Integrand size = 28, antiderivative size = 275 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {385}{192 a^4 d (d x)^{3/2}}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {5}{16 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {55}{64 a^3 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {385 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}}-\frac {385 b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{128 \sqrt {2} a^{19/4} d^{5/2}} \] Output:
-385/192/a^4/d/(d*x)^(3/2)+1/6/a/d/(d*x)^(3/2)/(b*x^2+a)^3+5/16/a^2/d/(d*x )^(3/2)/(b*x^2+a)^2+55/64/a^3/d/(d*x)^(3/2)/(b*x^2+a)+385/256*b^(3/4)*arct an(1-2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(19/4)/d^(5/2) -385/256*b^(3/4)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^( 1/2)/a^(19/4)/d^(5/2)-385/256*b^(3/4)*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^(19/4)/d^(5/2)
Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {x \left (-\frac {4 a^{3/4} \left (128 a^3+765 a^2 b x^2+990 a b^2 x^4+385 b^3 x^6\right )}{\left (a+b x^2\right )^3}+1155 \sqrt {2} b^{3/4} x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-1155 \sqrt {2} b^{3/4} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 a^{19/4} (d x)^{5/2}} \] Input:
Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
Output:
(x*((-4*a^(3/4)*(128*a^3 + 765*a^2*b*x^2 + 990*a*b^2*x^4 + 385*b^3*x^6))/( a + b*x^2)^3 + 1155*Sqrt[2]*b^(3/4)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/( Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 1155*Sqrt[2]*b^(3/4)*x^(3/2)*ArcTanh[( Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(768*a^(19/4)*(d *x)^(5/2))
Time = 1.04 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.44, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1380, 27, 253, 253, 253, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^4 \int \frac {1}{b^4 (d x)^{5/2} \left (b x^2+a\right )^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(d x)^{5/2} \left (a+b x^2\right )^4}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^3}dx}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \left (\frac {11 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \int \frac {1}{(d x)^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{a d^2}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \int \frac {1}{b x^2+a}d\sqrt {d x}}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {11 \left (\frac {7 \left (-\frac {2 b \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{a d^3}-\frac {2}{3 a d (d x)^{3/2}}\right )}{4 a}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a d (d x)^{3/2} \left (a+b x^2\right )^2}\right )}{4 a}+\frac {1}{6 a d (d x)^{3/2} \left (a+b x^2\right )^3}\) |
Input:
Int[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
Output:
1/(6*a*d*(d*x)^(3/2)*(a + b*x^2)^3) + (5*(1/(4*a*d*(d*x)^(3/2)*(a + b*x^2) ^2) + (11*(1/(2*a*d*(d*x)^(3/2)*(a + b*x^2)) + (7*(-2/(3*a*d*(d*x)^(3/2)) - (2*b*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(S qrt[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 + Sqrt[b]*d*x + Sqrt[ 2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) )/(2*Sqrt[a])))/(a*d^3)))/(4*a)))/(8*a)))/(4*a)
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[((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[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 = 0.56 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(-\frac {2}{3 a^{4} x \sqrt {d x}\, d^{2}}-\frac {2 b \left (\frac {\frac {257 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}+\frac {101 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}+\frac {127 \sqrt {d x}\, a^{2} d^{4}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{a^{4} d}\) | \(222\) |
| derivativedivides | \(2 d^{7} \left (-\frac {1}{3 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {257 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}+\frac {101 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}+\frac {127 \sqrt {d x}\, a^{2} d^{4}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{a^{4} d^{8}}\right )\) | \(224\) |
