\(\int \frac {(d x)^{11/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [478]

Optimal result

Integrand size = 28, antiderivative size = 314 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}} \] Output:

-1/10*d*(d*x)^(9/2)/b/(b*x^2+a)^5-9/160*d^3*(d*x)^(5/2)/b^2/(b*x^2+a)^4-3/ 
128*d^5*(d*x)^(1/2)/b^3/(b*x^2+a)^3+3/1024*d^5*(d*x)^(1/2)/a/b^3/(b*x^2+a) 
^2+21/4096*d^5*(d*x)^(1/2)/a^2/b^3/(b*x^2+a)-63/16384*d^(11/2)*arctan(1-2^ 
(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(11/4)/b^(13/4)+63/16 
384*d^(11/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2) 
/a^(11/4)/b^(13/4)+63/16384*d^(11/2)*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^(11/4)/b^(13/4)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.59 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^5 \sqrt {d x} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-315 a^4-1512 a^3 b x^2-2870 a^2 b^2 x^4+480 a b^3 x^6+105 b^4 x^8\right )}{\left (a+b x^2\right )^5}-315 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+315 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{11/4} b^{13/4} \sqrt {x}} \] Input:

Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

(d^5*Sqrt[d*x]*((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-315*a^4 - 1512*a^3*b*x^2 - 28 
70*a^2*b^2*x^4 + 480*a*b^3*x^6 + 105*b^4*x^8))/(a + b*x^2)^5 - 315*Sqrt[2] 
*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 315*Sqr 
t[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(8 
1920*a^(11/4)*b^(13/4)*Sqrt[x])
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.41, 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, 253, 253, 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.

\(\displaystyle \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^6 \int \frac {(d x)^{11/2}}{b^6 \left (b x^2+a\right )^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{11/2}}{\left (a+b x^2\right )^6}dx\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 d^2 \int \frac {(d x)^{7/2}}{\left (b x^2+a\right )^5}dx}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^3}dx}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {9 d^2 \left (\frac {5 d^2 \left (\frac {d^2 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d (d x)^{5/2}}{8 b \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}\)

Input:

Int[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/10*(d*(d*x)^(9/2))/(b*(a + b*x^2)^5) + (9*d^2*(-1/8*(d*(d*x)^(5/2))/(b* 
(a + b*x^2)^4) + (5*d^2*(-1/6*(d*Sqrt[d*x])/(b*(a + b*x^2)^3) + (d^2*(Sqrt 
[d*x]/(4*a*d*(a + b*x^2)^2) + (7*(Sqrt[d*x]/(2*a*d*(a + b*x^2)) + (3*((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)*Sqr 
t[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*a*d)))/(8*a)))/(12*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 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && 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 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 = 14.66 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {63 a^{2} d^{4} \sqrt {d x}}{8192 b^{3}}-\frac {189 a \,d^{2} \left (d x \right )^{\frac {5}{2}}}{5120 b^{2}}-\frac {287 \left (d x \right )^{\frac {9}{2}}}{4096 b}+\frac {3 \left (d x \right )^{\frac {13}{2}}}{256 a \,d^{2}}+\frac {21 b \left (d x \right )^{\frac {17}{2}}}{8192 a^{2} d^{4}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{3} d^{6} b^{3}}\right )\)	\(236\)
default	\(2 d^{11} \left (\frac {-\frac {63 a^{2} d^{4} \sqrt {d x}}{8192 b^{3}}-\frac {189 a \,d^{2} \left (d x \right )^{\frac {5}{2}}}{5120 b^{2}}-\frac {287 \left (d x \right )^{\frac {9}{2}}}{4096 b}+\frac {3 \left (d x \right )^{\frac {13}{2}}}{256 a \,d^{2}}+\frac {21 b \left (d x \right )^{\frac {17}{2}}}{8192 a^{2} d^{4}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{3} d^{6} b^{3}}\right )\)	\(236\)
pseudoelliptic	\(\frac {d^{5} \left (\left (840 a \,x^{8} b^{4}+3840 a^{2} x^{6} b^{3}-22960 a^{3} x^{4} b^{2}-12096 x^{2} a^{4} b -2520 a^{5}\right ) \sqrt {d x}+315 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{163840 a^{3} b^{3} \left (b \,x^{2}+a \right )^{5}}\)	\(240\)

