Integrand size = 28, antiderivative size = 312 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}-\frac {117 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}-\frac {117 d^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}} \] Output:
-1/10*d*(d*x)^(3/2)/b/(b*x^2+a)^5+3/160*d*(d*x)^(3/2)/a/b/(b*x^2+a)^4+13/6 40*d*(d*x)^(3/2)/a^2/b/(b*x^2+a)^3+117/5120*d*(d*x)^(3/2)/a^3/b/(b*x^2+a)^ 2+117/4096*d*(d*x)^(3/2)/a^4/b/(b*x^2+a)-117/16384*d^(5/2)*arctan(1-2^(1/2 )*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(17/4)/b^(7/4)+117/16384* d^(5/2)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(1 7/4)/b^(7/4)-117/16384*d^(5/2)*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^(17/4)/b^(7/4)
Time = 0.58 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.59 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{5/2} \left (\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-195 a^4+4960 a^3 b x^2+5330 a^2 b^2 x^4+2808 a b^3 x^6+585 b^4 x^8\right )}{\left (a+b x^2\right )^5}-585 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-585 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{17/4} b^{7/4} x^{5/2}} \] Input:
Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((d*x)^(5/2)*((4*a^(1/4)*b^(3/4)*x^(3/2)*(-195*a^4 + 4960*a^3*b*x^2 + 5330 *a^2*b^2*x^4 + 2808*a*b^3*x^6 + 585*b^4*x^8))/(a + b*x^2)^5 - 585*Sqrt[2]* ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 585*Sqrt [2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81 920*a^(17/4)*b^(7/4)*x^(5/2))
Time = 1.13 (sec) , antiderivative size = 438, 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, 253, 253, 253, 253, 266, 27, 826, 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.
\(\displaystyle \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^6 \int \frac {(d x)^{5/2}}{b^6 \left (b x^2+a\right )^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(d x)^{5/2}}{\left (a+b x^2\right )^6}dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {3 d^2 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^5}dx}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^4}dx}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^3}dx}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \int \frac {\sqrt {d x}}{\left (b x^2+a\right )^2}dx}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {\int \frac {\sqrt {d x}}{b x^2+a}dx}{4 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {\int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a d}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {3 d^2 \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 a}+\frac {(d x)^{3/2}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right )^2}\right )}{4 a}+\frac {(d x)^{3/2}}{6 a d \left (a+b x^2\right )^3}\right )}{16 a}+\frac {(d x)^{3/2}}{8 a d \left (a+b x^2\right )^4}\right )}{20 b}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}\) |
Input:
Int[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
-1/10*(d*(d*x)^(3/2))/(b*(a + b*x^2)^5) + (3*d^2*((d*x)^(3/2)/(8*a*d*(a + b*x^2)^4) + (13*((d*x)^(3/2)/(6*a*d*(a + b*x^2)^3) + (3*((d*x)^(3/2)/(4*a* d*(a + b*x^2)^2) + (5*((d*x)^(3/2)/(2*a*d*(a + b*x^2)) + (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[b]) - (-1/2*Log[Sqrt[a]*d + Sqrt[b]* d*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)* Sqrt[d]) + Log[Sqrt[a]*d + Sqrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*S qrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b])))/(2*a)))/(8*a) ))/(4*a)))/(16*a)))/(20*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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]]))
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]
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]
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]
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]
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]
Time = 14.54 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(2 d^{11} \left (\frac {-\frac {39 \left (d x \right )^{\frac {3}{2}}}{8192 b}+\frac {31 \left (d x \right )^{\frac {7}{2}}}{256 a \,d^{2}}+\frac {533 b \left (d x \right )^{\frac {11}{2}}}{4096 a^{2} d^{4}}+\frac {351 b^{2} \left (d x \right )^{\frac {15}{2}}}{5120 a^{3} d^{6}}+\frac {117 b^{3} \left (d x \right )^{\frac {19}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{4} d^{8} b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(238\) |