| default | \(2 d^{7} \left (-\frac {1}{3 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {257 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}+\frac {101 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}+\frac {127 \sqrt {d x}\, a^{2} d^{4}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {385 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{a^{4} d^{8}}\right )\) | \(224\) |
| pseudoelliptic | \(-\frac {385 \left (\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \left (d x \right )^{\frac {3}{2}}+\frac {1024 a \,d^{2} \left (\frac {385}{128} b^{3} x^{6}+\frac {495}{64} b^{2} x^{4} a +\frac {765}{128} a^{2} b \,x^{2}+a^{3}\right )}{1155}\right )}{512 \left (d x \right )^{\frac {3}{2}} d^{3} a^{5} \left (b \,x^{2}+a \right )^{3}}\) | \(231\) |
Input:
int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
-2/3/a^4/x/(d*x)^(1/2)/d^2-2*b/a^4/d*((257/384*b^2*(d*x)^(9/2)+101/64*a*b* d^2*(d*x)^(5/2)+127/128*(d*x)^(1/2)*a^2*d^4)/(b*d^2*x^2+a*d^2)^3+385/1024* (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)))
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.75 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1155 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (385 \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 1155 \, {\left (i \, a^{4} b^{3} d^{3} x^{8} + 3 i \, a^{5} b^{2} d^{3} x^{6} + 3 i \, a^{6} b d^{3} x^{4} + i \, a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (385 i \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 1155 \, {\left (-i \, a^{4} b^{3} d^{3} x^{8} - 3 i \, a^{5} b^{2} d^{3} x^{6} - 3 i \, a^{6} b d^{3} x^{4} - i \, a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (-385 i \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) - 1155 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} \log \left (-385 \, a^{5} d^{3} \left (-\frac {b^{3}}{a^{19} d^{10}}\right )^{\frac {1}{4}} + 385 \, \sqrt {d x} b\right ) + 4 \, {\left (385 \, b^{3} x^{6} + 990 \, a b^{2} x^{4} + 765 \, a^{2} b x^{2} + 128 \, a^{3}\right )} \sqrt {d x}}{768 \, {\left (a^{4} b^{3} d^{3} x^{8} + 3 \, a^{5} b^{2} d^{3} x^{6} + 3 \, a^{6} b d^{3} x^{4} + a^{7} d^{3} x^{2}\right )}} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
-1/768*(1155*(a^4*b^3*d^3*x^8 + 3*a^5*b^2*d^3*x^6 + 3*a^6*b*d^3*x^4 + a^7* d^3*x^2)*(-b^3/(a^19*d^10))^(1/4)*log(385*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) + 385*sqrt(d*x)*b) + 1155*(I*a^4*b^3*d^3*x^8 + 3*I*a^5*b^2*d^3*x^6 + 3*I* a^6*b*d^3*x^4 + I*a^7*d^3*x^2)*(-b^3/(a^19*d^10))^(1/4)*log(385*I*a^5*d^3* (-b^3/(a^19*d^10))^(1/4) + 385*sqrt(d*x)*b) + 1155*(-I*a^4*b^3*d^3*x^8 - 3 *I*a^5*b^2*d^3*x^6 - 3*I*a^6*b*d^3*x^4 - I*a^7*d^3*x^2)*(-b^3/(a^19*d^10)) ^(1/4)*log(-385*I*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) + 385*sqrt(d*x)*b) - 11 55*(a^4*b^3*d^3*x^8 + 3*a^5*b^2*d^3*x^6 + 3*a^6*b*d^3*x^4 + a^7*d^3*x^2)*( -b^3/(a^19*d^10))^(1/4)*log(-385*a^5*d^3*(-b^3/(a^19*d^10))^(1/4) + 385*sq rt(d*x)*b) + 4*(385*b^3*x^6 + 990*a*b^2*x^4 + 765*a^2*b*x^2 + 128*a^3)*sqr t(d*x))/(a^4*b^3*d^3*x^8 + 3*a^5*b^2*d^3*x^6 + 3*a^6*b*d^3*x^4 + a^7*d^3*x ^2)
\[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{4}}\, dx \] Input:
integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
Integral(1/((d*x)**(5/2)*(a + b*x**2)**4), x)
Time = 0.12 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {\frac {8 \, {\left (385 \, b^{3} d^{6} x^{6} + 990 \, a b^{2} d^{6} x^{4} + 765 \, a^{2} b d^{6} x^{2} + 128 \, a^{3} d^{6}\right )}}{\left (d x\right )^{\frac {15}{2}} a^{4} b^{3} + 3 \, \left (d x\right )^{\frac {11}{2}} a^{5} b^{2} d^{2} + 3 \, \left (d x\right )^{\frac {7}{2}} a^{6} b d^{4} + \left (d x\right )^{\frac {3}{2}} a^{7} d^{6}} + \frac {1155 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d}\right )}}{a^{4}}}{1536 \, d} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
