\(\int \frac {(d x)^{19/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [652]

Optimal result

Integrand size = 30, antiderivative size = 494 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:

-663/1024*d^7*(d*x)^(5/2)/b^4/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(17/2)/b/(b* 
x^2+a)^3/((b*x^2+a)^2)^(1/2)-17/96*d^3*(d*x)^(13/2)/b^2/(b*x^2+a)^2/((b*x^ 
2+a)^2)^(1/2)-221/768*d^5*(d*x)^(9/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)+33 
15/1024*d^9*(d*x)^(1/2)*(b*x^2+a)/b^5/((b*x^2+a)^2)^(1/2)+3315/4096*a^(1/4 
)*d^(19/2)*(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^(21/4)/((b*x^2+a)^2)^(1/2)-3315/4096*a^(1/4)*d^(19/2)*(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^(21/4)/(( 
b*x^2+a)^2)^(1/2)-3315/4096*a^(1/4)*d^(19/2)*(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^(21/4)/((b 
*x^2+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.44 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^9 \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (9945 a^4+37791 a^3 b x^2+52819 a^2 b^2 x^4+31501 a b^3 x^6+6144 b^4 x^8\right )-9945 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-9945 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 b^{21/4} \sqrt {x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \] Input:

Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(d^9*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(9945*a^4 + 37791*a^3*b*x^2 + 52819*a^2* 
b^2*x^4 + 31501*a*b^3*x^6 + 6144*b^4*x^8) - 9945*Sqrt[2]*a^(1/4)*(a + b*x^ 
2)^4*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 99 
45*Sqrt[2]*a^(1/4)*(a + b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]) 
/(Sqrt[a] + Sqrt[b]*x)]))/(12288*b^(21/4)*Sqrt[x]*(a + b*x^2)^3*Sqrt[(a + 
b*x^2)^2])
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 459, 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, 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.

\(\displaystyle \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {(d x)^{19/2}}{b^5 \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{19/2}}{\left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

((a + b*x^2)*(-1/8*(d*(d*x)^(17/2))/(b*(a + b*x^2)^4) + (17*d^2*(-1/6*(d*( 
d*x)^(13/2))/(b*(a + b*x^2)^3) + (13*d^2*(-1/4*(d*(d*x)^(9/2))/(b*(a + b*x 
^2)^2) + (9*d^2*(-1/2*(d*(d*x)^(5/2))/(b*(a + b*x^2)) + (5*d^2*((2*d*Sqrt[ 
d*x])/b - (2*a*d*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sq 
rt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*S 
qrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a 
]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d 
]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Log[Sqrt[a]*d + Sqrt[b]*d 
*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4) 
*Sqrt[d])))/(2*Sqrt[a])))/b))/(4*b)))/(8*b)))/(12*b)))/(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 = 0.20 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {2 x \,d^{10} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{5} \sqrt {d x}\, \left (b \,x^{2}+a \right )}-\frac {2 a \,d^{11} \left (\frac {-\frac {1267 a^{3} d^{6} \sqrt {d x}}{2048}-\frac {4405 b \,d^{4} a^{2} \left (d x \right )^{\frac {5}{2}}}{2048}-\frac {15955 a \,d^{2} b^{2} \left (d x \right )^{\frac {9}{2}}}{6144}-\frac {6925 b^{3} \left (d x \right )^{\frac {13}{2}}}{6144}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {3315 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16384 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{5} \left (b \,x^{2}+a \right )}\)	\(276\)
default	\(\text {Expression too large to display}\)	\(1202\)

Input:

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/b^5*x/(d*x)^(1/2)*d^10*((b*x^2+a)^2)^(1/2)/(b*x^2+a)-2*a/b^5*d^11*((-126 
7/2048*a^3*d^6*(d*x)^(1/2)-4405/2048*b*d^4*a^2*(d*x)^(5/2)-15955/6144*a*d^ 
2*b^2*(d*x)^(9/2)-6925/6144*b^3*(d*x)^(13/2))/(b*d^2*x^2+a*d^2)^4+3315/163 
84*(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 = 0.09 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.95 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} + 3315 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) + 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (i \, b^{9} x^{8} + 4 i \, a b^{8} x^{6} + 6 i \, a^{2} b^{7} x^{4} + 4 i \, a^{3} b^{6} x^{2} + i \, a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} + 3315 i \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) + 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (-i \, b^{9} x^{8} - 4 i \, a b^{8} x^{6} - 6 i \, a^{2} b^{7} x^{4} - 4 i \, a^{3} b^{6} x^{2} - i \, a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} - 3315 i \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} - 3315 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 4 \, {\left (6144 \, b^{4} d^{9} x^{8} + 31501 \, a b^{3} d^{9} x^{6} + 52819 \, a^{2} b^{2} d^{9} x^{4} + 37791 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{12288 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \] Input:

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas" 
)
 

Output:

-1/12288*(9945*(-a*d^38/b^21)^(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(3315*sqrt(d*x)*d^9 + 3315*(-a*d^38/b^21)^( 
1/4)*b^5) + 9945*(-a*d^38/b^21)^(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(3315*sqrt(d*x)*d^9 + 3315*I*(- 
a*d^38/b^21)^(1/4)*b^5) + 9945*(-a*d^38/b^21)^(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(3315*sqrt(d*x)* 
d^9 - 3315*I*(-a*d^38/b^21)^(1/4)*b^5) - 9945*(-a*d^38/b^21)^(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(3315*sqrt(d 
*x)*d^9 - 3315*(-a*d^38/b^21)^(1/4)*b^5) - 4*(6144*b^4*d^9*x^8 + 31501*a*b 
^3*d^9*x^6 + 52819*a^2*b^2*d^9*x^4 + 37791*a^3*b*d^9*x^2 + 9945*a^4*d^9)*s 
qrt(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. \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {19}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima" 
)
 

Output:

d^(19/2)*integrate(x^(3/2)/(b^5*x^2 + a*b^4), x) - 1267/8192*(2*sqrt(2)*sq 
rt(a)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqr 
t(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*sqrt(a)*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(2)*a^(1/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqr 
t(x) + sqrt(b)*x + sqrt(a))/b^(1/4) - sqrt(2)*a^(1/4)*log(-sqrt(2)*a^(1/4) 
*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))*d^(19/2)/b^5 + 1/3072*(18 
53*a*b^3*d^(19/2)*x^(13/2) + 6515*a^2*b^2*d^(19/2)*x^(9/2) + 8079*a^3*b*d^ 
(19/2)*x^(5/2) + 3801*a^4*d^(19/2)*sqrt(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) + 1/192*((317*a*b^4*d^(19/2)*x^5 + 738 
*a^2*b^3*d^(19/2)*x^3 + 453*a^3*b^2*d^(19/2)*x)*x^(11/2) + 2*(243*a^2*b^3* 
d^(19/2)*x^5 + 582*a^3*b^2*d^(19/2)*x^3 + 371*a^4*b*d^(19/2)*x)*x^(7/2) + 
(201*a^3*b^2*d^(19/2)*x^5 + 490*a^4*b*d^(19/2)*x^3 + 321*a^5*d^(19/2)*x)*x 
^(3/2))/(a^3*b^7*x^6 + 3*a^4*b^6*x^4 + 3*a^5*b^5*x^2 + a^6*b^4 + (b^10*x^6 
 + 3*a*b^9*x^4 + 3*a^2*b^8*x^2 + a^3*b^7)*x^6 + 3*(a*b^9*x^6 + 3*a^2*b^8*x 
^4 + 3*a^3*b^7*x^2 + a^4*b^6)*x^4 + 3*(a^2*b^8*x^6 + 3*a^3*b^7*x^4 + 3*a^4 
*b^6*x^2 + a^5*b^5)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.80 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{10} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {49152 \, \sqrt {d x} d^{10}}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {8 \, {\left (6925 \, \sqrt {d x} a b^{3} d^{18} x^{6} + 15955 \, \sqrt {d x} a^{2} b^{2} d^{18} x^{4} + 13215 \, \sqrt {d x} a^{3} b d^{18} x^{2} + 3801 \, \sqrt {d x} a^{4} d^{18}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \mathrm {sgn}\left (b x^{2} + a\right )}}{24576 \, d} \] Input:

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

-1/24576*(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))/(b^6*sgn(b*x^2 + a)) + 19 
890*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))/(b^6*sgn(b*x^2 + a)) + 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))/(b^6*sgn(b*x^2 + a)) - 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))/(b^6*sgn(b*x^2 + a 
)) - 49152*sqrt(d*x)*d^10/(b^5*sgn(b*x^2 + a)) - 8*(6925*sqrt(d*x)*a*b^3*d 
^18*x^6 + 15955*sqrt(d*x)*a^2*b^2*d^18*x^4 + 13215*sqrt(d*x)*a^3*b*d^18*x^ 
2 + 3801*sqrt(d*x)*a^4*d^18)/((b*d^2*x^2 + a*d^2)^4*b^5*sgn(b*x^2 + a)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{19/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d*x)^(19/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
 

Output:

int((d*x)^(19/2)/(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 = 831, normalized size of antiderivative = 1.68 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(sqrt(d)*d**9*(19890*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**4 + 79560*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*x**2 + 119340*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)*s 
qrt(2)))*a**2*b**2*x**4 + 79560*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**3*x 
**6 + 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**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)))*a**4 - 79560*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*x**2 - 119340*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**2*x**4 - 79560 
*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**3*x**6 - 19890*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)))*b**4*x**8 + 9945*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b 
**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**4 + 39780*b**(3/4)*a**( 
1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(...