\(\int \frac {(d x)^{7/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [480]

Optimal result

Integrand size = 28, antiderivative size = 317 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}-\frac {77 d^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}} \] Output:

-1/10*d*(d*x)^(5/2)/b/(b*x^2+a)^5-1/32*d^3*(d*x)^(1/2)/b^2/(b*x^2+a)^4+1/3 
84*d^3*(d*x)^(1/2)/a/b^2/(b*x^2+a)^3+11/3072*d^3*(d*x)^(1/2)/a^2/b^2/(b*x^ 
2+a)^2+77/12288*d^3*(d*x)^(1/2)/a^3/b^2/(b*x^2+a)-77/16384*d^(7/2)*arctan( 
1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(15/4)/b^(9/4)+77 
/16384*d^(7/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/ 
2)/a^(15/4)/b^(9/4)+77/16384*d^(7/2)*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^(15/4)/b^(9/4)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.59 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^3 \sqrt {d x} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-1155 a^4-5544 a^3 b x^2+3130 a^2 b^2 x^4+1760 a b^3 x^6+385 b^4 x^8\right )}{\left (a+b x^2\right )^5}-1155 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+1155 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{245760 a^{15/4} b^{9/4} \sqrt {x}} \] Input:

Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

(d^3*Sqrt[d*x]*((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-1155*a^4 - 5544*a^3*b*x^2 + 3 
130*a^2*b^2*x^4 + 1760*a*b^3*x^6 + 385*b^4*x^8))/(a + b*x^2)^5 - 1155*Sqrt 
[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 1155 
*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]) 
)/(245760*a^(15/4)*b^(9/4)*Sqrt[x])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.40, 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, 253, 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.

\(\displaystyle \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle b^6 \int \frac {(d x)^{7/2}}{b^6 \left (b x^2+a\right )^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d x)^{7/2}}{\left (a+b x^2\right )^6}dx\)

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^3}dx}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\right )}{16 b}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^4}\right )}{4 b}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}\)

Input:

Int[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

-1/10*(d*(d*x)^(5/2))/(b*(a + b*x^2)^5) + (d^2*(-1/8*(d*Sqrt[d*x])/(b*(a + 
 b*x^2)^4) + (d^2*(Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*(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)*S 
qrt[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*a*d)) 
)/(8*a)))/(12*a)))/(16*b)))/(4*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 = 14.57 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {77 a \,d^{2} \sqrt {d x}}{8192 b^{2}}-\frac {231 \left (d x \right )^{\frac {5}{2}}}{5120 b}+\frac {313 \left (d x \right )^{\frac {9}{2}}}{12288 a \,d^{2}}+\frac {11 b \left (d x \right )^{\frac {13}{2}}}{768 a^{2} d^{4}}+\frac {77 b^{2} \left (d x \right )^{\frac {17}{2}}}{24576 a^{3} d^{6}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{4} d^{8} b^{2}}\right )\)	\(236\)
default	\(2 d^{11} \left (\frac {-\frac {77 a \,d^{2} \sqrt {d x}}{8192 b^{2}}-\frac {231 \left (d x \right )^{\frac {5}{2}}}{5120 b}+\frac {313 \left (d x \right )^{\frac {9}{2}}}{12288 a \,d^{2}}+\frac {11 b \left (d x \right )^{\frac {13}{2}}}{768 a^{2} d^{4}}+\frac {77 b^{2} \left (d x \right )^{\frac {17}{2}}}{24576 a^{3} d^{6}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{4} d^{8} b^{2}}\right )\)	\(236\)
pseudoelliptic	\(\frac {d^{3} \left (\left (3080 a \,x^{8} b^{4}+14080 a^{2} x^{6} b^{3}+25040 a^{3} x^{4} b^{2}-44352 x^{2} a^{4} b -9240 a^{5}\right ) \sqrt {d x}+1155 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{491520 a^{4} b^{2} \left (b \,x^{2}+a \right )^{5}}\)	\(240\)

