Integrand size = 30, antiderivative size = 445 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \] Output:
-13/16*d^3*(d*x)^(9/2)/b^2/((b*x^2+a)^2)^(1/2)-1/4*d*(d*x)^(13/2)/b/(b*x^2 +a)/((b*x^2+a)^2)^(1/2)-117/16*a*d^7*(d*x)^(1/2)*(b*x^2+a)/b^4/((b*x^2+a)^ 2)^(1/2)+117/80*d^5*(d*x)^(5/2)*(b*x^2+a)/b^3/((b*x^2+a)^2)^(1/2)-117/64*a ^(5/4)*d^(15/2)*(b*x^2+a)*arctan(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^( 1/2))*2^(1/2)/b^(17/4)/((b*x^2+a)^2)^(1/2)+117/64*a^(5/4)*d^(15/2)*(b*x^2+ a)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/b^(17/4)/ ((b*x^2+a)^2)^(1/2)+117/64*a^(5/4)*d^(15/2)*(b*x^2+a)*arctanh(2^(1/2)*a^(1 /4)*b^(1/4)*(d*x)^(1/2)/d^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/b^(17/4)/((b* x^2+a)^2)^(1/2)
Time = 0.43 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.46 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {d^7 \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-585 a^3-1053 a^2 b x^2-416 a b^2 x^4+32 b^3 x^6\right )-585 \sqrt {2} a^{5/4} \left (a+b x^2\right )^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+585 \sqrt {2} a^{5/4} \left (a+b x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{320 b^{17/4} \sqrt {x} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \] Input:
Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
Output:
(d^7*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(-585*a^3 - 1053*a^2*b*x^2 - 416*a*b^2*x ^4 + 32*b^3*x^6) - 585*Sqrt[2]*a^(5/4)*(a + b*x^2)^2*ArcTan[(Sqrt[a] - Sqr t[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 585*Sqrt[2]*a^(5/4)*(a + b*x^ 2)^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(3 20*b^(17/4)*Sqrt[x]*(a + b*x^2)*Sqrt[(a + b*x^2)^2])
Time = 1.10 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.93, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1384, 27, 252, 252, 262, 262, 266, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^3 \left (a+b x^2\right ) \int \frac {(d x)^{15/2}}{b^3 \left (b x^2+a\right )^3}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {(d x)^{15/2}}{\left (b x^2+a\right )^3}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \int \frac {(d x)^{11/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \int \frac {(d x)^{7/2}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \int \frac {(d x)^{3/2}}{b x^2+a}dx}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {a d^2 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \int \frac {1}{b x^2+a}d\sqrt {d x}}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {13 d^2 \left (\frac {9 d^2 \left (\frac {2 d (d x)^{5/2}}{5 b}-\frac {a d^2 \left (\frac {2 d \sqrt {d x}}{b}-\frac {2 a d \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{b}\right )}{b}\right )}{4 b}-\frac {d (d x)^{9/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right )^2}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
Input:
Int[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
Output:
((a + b*x^2)*(-1/4*(d*(d*x)^(13/2))/(b*(a + b*x^2)^2) + (13*d^2*(-1/2*(d*( d*x)^(9/2))/(b*(a + b*x^2)) + (9*d^2*((2*d*(d*x)^(5/2))/(5*b) - (a*d^2*((2 *d*Sqrt[d*x])/b - (2*a*d*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^ (1/4)*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[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 + S qrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4) *b^(1/4)*Sqrt[d])))/(2*Sqrt[a])))/b))/b))/(4*b)))/(8*b)))/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*(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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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[((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]]))
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 = 0.19 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {2 \left (-b \,x^{2}+15 a \right ) x \,d^{8} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 b^{4} \sqrt {d x}\, \left (b \,x^{2}+a \right )}+\frac {2 a^{2} d^{9} \left (\frac {-\frac {25 b \left (d x \right )^{\frac {5}{2}}}{32}-\frac {21 \sqrt {d x}\, a \,d^{2}}{32}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{2}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{4} \left (b \,x^{2}+a \right )}\) | \(256\) |
| default | \(-\frac {\left (-256 \left (d x \right )^{\frac {5}{2}} b^{3} x^{4}-585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \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 ) a \,b^{2} d^{2} x^{4}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a \,b^{2} d^{2} x^{4}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a \,b^{2} d^{2} x^{4}-512 \left (d x \right )^{\frac {5}{2}} a \,b^{2} x^{2}+3840 \sqrt {d x}\, a \,b^{2} d^{2} x^{4}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \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 ) a^{2} b \,d^{2} x^{2}-2340 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a^{2} b \,d^{2} x^{2}-2340 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a^{2} b \,d^{2} x^{2}+744 \left (d x \right )^{\frac {5}{2}} a^{2} b +7680 \sqrt {d x}\, a^{2} b \,d^{2} x^{2}-585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \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 ) a^{3} d^{2}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a^{3} d^{2}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a^{3} d^{2}+4680 \sqrt {d x}\, a^{3} d^{2}\right ) d^{5} \left (b \,x^{2}+a \right )}{640 b^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(737\) |
Input:
int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/5*(-b*x^2+15*a)*x/b^4/(d*x)^(1/2)*d^8*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+2*a ^2/b^4*d^9*((-25/32*b*(d*x)^(5/2)-21/32*(d*x)^(1/2)*a*d^2)/(b*d^2*x^2+a*d^ 2)^2+117/256*(a*d^2/b)^(1/4)/a/d^2*2^(1/2)*(ln((d*x+(a*d^2/b)^(1/4)*(d*x)^ (1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a *d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2 ^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))*((b*x^2+a)^2)^(1/2)/(b*x^2+a)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.82 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} + 117 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (-i \, b^{6} x^{4} - 2 i \, a b^{5} x^{2} - i \, a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} + 117 i \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (i \, b^{6} x^{4} + 2 i \, a b^{5} x^{2} + i \, a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} - 117 i \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} - 117 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 4 \, {\left (32 \, b^{3} d^{7} x^{6} - 416 \, a b^{2} d^{7} x^{4} - 1053 \, a^{2} b d^{7} x^{2} - 585 \, a^{3} d^{7}\right )} \sqrt {d x}}{320 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \] Input:
integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas" )
Output:
1/320*(585*(-a^5*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*log(11 7*sqrt(d*x)*a*d^7 + 117*(-a^5*d^30/b^17)^(1/4)*b^4) - 585*(-a^5*d^30/b^17) ^(1/4)*(-I*b^6*x^4 - 2*I*a*b^5*x^2 - I*a^2*b^4)*log(117*sqrt(d*x)*a*d^7 + 117*I*(-a^5*d^30/b^17)^(1/4)*b^4) - 585*(-a^5*d^30/b^17)^(1/4)*(I*b^6*x^4 + 2*I*a*b^5*x^2 + I*a^2*b^4)*log(117*sqrt(d*x)*a*d^7 - 117*I*(-a^5*d^30/b^ 17)^(1/4)*b^4) - 585*(-a^5*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a^2*b ^4)*log(117*sqrt(d*x)*a*d^7 - 117*(-a^5*d^30/b^17)^(1/4)*b^4) + 4*(32*b^3* d^7*x^6 - 416*a*b^2*d^7*x^4 - 1053*a^2*b*d^7*x^2 - 585*a^3*d^7)*sqrt(d*x)) /(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)
Timed out. \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
Output:
Timed out
\[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {15}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima" )
Output:
-1/2*a^2*d^(15/2)*x^(5/2)/(a*b^4*x^2 + a^2*b^3 + (b^5*x^2 + a*b^4)*x^2) - 2*a*d^(15/2)*integrate(x^(3/2)/(b^4*x^2 + a*b^3), x) + d^(15/2)*integrate( x^(7/2)/(b^3*x^2 + a*b^2), x) + 21/128*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqrt( 2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sq rt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(3/2)*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(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqr t(a))/b^(1/4) - sqrt(2)*a^(5/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqr t(b)*x + sqrt(a))/b^(1/4))*d^(15/2)/b^4 - 1/16*(17*a^2*b*d^(15/2)*x^(5/2) + 21*a^3*d^(15/2)*sqrt(x))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)
Time = 0.19 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.87 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{8} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{8} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{8} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a d^{8} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {40 \, {\left (25 \, \sqrt {d x} a^{2} b d^{12} x^{2} + 21 \, \sqrt {d x} a^{3} d^{12}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {256 \, {\left (\sqrt {d x} b^{12} d^{8} x^{2} - 15 \, \sqrt {d x} a b^{11} d^{8}\right )}}{b^{15} \mathrm {sgn}\left (b x^{2} + a\right )}}{640 \, d} \] Input:
integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")
Output:
1/640*(1170*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^8*arctan(1/2*sqrt(2)*(sqrt(2)*(a *d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^5*sgn(b*x^2 + a)) + 1170* sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^8*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/ 4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^5*sgn(b*x^2 + a)) + 585*sqrt(2)*(a*b ^3*d^2)^(1/4)*a*d^8*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d ^2/b))/(b^5*sgn(b*x^2 + a)) - 585*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d^8*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^5*sgn(b*x^2 + a)) - 40*(25*sqrt(d*x)*a^2*b*d^12*x^2 + 21*sqrt(d*x)*a^3*d^12)/((b*d^2*x^2 + a *d^2)^2*b^4*sgn(b*x^2 + a)) + 256*(sqrt(d*x)*b^12*d^8*x^2 - 15*sqrt(d*x)*a *b^11*d^8)/(b^15*sgn(b*x^2 + a)))/d
Timed out. \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^{15/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \] Input:
int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2),x)
Output:
int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.13 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
Output:
(sqrt(d)*d**7*( - 1170*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**3 - 2340*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*x**2 - 1170*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)*sq rt(2)))*a*b**2*x**4 + 1170*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 + 2340*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*x**2 + 1170*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**2*x**4 - 585*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b** (1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3 - 1170*b**(3/4)*a**(1/4 )*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)* a**2*b*x**2 - 585*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/ 4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b**2*x**4 + 585*b**(3/4)*a**(1/4)*sqrt (2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3 + 11 70*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt( a) + sqrt(b)*x)*a**2*b*x**2 + 585*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b* *(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b**2*x**4 - 4680*sqrt(x)* a**3*b - 8424*sqrt(x)*a**2*b**2*x**2 - 3328*sqrt(x)*a*b**3*x**4 + 256*s...