-1/1536*(8*(385*b^3*d^6*x^6 + 990*a*b^2*d^6*x^4 + 765*a^2*b*d^6*x^2 + 128* a^3*d^6)/((d*x)^(15/2)*a^4*b^3 + 3*(d*x)^(11/2)*a^5*b^2*d^2 + 3*(d*x)^(7/2 )*a^6*b*d^4 + (d*x)^(3/2)*a^7*d^6) + 1155*(sqrt(2)*b^(3/4)*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/(a*d^2)^(3/4) - sq rt(2)*b^(3/4)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/(a*d^2)^(3/4) + 2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2) ^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt( a)*sqrt(b)*d)*sqrt(a)*d) + 2*sqrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2 )^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt (a)*sqrt(b)*d)*sqrt(a)*d))/a^4)/d
Time = 0.16 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{5} d^{3}} - \frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{5} d^{3}} - \frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{5} d^{3}} + \frac {385 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{5} d^{3}} - \frac {385 \, b^{3} d^{6} x^{6} + 990 \, a b^{2} d^{6} x^{4} + 765 \, a^{2} b d^{6} x^{2} + 128 \, a^{3} d^{6}}{192 \, {\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )}^{3} a^{4} d} \] Input:
integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
-385/256*sqrt(2)*(a*b^3*d^2)^(1/4)*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*d^3) - 385/256*sqrt(2)*(a*b^3*d^ 2)^(1/4)*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*d^3) - 385/512*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2 )*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^5*d^3) + 385/512*sqrt(2)*( a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/ b))/(a^5*d^3) - 1/192*(385*b^3*d^6*x^6 + 990*a*b^2*d^6*x^4 + 765*a^2*b*d^6 *x^2 + 128*a^3*d^6)/((sqrt(d*x)*b*d^2*x^2 + sqrt(d*x)*a*d^2)^3*a^4*d)
Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {385\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{19/4}\,d^{5/2}}-\frac {\frac {2\,d^5}{3\,a}+\frac {255\,b\,d^5\,x^2}{64\,a^2}+\frac {165\,b^2\,d^5\,x^4}{32\,a^3}+\frac {385\,b^3\,d^5\,x^6}{192\,a^4}}{b^3\,{\left (d\,x\right )}^{15/2}+a^3\,d^6\,{\left (d\,x\right )}^{3/2}+3\,a^2\,b\,d^4\,{\left (d\,x\right )}^{7/2}+3\,a\,b^2\,d^2\,{\left (d\,x\right )}^{11/2}}+\frac {385\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{128\,a^{19/4}\,d^{5/2}} \] Input:
int(1/((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^2),x)
Output:
(385*(-b)^(3/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^( 19/4)*d^(5/2)) - ((2*d^5)/(3*a) + (255*b*d^5*x^2)/(64*a^2) + (165*b^2*d^5* x^4)/(32*a^3) + (385*b^3*d^5*x^6)/(192*a^4))/(b^3*(d*x)^(15/2) + a^3*d^6*( d*x)^(3/2) + 3*a^2*b*d^4*(d*x)^(7/2) + 3*a*b^2*d^2*(d*x)^(11/2)) + (385*(- b)^(3/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(128*a^(19/4)* d^(5/2))
Time = 0.18 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.53 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
(sqrt(d)*(2310*sqrt(x)*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*x + 6930*sqr t(x)*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**3 + 6930*sqrt(x)*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**5 + 2310*sqrt(x)*b**(3/4)*a**(1/4)*sqr t(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/ 4)*sqrt(2)))*b**3*x**7 - 2310*sqrt(x)*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*x - 6930*sqrt(x)*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**3 - 6930*s qrt(x)*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**5 - 2310*sqrt(x)*b**(3/ 4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/( b**(1/4)*a**(1/4)*sqrt(2)))*b**3*x**7 + 1155*sqrt(x)*b**(3/4)*a**(1/4)*sqr t(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3* x + 3465*sqrt(x)*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**3 + 3465*sqrt(x)*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**5 + 1155*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b...