Input:

int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

2*d^11*((-77/8192*a*d^2/b^2*(d*x)^(1/2)-231/5120/b*(d*x)^(5/2)+313/12288/a 
/d^2*(d*x)^(9/2)+11/768/a^2/d^4*b*(d*x)^(13/2)+77/24576/a^3/d^6*b^2*(d*x)^ 
(17/2))/(b*d^2*x^2+a*d^2)^5+77/65536/a^4/d^8/b^2*(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 = 0.19 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.82 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1155 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \log \left (77 \, a^{4} b^{2} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d^{3}\right ) - 1155 \, {\left (-i \, a^{3} b^{7} x^{10} - 5 i \, a^{4} b^{6} x^{8} - 10 i \, a^{5} b^{5} x^{6} - 10 i \, a^{6} b^{4} x^{4} - 5 i \, a^{7} b^{3} x^{2} - i \, a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \log \left (77 i \, a^{4} b^{2} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d^{3}\right ) - 1155 \, {\left (i \, a^{3} b^{7} x^{10} + 5 i \, a^{4} b^{6} x^{8} + 10 i \, a^{5} b^{5} x^{6} + 10 i \, a^{6} b^{4} x^{4} + 5 i \, a^{7} b^{3} x^{2} + i \, a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \log \left (-77 i \, a^{4} b^{2} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d^{3}\right ) - 1155 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \log \left (-77 \, a^{4} b^{2} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d^{3}\right ) + 4 \, {\left (385 \, b^{4} d^{3} x^{8} + 1760 \, a b^{3} d^{3} x^{6} + 3130 \, a^{2} b^{2} d^{3} x^{4} - 5544 \, a^{3} b d^{3} x^{2} - 1155 \, a^{4} d^{3}\right )} \sqrt {d x}}{245760 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}} \] Input:

integrate((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

1/245760*(1155*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4 
*x^4 + 5*a^7*b^3*x^2 + a^8*b^2)*(-d^14/(a^15*b^9))^(1/4)*log(77*a^4*b^2*(- 
d^14/(a^15*b^9))^(1/4) + 77*sqrt(d*x)*d^3) - 1155*(-I*a^3*b^7*x^10 - 5*I*a 
^4*b^6*x^8 - 10*I*a^5*b^5*x^6 - 10*I*a^6*b^4*x^4 - 5*I*a^7*b^3*x^2 - I*a^8 
*b^2)*(-d^14/(a^15*b^9))^(1/4)*log(77*I*a^4*b^2*(-d^14/(a^15*b^9))^(1/4) + 
 77*sqrt(d*x)*d^3) - 1155*(I*a^3*b^7*x^10 + 5*I*a^4*b^6*x^8 + 10*I*a^5*b^5 
*x^6 + 10*I*a^6*b^4*x^4 + 5*I*a^7*b^3*x^2 + I*a^8*b^2)*(-d^14/(a^15*b^9))^ 
(1/4)*log(-77*I*a^4*b^2*(-d^14/(a^15*b^9))^(1/4) + 77*sqrt(d*x)*d^3) - 115 
5*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7* 
b^3*x^2 + a^8*b^2)*(-d^14/(a^15*b^9))^(1/4)*log(-77*a^4*b^2*(-d^14/(a^15*b 
^9))^(1/4) + 77*sqrt(d*x)*d^3) + 4*(385*b^4*d^3*x^8 + 1760*a*b^3*d^3*x^6 + 
 3130*a^2*b^2*d^3*x^4 - 5544*a^3*b*d^3*x^2 - 1155*a^4*d^3)*sqrt(d*x))/(a^3 
*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7*b^3*x^ 
2 + a^8*b^2)
 

Sympy [F]

\[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{\frac {7}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \] Input:

integrate((d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

Integral((d*x)**(7/2)/(a + b*x**2)**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.24 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (385 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{6} + 1760 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{8} + 3130 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{10} - 5544 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{12} - 1155 \, \sqrt {d x} a^{4} d^{14}\right )}}{a^{3} b^{7} d^{10} x^{10} + 5 \, a^{4} b^{6} d^{10} x^{8} + 10 \, a^{5} b^{5} d^{10} x^{6} + 10 \, a^{6} b^{4} d^{10} x^{4} + 5 \, a^{7} b^{3} d^{10} x^{2} + a^{8} b^{2} d^{10}} + \frac {1155 \, {\left (\frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{3} b^{2}}}{491520 \, d} \] Input:

integrate((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

1/491520*(8*(385*(d*x)^(17/2)*b^4*d^6 + 1760*(d*x)^(13/2)*a*b^3*d^8 + 3130 
*(d*x)^(9/2)*a^2*b^2*d^10 - 5544*(d*x)^(5/2)*a^3*b*d^12 - 1155*sqrt(d*x)*a 
^4*d^14)/(a^3*b^7*d^10*x^10 + 5*a^4*b^6*d^10*x^8 + 10*a^5*b^5*d^10*x^6 + 1 
0*a^6*b^4*d^10*x^4 + 5*a^7*b^3*d^10*x^2 + a^8*b^2*d^10) + 1155*(sqrt(2)*d^ 
6*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^6*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^5* 
arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/s 
qrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2)*d^5* 
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)))/(a^3*b^2))/d
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{3}} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{3}} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{3}} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} d^{4} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{3}} + \frac {8 \, {\left (385 \, \sqrt {d x} b^{4} d^{14} x^{8} + 1760 \, \sqrt {d x} a b^{3} d^{14} x^{6} + 3130 \, \sqrt {d x} a^{2} b^{2} d^{14} x^{4} - 5544 \, \sqrt {d x} a^{3} b d^{14} x^{2} - 1155 \, \sqrt {d x} a^{4} d^{14}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{3} b^{2}}}{491520 \, d} \] Input:

integrate((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

1/491520*(2310*sqrt(2)*(a*b^3*d^2)^(1/4)*d^4*arctan(1/2*sqrt(2)*(sqrt(2)*( 
a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b^3) + 2310*sqrt(2)*(a 
*b^3*d^2)^(1/4)*d^4*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt( 
d*x))/(a*d^2/b)^(1/4))/(a^4*b^3) + 1155*sqrt(2)*(a*b^3*d^2)^(1/4)*d^4*log( 
d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^3) - 1155* 
sqrt(2)*(a*b^3*d^2)^(1/4)*d^4*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) 
+ sqrt(a*d^2/b))/(a^4*b^3) + 8*(385*sqrt(d*x)*b^4*d^14*x^8 + 1760*sqrt(d*x 
)*a*b^3*d^14*x^6 + 3130*sqrt(d*x)*a^2*b^2*d^14*x^4 - 5544*sqrt(d*x)*a^3*b* 
d^14*x^2 - 1155*sqrt(d*x)*a^4*d^14)/((b*d^2*x^2 + a*d^2)^5*a^3*b^2))/d
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.65 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {313\,d^9\,{\left (d\,x\right )}^{9/2}}{6144\,a}-\frac {231\,d^{11}\,{\left (d\,x\right )}^{5/2}}{2560\,b}+\frac {77\,b^2\,d^5\,{\left (d\,x\right )}^{17/2}}{12288\,a^3}-\frac {77\,a\,d^{13}\,\sqrt {d\,x}}{4096\,b^2}+\frac {11\,b\,d^7\,{\left (d\,x\right )}^{13/2}}{384\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {77\,d^{7/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{15/4}\,b^{9/4}}+\frac {77\,d^{7/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{15/4}\,b^{9/4}} \] Input:

int((d*x)^(7/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
 

Output:

((313*d^9*(d*x)^(9/2))/(6144*a) - (231*d^11*(d*x)^(5/2))/(2560*b) + (77*b^ 
2*d^5*(d*x)^(17/2))/(12288*a^3) - (77*a*d^13*(d*x)^(1/2))/(4096*b^2) + (11 
*b*d^7*(d*x)^(13/2))/(384*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) + (77*d 
^(7/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(15/4) 
*b^(9/4)) + (77*d^(7/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2)))) 
/(8192*(-a)^(15/4)*b^(9/4))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 995, normalized size of antiderivative = 3.14 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

(sqrt(d)*d**3*( - 2310*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s 
qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**5 - 11550*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 - 23100*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 - 23100*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 - 11550*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 - 2310*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 + 2310*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**5 + 11550*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 + 23100* 
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 + 23100*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 + 11550*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 + 2310*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq...