Integrand size = 30, antiderivative size = 543 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1547}{1024 a^4 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {7}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {119}{256 a^3 d (d x)^{5/2} \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 \left (a+b x^2\right )}{5120 a^5 d (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b \left (a+b x^2\right )}{1024 a^6 d^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 b^{5/4} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 b^{5/4} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{2048 \sqrt {2} a^{25/4} d^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:
1547/1024/a^4/d/(d*x)^(5/2)/((b*x^2+a)^2)^(1/2)+1/8/a/d/(d*x)^(5/2)/(b*x^2 +a)^3/((b*x^2+a)^2)^(1/2)+7/32/a^2/d/(d*x)^(5/2)/(b*x^2+a)^2/((b*x^2+a)^2) ^(1/2)+119/256/a^3/d/(d*x)^(5/2)/(b*x^2+a)/((b*x^2+a)^2)^(1/2)-13923/5120* (b*x^2+a)/a^5/d/(d*x)^(5/2)/((b*x^2+a)^2)^(1/2)+13923/1024*b*(b*x^2+a)/a^6 /d^3/(d*x)^(1/2)/((b*x^2+a)^2)^(1/2)-13923/4096*b^(5/4)*(b*x^2+a)*arctan(1 -2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(25/4)/d^(7/2)/((b *x^2+a)^2)^(1/2)+13923/4096*b^(5/4)*(b*x^2+a)*arctan(1+2^(1/2)*b^(1/4)*(d* x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(25/4)/d^(7/2)/((b*x^2+a)^2)^(1/2)-139 23/4096*b^(5/4)*(b*x^2+a)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(d*x)^(1/2)/d^(1 /2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(25/4)/d^(7/2)/((b*x^2+a)^2)^(1/2)
Time = 0.70 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {x \left (a+b x^2\right ) \left (4 \sqrt [4]{a} \left (-2048 a^5+43008 a^4 b x^2+220507 a^3 b^2 x^4+369733 a^2 b^3 x^6+264537 a b^4 x^8+69615 b^5 x^{10}\right )-69615 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-69615 \sqrt {2} b^{5/4} x^{5/2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{20480 a^{25/4} (d x)^{7/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \] Input:
Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
Output:
(x*(a + b*x^2)*(4*a^(1/4)*(-2048*a^5 + 43008*a^4*b*x^2 + 220507*a^3*b^2*x^ 4 + 369733*a^2*b^3*x^6 + 264537*a*b^4*x^8 + 69615*b^5*x^10) - 69615*Sqrt[2 ]*b^(5/4)*x^(5/2)*(a + b*x^2)^4*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1 /4)*b^(1/4)*Sqrt[x])] - 69615*Sqrt[2]*b^(5/4)*x^(5/2)*(a + b*x^2)^4*ArcTan h[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(20480*a^(25/ 4)*(d*x)^(7/2)*((a + b*x^2)^2)^(5/2))
Time = 1.21 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.89, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1384, 27, 253, 253, 253, 253, 264, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^2\right ) \int \frac {1}{b^5 (d x)^{7/2} \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^5}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^4}dx}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^3}dx}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \int \frac {1}{(d x)^{7/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )}dx}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {b \int \frac {\sqrt {d x}}{b x^2+a}dx}{a d^2}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d^3}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {21 \left (\frac {17 \left (\frac {13 \left (\frac {9 \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{a d^2}-\frac {2}{5 a d (d x)^{5/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{5/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{5/2} \left (a+b x^2\right )^2}\right )}{12 a}+\frac {1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3}\right )}{16 a}+\frac {1}{8 a d (d x)^{5/2} \left (a+b x^2\right )^4}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
Input:
Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
Output:
((a + b*x^2)*(1/(8*a*d*(d*x)^(5/2)*(a + b*x^2)^4) + (21*(1/(6*a*d*(d*x)^(5 /2)*(a + b*x^2)^3) + (17*(1/(4*a*d*(d*x)^(5/2)*(a + b*x^2)^2) + (13*(1/(2* a*d*(d*x)^(5/2)*(a + b*x^2)) + (9*(-2/(5*a*d*(d*x)^(5/2)) - (b*(-2/(a*d*Sq rt[d*x]) - (2*b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d ])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[ d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[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 + Sqr t[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[b])))/(a*d)))/(a*d^2)))/(4*a)))/(8*a)))/(12*a)))/(16*a)))/Sqrt[ a^2 + 2*a*b*x^2 + b^2*x^4]
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*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] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -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[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
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 = 0.22 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {2 \left (-25 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 a^{6} \sqrt {d x}\, x^{2} d^{3} \left (b \,x^{2}+a \right )}+\frac {b^{2} \left (\frac {\frac {5599 a^{3} d^{6} \left (d x \right )^{\frac {3}{2}}}{1024}+\frac {14145 a^{2} d^{4} b \left (d x \right )^{\frac {7}{2}}}{1024}+\frac {12357 a \,d^{2} b^{2} \left (d x \right )^{\frac {11}{2}}}{1024}+\frac {3683 b^{3} \left (d x \right )^{\frac {15}{2}}}{1024}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {13923 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8192 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{a^{6} d^{3} \left (b \,x^{2}+a \right )}\) | \(285\) |
| default | \(\text {Expression too large to display}\) | \(1129\) |
Input:
int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/5*(-25*b*x^2+a)/a^6/(d*x)^(1/2)/x^2/d^3*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+1 /a^6*b^2*(2*(5599/2048*a^3*d^6*(d*x)^(3/2)+14145/2048*a^2*d^4*b*(d*x)^(7/2 )+12357/2048*a*d^2*b^2*(d*x)^(11/2)+3683/2048*b^3*(d*x)^(15/2))/(b*d^2*x^2 +a*d^2)^4+13923/8192/b/(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*arct an(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))/d^3*((b*x^2+a)^2)^(1/2)/(b*x^2 +a)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {69615 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (2698972561467 \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) - 69615 \, {\left (i \, a^{6} b^{4} d^{4} x^{11} + 4 i \, a^{7} b^{3} d^{4} x^{9} + 6 i \, a^{8} b^{2} d^{4} x^{7} + 4 i \, a^{9} b d^{4} x^{5} + i \, a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (2698972561467 i \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) - 69615 \, {\left (-i \, a^{6} b^{4} d^{4} x^{11} - 4 i \, a^{7} b^{3} d^{4} x^{9} - 6 i \, a^{8} b^{2} d^{4} x^{7} - 4 i \, a^{9} b d^{4} x^{5} - i \, a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (-2698972561467 i \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) - 69615 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {1}{4}} \log \left (-2698972561467 \, a^{19} d^{11} \left (-\frac {b^{5}}{a^{25} d^{14}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b^{4}\right ) + 4 \, {\left (69615 \, b^{5} x^{10} + 264537 \, a b^{4} x^{8} + 369733 \, a^{2} b^{3} x^{6} + 220507 \, a^{3} b^{2} x^{4} + 43008 \, a^{4} b x^{2} - 2048 \, a^{5}\right )} \sqrt {d x}}{20480 \, {\left (a^{6} b^{4} d^{4} x^{11} + 4 \, a^{7} b^{3} d^{4} x^{9} + 6 \, a^{8} b^{2} d^{4} x^{7} + 4 \, a^{9} b d^{4} x^{5} + a^{10} d^{4} x^{3}\right )}} \] Input:
integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas ")
Output:
1/20480*(69615*(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^8*b^2*d^4*x^7 + 4*a^9*b*d^4*x^5 + a^10*d^4*x^3)*(-b^5/(a^25*d^14))^(1/4)*log(269897256146 7*a^19*d^11*(-b^5/(a^25*d^14))^(3/4) + 2698972561467*sqrt(d*x)*b^4) - 6961 5*(I*a^6*b^4*d^4*x^11 + 4*I*a^7*b^3*d^4*x^9 + 6*I*a^8*b^2*d^4*x^7 + 4*I*a^ 9*b*d^4*x^5 + I*a^10*d^4*x^3)*(-b^5/(a^25*d^14))^(1/4)*log(2698972561467*I *a^19*d^11*(-b^5/(a^25*d^14))^(3/4) + 2698972561467*sqrt(d*x)*b^4) - 69615 *(-I*a^6*b^4*d^4*x^11 - 4*I*a^7*b^3*d^4*x^9 - 6*I*a^8*b^2*d^4*x^7 - 4*I*a^ 9*b*d^4*x^5 - I*a^10*d^4*x^3)*(-b^5/(a^25*d^14))^(1/4)*log(-2698972561467* I*a^19*d^11*(-b^5/(a^25*d^14))^(3/4) + 2698972561467*sqrt(d*x)*b^4) - 6961 5*(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^8*b^2*d^4*x^7 + 4*a^9*b*d^4* x^5 + a^10*d^4*x^3)*(-b^5/(a^25*d^14))^(1/4)*log(-2698972561467*a^19*d^11* (-b^5/(a^25*d^14))^(3/4) + 2698972561467*sqrt(d*x)*b^4) + 4*(69615*b^5*x^1 0 + 264537*a*b^4*x^8 + 369733*a^2*b^3*x^6 + 220507*a^3*b^2*x^4 + 43008*a^4 *b*x^2 - 2048*a^5)*sqrt(d*x))/(a^6*b^4*d^4*x^11 + 4*a^7*b^3*d^4*x^9 + 6*a^ 8*b^2*d^4*x^7 + 4*a^9*b*d^4*x^5 + a^10*d^4*x^3)
\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (d x\right )^{\frac {7}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
Output:
Integral(1/((d*x)**(7/2)*((a + b*x**2)**2)**(5/2)), x)
\[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}} \left (d x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima ")
Output:
-4*b*integrate(1/((a^5*b*d^(7/2)*x^2 + a^6*d^(7/2))*x^(3/2)), x) + 1/3072* (11049*b^5*x^(15/2) + 27135*a*b^4*x^(11/2) + 23395*a^2*b^3*x^(7/2) + 6925* a^3*b^2*x^(3/2))/(a^6*b^4*d^(7/2)*x^8 + 4*a^7*b^3*d^(7/2)*x^6 + 6*a^8*b^2* d^(7/2)*x^4 + 4*a^9*b*d^(7/2)*x^2 + a^10*d^(7/2)) + 1/192*((621*b^6*x^5 + 1042*a*b^5*x^3 + 453*a^2*b^4*x)*x^(9/2) + 2*(695*a*b^5*x^5 + 1182*a^2*b^4* x^3 + 519*a^3*b^3*x)*x^(5/2) + (801*a^2*b^4*x^5 + 1386*a^3*b^3*x^3 + 617*a ^4*b^2*x)*sqrt(x))/(a^8*b^3*d^(7/2)*x^6 + 3*a^9*b^2*d^(7/2)*x^4 + 3*a^10*b *d^(7/2)*x^2 + a^11*d^(7/2) + (a^5*b^6*d^(7/2)*x^6 + 3*a^6*b^5*d^(7/2)*x^4 + 3*a^7*b^4*d^(7/2)*x^2 + a^8*b^3*d^(7/2))*x^6 + 3*(a^6*b^5*d^(7/2)*x^6 + 3*a^7*b^4*d^(7/2)*x^4 + 3*a^8*b^3*d^(7/2)*x^2 + a^9*b^2*d^(7/2))*x^4 + 3* (a^7*b^4*d^(7/2)*x^6 + 3*a^8*b^3*d^(7/2)*x^4 + 3*a^9*b^2*d^(7/2)*x^2 + a^1 0*b*d^(7/2))*x^2) + 3683/8192*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a ^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*s qrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b) ) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^ (1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^6*d^(7/2)) + integrate(1/((a^4*b*d^(7/2)* x^2 + a^5*d^(7/2))*x^(7/2)), x)
Time = 0.17 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {13923 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4096 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {13923 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{4096 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {13923 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{8192 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {13923 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{8192 \, a^{7} b d^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3683 \, \sqrt {d x} b^{5} d^{7} x^{7} + 12357 \, \sqrt {d x} a b^{4} d^{7} x^{5} + 14145 \, \sqrt {d x} a^{2} b^{3} d^{7} x^{3} + 5599 \, \sqrt {d x} a^{3} b^{2} d^{7} x}{1024 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{6} d^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {2 \, {\left (25 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt {d x} a^{6} d^{5} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \] Input:
integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Output:
13923/4096*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^7*b*d^5*sgn(b*x^2 + a)) + 13923/ 4096*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^7*b*d^5*sgn(b*x^2 + a)) - 13923/8192* sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq rt(a*d^2/b))/(a^7*b*d^5*sgn(b*x^2 + a)) + 13923/8192*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^7*b*d ^5*sgn(b*x^2 + a)) + 1/1024*(3683*sqrt(d*x)*b^5*d^7*x^7 + 12357*sqrt(d*x)* a*b^4*d^7*x^5 + 14145*sqrt(d*x)*a^2*b^3*d^7*x^3 + 5599*sqrt(d*x)*a^3*b^2*d ^7*x)/((b*d^2*x^2 + a*d^2)^4*a^6*d^3*sgn(b*x^2 + a)) + 2/5*(25*b*d^2*x^2 - a*d^2)/(sqrt(d*x)*a^6*d^5*x^2*sgn(b*x^2 + a))
Timed out. \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \] Input:
int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)
Output:
int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)
Time = 0.19 (sec) , antiderivative size = 891, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
Output:
(sqrt(d)*( - 139230*sqrt(x)*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 - 556920*sqrt(x)*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 - 835380 *sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqr t(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**3*x**6 - 556920*sqrt(x) *b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqr t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**4*x**8 - 139230*sqrt(x)*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 + 139230*sqrt(x)*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 + 556920*sqrt(x)*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 + 835380*sqrt(x)*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 + 556920*sqrt(x)*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 + 1 39230*sqrt(x)*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 + 69615*sqrt(x) *b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + s...