Input:

int((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

2*d^11*((-63/8192*a^2*d^4/b^3*(d*x)^(1/2)-189/5120*a*d^2/b^2*(d*x)^(5/2)-2 
87/4096/b*(d*x)^(9/2)+3/256/a/d^2*(d*x)^(13/2)+21/8192/a^2/d^4*b*(d*x)^(17 
/2))/(b*d^2*x^2+a*d^2)^5+63/65536/a^3/d^6/b^3*(a*d^2/b)^(1/4)*2^(1/2)*(ln( 
(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^( 
1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4 
)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.83 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {315 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (63 \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\left (-i \, a^{2} b^{8} x^{10} - 5 i \, a^{3} b^{7} x^{8} - 10 i \, a^{4} b^{6} x^{6} - 10 i \, a^{5} b^{5} x^{4} - 5 i \, a^{6} b^{4} x^{2} - i \, a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (63 i \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\left (i \, a^{2} b^{8} x^{10} + 5 i \, a^{3} b^{7} x^{8} + 10 i \, a^{4} b^{6} x^{6} + 10 i \, a^{5} b^{5} x^{4} + 5 i \, a^{6} b^{4} x^{2} + i \, a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (-63 i \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (-63 \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) + 4 \, {\left (105 \, b^{4} d^{5} x^{8} + 480 \, a b^{3} d^{5} x^{6} - 2870 \, a^{2} b^{2} d^{5} x^{4} - 1512 \, a^{3} b d^{5} x^{2} - 315 \, a^{4} d^{5}\right )} \sqrt {d x}}{81920 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )}} \] Input:

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

1/81920*(315*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x 
^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d^22/(a^11*b^13))^(1/4)*log(63*a^3*b^3*(-d 
^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) - 315*(-I*a^2*b^8*x^10 - 5*I*a^ 
3*b^7*x^8 - 10*I*a^4*b^6*x^6 - 10*I*a^5*b^5*x^4 - 5*I*a^6*b^4*x^2 - I*a^7* 
b^3)*(-d^22/(a^11*b^13))^(1/4)*log(63*I*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) 
+ 63*sqrt(d*x)*d^5) - 315*(I*a^2*b^8*x^10 + 5*I*a^3*b^7*x^8 + 10*I*a^4*b^6 
*x^6 + 10*I*a^5*b^5*x^4 + 5*I*a^6*b^4*x^2 + I*a^7*b^3)*(-d^22/(a^11*b^13)) 
^(1/4)*log(-63*I*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) - 3 
15*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6 
*b^4*x^2 + a^7*b^3)*(-d^22/(a^11*b^13))^(1/4)*log(-63*a^3*b^3*(-d^22/(a^11 
*b^13))^(1/4) + 63*sqrt(d*x)*d^5) + 4*(105*b^4*d^5*x^8 + 480*a*b^3*d^5*x^6 
 - 2870*a^2*b^2*d^5*x^4 - 1512*a^3*b*d^5*x^2 - 315*a^4*d^5)*sqrt(d*x))/(a^ 
2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x 
^2 + a^7*b^3)
 

Sympy [F]

\[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{\frac {11}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \] Input:

integrate((d*x)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

Integral((d*x)**(11/2)/(a + b*x**2)**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.25 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (105 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{8} + 480 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{10} - 2870 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{12} - 1512 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{14} - 315 \, \sqrt {d x} a^{4} d^{16}\right )}}{a^{2} b^{8} d^{10} x^{10} + 5 \, a^{3} b^{7} d^{10} x^{8} + 10 \, a^{4} b^{6} d^{10} x^{6} + 10 \, a^{5} b^{5} d^{10} x^{4} + 5 \, a^{6} b^{4} d^{10} x^{2} + a^{7} b^{3} d^{10}} + \frac {315 \, {\left (\frac {\sqrt {2} d^{8} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{8} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{7} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{7} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{2} b^{3}}}{163840 \, d} \] Input:

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

1/163840*(8*(105*(d*x)^(17/2)*b^4*d^8 + 480*(d*x)^(13/2)*a*b^3*d^10 - 2870 
*(d*x)^(9/2)*a^2*b^2*d^12 - 1512*(d*x)^(5/2)*a^3*b*d^14 - 315*sqrt(d*x)*a^ 
4*d^16)/(a^2*b^8*d^10*x^10 + 5*a^3*b^7*d^10*x^8 + 10*a^4*b^6*d^10*x^6 + 10 
*a^5*b^5*d^10*x^4 + 5*a^6*b^4*d^10*x^2 + a^7*b^3*d^10) + 315*(sqrt(2)*d^8* 
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^8*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^7*ar 
ctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqr 
t(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2)*d^7*ar 
ctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sq 
rt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)))/(a^2*b^3))/d
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.13 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{6} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4}} + \frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{6} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4}} + \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{6} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{4}} - \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{6} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{4}} + \frac {8 \, {\left (105 \, \sqrt {d x} b^{4} d^{16} x^{8} + 480 \, \sqrt {d x} a b^{3} d^{16} x^{6} - 2870 \, \sqrt {d x} a^{2} b^{2} d^{16} x^{4} - 1512 \, \sqrt {d x} a^{3} b d^{16} x^{2} - 315 \, \sqrt {d x} a^{4} d^{16}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{2} b^{3}}}{163840 \, d} \] Input:

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

1/163840*(630*sqrt(2)*(a*b^3*d^2)^(1/4)*d^6*arctan(1/2*sqrt(2)*(sqrt(2)*(a 
*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^3*b^4) + 630*sqrt(2)*(a*b 
^3*d^2)^(1/4)*d^6*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d* 
x))/(a*d^2/b)^(1/4))/(a^3*b^4) + 315*sqrt(2)*(a*b^3*d^2)^(1/4)*d^6*log(d*x 
 + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^3*b^4) - 315*sqrt 
(2)*(a*b^3*d^2)^(1/4)*d^6*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq 
rt(a*d^2/b))/(a^3*b^4) + 8*(105*sqrt(d*x)*b^4*d^16*x^8 + 480*sqrt(d*x)*a*b 
^3*d^16*x^6 - 2870*sqrt(d*x)*a^2*b^2*d^16*x^4 - 1512*sqrt(d*x)*a^3*b*d^16* 
x^2 - 315*sqrt(d*x)*a^4*d^16)/((b*d^2*x^2 + a*d^2)^5*a^2*b^3))/d
 

Mupad [B] (verification not implemented)

Time = 17.76 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.66 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {287\,d^{11}\,{\left (d\,x\right )}^{9/2}}{2048\,b}-\frac {3\,d^9\,{\left (d\,x\right )}^{13/2}}{128\,a}+\frac {63\,a^2\,d^{15}\,\sqrt {d\,x}}{4096\,b^3}+\frac {189\,a\,d^{13}\,{\left (d\,x\right )}^{5/2}}{2560\,b^2}-\frac {21\,b\,d^7\,{\left (d\,x\right )}^{17/2}}{4096\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {63\,d^{11/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{11/4}\,b^{13/4}}-\frac {63\,d^{11/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{11/4}\,b^{13/4}} \] Input:

int((d*x)^(11/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
 

Output:

- ((287*d^11*(d*x)^(9/2))/(2048*b) - (3*d^9*(d*x)^(13/2))/(128*a) + (63*a^ 
2*d^15*(d*x)^(1/2))/(4096*b^3) + (189*a*d^13*(d*x)^(5/2))/(2560*b^2) - (21 
*b*d^7*(d*x)^(17/2))/(4096*a^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) - (63* 
d^(11/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(11/ 
4)*b^(13/4)) - (63*d^(11/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2 
))))/(8192*(-a)^(11/4)*b^(13/4))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 995, normalized size of antiderivative = 3.17 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

(sqrt(d)*d**5*( - 630*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq 
rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5 - 3150*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 - 6300*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**3*b**2*x**4 - 6300*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 - 3150*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*s 
qrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**4*x**8 - 630*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 + 630*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 + 3150*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 + 6300*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 + 6300*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**2*b**3*x**6 + 3150*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 
 + 630*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sq...