5*d^2*(-1/2*(d*Sqrt[d*x])/(b 
*(a + b*x^2)) + (d*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)* 
Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4) 
*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt 
[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt 
[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Log[Sqrt[a]*d + Sqrt[b] 
*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/ 
4)*Sqrt[d])))/(2*Sqrt[a])))/(2*b)))/(8*b)))/(4*b)))/(16*b)))/(20*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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 14.63 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(\frac {\left (\left (-301160 a \,x^{8} b^{4}-674560 a^{2} x^{6} b^{3}-724880 a^{3} x^{4} b^{2}-381888 x^{2} a^{4} b -79560 a^{5}\right ) \sqrt {d x}+9945 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right ) d^{9}}{491520 b^{5} \left (b \,x^{2}+a \right )^{5} a}\)	\(240\)
derivativedivides	\(2 d^{11} \left (\frac {-\frac {663 a^{4} d^{8} \sqrt {d x}}{8192 b^{5}}-\frac {1989 a^{3} d^{6} \left (d x \right )^{\frac {5}{2}}}{5120 b^{4}}-\frac {9061 a^{2} d^{4} \left (d x \right )^{\frac {9}{2}}}{12288 b^{3}}-\frac {527 a \,d^{2} \left (d x \right )^{\frac {13}{2}}}{768 b^{2}}-\frac {7529 \left (d x \right )^{\frac {17}{2}}}{24576 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b^{5} a \,d^{2}}\right )\)	\(241\)
default	\(2 d^{11} \left (\frac {-\frac {663 a^{4} d^{8} \sqrt {d x}}{8192 b^{5}}-\frac {1989 a^{3} d^{6} \left (d x \right )^{\frac {5}{2}}}{5120 b^{4}}-\frac {9061 a^{2} d^{4} \left (d x \right )^{\frac {9}{2}}}{12288 b^{3}}-\frac {527 a \,d^{2} \left (d x \right )^{\frac {13}{2}}}{768 b^{2}}-\frac {7529 \left (d x \right )^{\frac {17}{2}}}{24576 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b^{5} a \,d^{2}}\right )\)	\(241\)

Input:

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/491520*((-301160*a*b^4*x^8-674560*a^2*b^3*x^6-724880*a^3*b^2*x^4-381888* 
a^4*b*x^2-79560*a^5)*(d*x)^(1/2)+9945*(a*d^2/b)^(1/4)*2^(1/2)*(b*x^2+a)^5* 
(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)*(d*x)^(1/ 
2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))+2*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/ 
b)^(1/4))/(a*d^2/b)^(1/4))))*d^9/b^5/(b*x^2+a)^5/a
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.76 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {9945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} + 663 \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 9945 \, {\left (-i \, b^{10} x^{10} - 5 i \, a b^{9} x^{8} - 10 i \, a^{2} b^{8} x^{6} - 10 i \, a^{3} b^{7} x^{4} - 5 i \, a^{4} b^{6} x^{2} - i \, a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} + 663 i \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 9945 \, {\left (i \, b^{10} x^{10} + 5 i \, a b^{9} x^{8} + 10 i \, a^{2} b^{8} x^{6} + 10 i \, a^{3} b^{7} x^{4} + 5 i \, a^{4} b^{6} x^{2} + i \, a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} - 663 i \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 9945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} - 663 \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 4 \, {\left (37645 \, b^{4} d^{9} x^{8} + 84320 \, a b^{3} d^{9} x^{6} + 90610 \, a^{2} b^{2} d^{9} x^{4} + 47736 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{245760 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \] Input:

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

1/245760*(9945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 
+ 5*a^4*b^6*x^2 + a^5*b^5)*(-d^38/(a^3*b^21))^(1/4)*log(663*sqrt(d*x)*d^9 
+ 663*(-d^38/(a^3*b^21))^(1/4)*a*b^5) - 9945*(-I*b^10*x^10 - 5*I*a*b^9*x^8 
 - 10*I*a^2*b^8*x^6 - 10*I*a^3*b^7*x^4 - 5*I*a^4*b^6*x^2 - I*a^5*b^5)*(-d^ 
38/(a^3*b^21))^(1/4)*log(663*sqrt(d*x)*d^9 + 663*I*(-d^38/(a^3*b^21))^(1/4 
)*a*b^5) - 9945*(I*b^10*x^10 + 5*I*a*b^9*x^8 + 10*I*a^2*b^8*x^6 + 10*I*a^3 
*b^7*x^4 + 5*I*a^4*b^6*x^2 + I*a^5*b^5)*(-d^38/(a^3*b^21))^(1/4)*log(663*s 
qrt(d*x)*d^9 - 663*I*(-d^38/(a^3*b^21))^(1/4)*a*b^5) - 9945*(b^10*x^10 + 5 
*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(- 
d^38/(a^3*b^21))^(1/4)*log(663*sqrt(d*x)*d^9 - 663*(-d^38/(a^3*b^21))^(1/4 
)*a*b^5) - 4*(37645*b^4*d^9*x^8 + 84320*a*b^3*d^9*x^6 + 90610*a^2*b^2*d^9* 
x^4 + 47736*a^3*b*d^9*x^2 + 9945*a^4*d^9)*sqrt(d*x))/(b^10*x^10 + 5*a*b^9* 
x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.25 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {8 \, {\left (37645 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{12} + 84320 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{14} + 90610 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{16} + 47736 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{18} + 9945 \, \sqrt {d x} a^{4} d^{20}\right )}}{b^{10} d^{10} x^{10} + 5 \, a b^{9} d^{10} x^{8} + 10 \, a^{2} b^{8} d^{10} x^{6} + 10 \, a^{3} b^{7} d^{10} x^{4} + 5 \, a^{4} b^{6} d^{10} x^{2} + a^{5} b^{5} d^{10}} - \frac {9945 \, {\left (\frac {\sqrt {2} d^{12} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{12} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{11} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{11} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{b^{5}}}{491520 \, d} \] Input:

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

-1/491520*(8*(37645*(d*x)^(17/2)*b^4*d^12 + 84320*(d*x)^(13/2)*a*b^3*d^14 
+ 90610*(d*x)^(9/2)*a^2*b^2*d^16 + 47736*(d*x)^(5/2)*a^3*b*d^18 + 9945*sqr 
t(d*x)*a^4*d^20)/(b^10*d^10*x^10 + 5*a*b^9*d^10*x^8 + 10*a^2*b^8*d^10*x^6 
+ 10*a^3*b^7*d^10*x^4 + 5*a^4*b^6*d^10*x^2 + a^5*b^5*d^10) - 9945*(sqrt(2) 
*d^12*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)* 
d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^12*log(sqrt(b)*d*x - sqrt(2)*(a*d^2 
)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2) 
*d^11*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt 
(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2 
)*d^11*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sq 
rt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)))/b^5)/d
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.14 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{6}} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{6}} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{6}} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{6}} - \frac {8 \, {\left (37645 \, \sqrt {d x} b^{4} d^{20} x^{8} + 84320 \, \sqrt {d x} a b^{3} d^{20} x^{6} + 90610 \, \sqrt {d x} a^{2} b^{2} d^{20} x^{4} + 47736 \, \sqrt {d x} a^{3} b d^{20} x^{2} + 9945 \, \sqrt {d x} a^{4} d^{20}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}}{491520 \, d} \] Input:

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/491520*(19890*sqrt(2)*(a*b^3*d^2)^(1/4)*d^10*arctan(1/2*sqrt(2)*(sqrt(2) 
*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^6) + 19890*sqrt(2)*( 
a*b^3*d^2)^(1/4)*d^10*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqr 
t(d*x))/(a*d^2/b)^(1/4))/(a*b^6) + 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*d^10*log 
(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^6) - 9945*s 
qrt(2)*(a*b^3*d^2)^(1/4)*d^10*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) 
+ sqrt(a*d^2/b))/(a*b^6) - 8*(37645*sqrt(d*x)*b^4*d^20*x^8 + 84320*sqrt(d* 
x)*a*b^3*d^20*x^6 + 90610*sqrt(d*x)*a^2*b^2*d^20*x^4 + 47736*sqrt(d*x)*a^3 
*b*d^20*x^2 + 9945*sqrt(d*x)*a^4*d^20)/((b*d^2*x^2 + a*d^2)^5*b^5))/d
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.69 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {7529\,d^{11}\,{\left (d\,x\right )}^{17/2}}{12288\,b}+\frac {9061\,a^2\,d^{15}\,{\left (d\,x\right )}^{9/2}}{6144\,b^3}+\frac {1989\,a^3\,d^{17}\,{\left (d\,x\right )}^{5/2}}{2560\,b^4}+\frac {663\,a^4\,d^{19}\,\sqrt {d\,x}}{4096\,b^5}+\frac {527\,a\,d^{13}\,{\left (d\,x\right )}^{13/2}}{384\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {663\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{3/4}\,b^{21/4}}-\frac {663\,d^{19/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{3/4}\,b^{21/4}} \] Input:

int((d*x)^(19/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
 

Output:

- ((7529*d^11*(d*x)^(17/2))/(12288*b) + (9061*a^2*d^15*(d*x)^(9/2))/(6144* 
b^3) + (1989*a^3*d^17*(d*x)^(5/2))/(2560*b^4) + (663*a^4*d^19*(d*x)^(1/2)) 
/(4096*b^5) + (527*a*d^13*(d*x)^(13/2))/(384*b^2))/(a^5*d^10 + b^5*d^10*x^ 
10 + 5*a^4*b*d^10*x^2 + 5*a*b^4*d^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^ 
3*d^10*x^6) - (663*d^(19/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2) 
)))/(8192*(-a)^(3/4)*b^(21/4)) - (663*d^(19/2)*atanh((b^(1/4)*(d*x)^(1/2)) 
/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(3/4)*b^(21/4))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 995, normalized size of antiderivative = 3.23 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

(sqrt(d)*d**9*( - 19890*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 - 99450*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 - 198900*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 - 198900*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 - 99450*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt( 
2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**4*x**8 - 19890*b 
**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt( 
b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**5*x**10 + 19890*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 + 99450*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 + 1 
98900*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 + 198900*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 + 99450*b**(3/4)*a**(1/4)*sqrt(2)*a 
tan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr 
t(2)))*a*b**4*x**8 + 19890*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**...