default | \(2 d^{11} \left (\frac {-\frac {39 \left (d x \right )^{\frac {3}{2}}}{8192 b}+\frac {31 \left (d x \right )^{\frac {7}{2}}}{256 a \,d^{2}}+\frac {533 b \left (d x \right )^{\frac {11}{2}}}{4096 a^{2} d^{4}}+\frac {351 b^{2} \left (d x \right )^{\frac {15}{2}}}{5120 a^{3} d^{6}}+\frac {117 b^{3} \left (d x \right )^{\frac {19}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{4} d^{8} b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(238\) |
pseudoelliptic | \(\frac {39 d^{2} \left (-8 \sqrt {d x}\, \left (-3 b^{4} x^{8}-\frac {72}{5} a \,b^{3} x^{6}-\frac {82}{3} a^{2} b^{2} x^{4}-\frac {992}{39} a^{3} b \,x^{2}+a^{4}\right ) b x \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+3 d \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{4} b^{2} \left (b \,x^{2}+a \right )^{5}}\) | \(249\) |
Input:
int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
2*d^11*((-39/8192/b*(d*x)^(3/2)+31/256/a/d^2*(d*x)^(7/2)+533/4096/a^2/d^4* b*(d*x)^(11/2)+351/5120/a^3*b^2/d^6*(d*x)^(15/2)+117/8192/a^4/d^8*b^3*(d*x )^(19/2))/(b*d^2*x^2+a*d^2)^5+117/65536/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)))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.82 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {585 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (1601613 \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 585 \, {\left (i \, a^{4} b^{6} x^{10} + 5 i \, a^{5} b^{5} x^{8} + 10 i \, a^{6} b^{4} x^{6} + 10 i \, a^{7} b^{3} x^{4} + 5 i \, a^{8} b^{2} x^{2} + i \, a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (1601613 i \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 585 \, {\left (-i \, a^{4} b^{6} x^{10} - 5 i \, a^{5} b^{5} x^{8} - 10 i \, a^{6} b^{4} x^{6} - 10 i \, a^{7} b^{3} x^{4} - 5 i \, a^{8} b^{2} x^{2} - i \, a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (-1601613 i \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 585 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (-1601613 \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) + 4 \, {\left (585 \, b^{4} d^{2} x^{9} + 2808 \, a b^{3} d^{2} x^{7} + 5330 \, a^{2} b^{2} d^{2} x^{5} + 4960 \, a^{3} b d^{2} x^{3} - 195 \, a^{4} d^{2} x\right )} \sqrt {d x}}{81920 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \] Input:
integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/81920*(585*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x ^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10/(a^17*b^7))^(1/4)*log(1601613*a^13*b^5* (-d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) - 585*(I*a^4*b^6*x^10 + 5*I*a^5*b^5*x^8 + 10*I*a^6*b^4*x^6 + 10*I*a^7*b^3*x^4 + 5*I*a^8*b^2*x^2 + I*a^9*b)*(-d^10/(a^17*b^7))^(1/4)*log(1601613*I*a^13*b^5*(-d^10/(a^17*b^7) )^(3/4) + 1601613*sqrt(d*x)*d^7) - 585*(-I*a^4*b^6*x^10 - 5*I*a^5*b^5*x^8 - 10*I*a^6*b^4*x^6 - 10*I*a^7*b^3*x^4 - 5*I*a^8*b^2*x^2 - I*a^9*b)*(-d^10/ (a^17*b^7))^(1/4)*log(-1601613*I*a^13*b^5*(-d^10/(a^17*b^7))^(3/4) + 16016 13*sqrt(d*x)*d^7) - 585*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 1 0*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10/(a^17*b^7))^(1/4)*log(-16016 13*a^13*b^5*(-d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) + 4*(585*b^4 *d^2*x^9 + 2808*a*b^3*d^2*x^7 + 5330*a^2*b^2*d^2*x^5 + 4960*a^3*b*d^2*x^3 - 195*a^4*d^2*x)*sqrt(d*x))/(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)
\[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \] Input:
integrate((d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
Integral((d*x)**(5/2)/(a + b*x**2)**6, x)
Time = 0.12 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.23 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (585 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{4} + 2808 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{6} + 5330 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{8} + 4960 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{10} - 195 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{12}\right )}}{a^{4} b^{6} d^{10} x^{10} + 5 \, a^{5} b^{5} d^{10} x^{8} + 10 \, a^{6} b^{4} d^{10} x^{6} + 10 \, a^{7} b^{3} d^{10} x^{4} + 5 \, a^{8} b^{2} d^{10} x^{2} + a^{9} b d^{10}} + \frac {585 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{4} b}}{163840 \, d} \] Input:
integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/163840*(8*(585*(d*x)^(19/2)*b^4*d^4 + 2808*(d*x)^(15/2)*a*b^3*d^6 + 5330 *(d*x)^(11/2)*a^2*b^2*d^8 + 4960*(d*x)^(7/2)*a^3*b*d^10 - 195*(d*x)^(3/2)* a^4*d^12)/(a^4*b^6*d^10*x^10 + 5*a^5*b^5*d^10*x^8 + 10*a^6*b^4*d^10*x^6 + 10*a^7*b^3*d^10*x^4 + 5*a^8*b^2*d^10*x^2 + a^9*b*d^10) + 585*d^4*(2*sqrt(2 )*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(b)) + 2*sqrt(2)*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(b)) - sqrt(2)*log(sqr t(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^( 1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)* b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/(a^4*b))/d
Time = 0.12 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.14 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d^{2} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{4} d} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{4} d} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{4} d} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{4} d} + \frac {8 \, {\left (585 \, \sqrt {d x} b^{4} d^{10} x^{9} + 2808 \, \sqrt {d x} a b^{3} d^{10} x^{7} + 5330 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} + 4960 \, \sqrt {d x} a^{3} b d^{10} x^{3} - 195 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{4} b}\right )} \] Input:
integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
1/163840*d^2*(1170*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*( a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^5*b^4*d) + 1170*sqrt(2)* (a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d* x))/(a*d^2/b)^(1/4))/(a^5*b^4*d) - 585*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*b^4*d) + 585*sqrt (2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a *d^2/b))/(a^5*b^4*d) + 8*(585*sqrt(d*x)*b^4*d^10*x^9 + 2808*sqrt(d*x)*a*b^ 3*d^10*x^7 + 5330*sqrt(d*x)*a^2*b^2*d^10*x^5 + 4960*sqrt(d*x)*a^3*b*d^10*x ^3 - 195*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*a^4*b))
Time = 17.75 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.67 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {31\,d^9\,{\left (d\,x\right )}^{7/2}}{128\,a}-\frac {39\,d^{11}\,{\left (d\,x\right )}^{3/2}}{4096\,b}+\frac {351\,b^2\,d^5\,{\left (d\,x\right )}^{15/2}}{2560\,a^3}+\frac {117\,b^3\,d^3\,{\left (d\,x\right )}^{19/2}}{4096\,a^4}+\frac {533\,b\,d^7\,{\left (d\,x\right )}^{11/2}}{2048\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {117\,d^{5/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{17/4}\,b^{7/4}}-\frac {117\,d^{5/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{17/4}\,b^{7/4}} \] Input:
int((d*x)^(5/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((31*d^9*(d*x)^(7/2))/(128*a) - (39*d^11*(d*x)^(3/2))/(4096*b) + (351*b^2* d^5*(d*x)^(15/2))/(2560*a^3) + (117*b^3*d^3*(d*x)^(19/2))/(4096*a^4) + (53 3*b*d^7*(d*x)^(11/2))/(2048*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) + (11 7*d^(5/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(17 /4)*b^(7/4)) - (117*d^(5/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2 ))))/(8192*(-a)^(17/4)*b^(7/4))
Time = 0.19 (sec) , antiderivative size = 996, normalized size of antiderivative = 3.19 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(sqrt(d)*d**2*( - 1170*b**(1/4)*a**(3/4)*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 - 5850*b**(1 /4)*a**(3/4)*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 - 11700*b**(1/4)*a**(3/4)*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**3*b**2*x**4 - 11700*b**(1/4)*a**(3/4)*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 - 5850*b**(1/4)*a**(3/4)*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 - 1170*b**(1/4 )*a**(3/4)*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 + 1170*b**(1/4)*a**(3/4)*sqrt(2)*ata n((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 2)))*a**5 + 5850*b**(1/4)*a**(3/4)*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 + 11700*b** (1/4)*a**(3/4)*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 + 11700*b**(1/4)*a**(3/4)*sq rt(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 + 5850*b**(1/4)*a**(3/4)*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 + 1170*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2...