Integrand size = 24, antiderivative size = 399 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}-\frac {\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) x^{3/2}}{16 c^3 d \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{13/4} d^{7/4}}+\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}}-\frac {\left (3 b^2 c^2+5 a d (2 b c-9 a d)\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{13/4} d^{7/4}} \]
-1/4*(9*a^2*d^2-2*a*b*c*d+b^2*c^2)*x^(3/2)/c^2/d/(d*x^2+c)^2+1/16*(3*b^2*c ^2+5*a*d*(-9*a*d+2*b*c))*x^(3/2)/c^3/d/(d*x^2+c)-1/64*(3*b^2*c^2+5*a*d*(-9 *a*d+2*b*c))*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/d^(7/4)*2^ (1/2)+1/64*(3*b^2*c^2+5*a*d*(-9*a*d+2*b*c))*arctan(1+d^(1/4)*2^(1/2)*x^(1/ 2)/c^(1/4))/c^(13/4)/d^(7/4)*2^(1/2)+1/128*(3*b^2*c^2+5*a*d*(-9*a*d+2*b*c) )*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(13/4)/d^(7/4)*2 ^(1/2)-1/128*(3*b^2*c^2+5*a*d*(-9*a*d+2*b*c))*ln(c^(1/2)+x*d^(1/2)+c^(1/4) *d^(1/4)*2^(1/2)*x^(1/2))/c^(13/4)/d^(7/4)*2^(1/2)-2*a^2/c/(d*x^2+c)^2/x^( 1/2)
Time = 0.78 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{c} d^{3/4} \left (b^2 c^2 x^2 \left (c-3 d x^2\right )-2 a b c d x^2 \left (9 c+5 d x^2\right )+a^2 d \left (32 c^2+81 c d x^2+45 d^2 x^4\right )\right )}{\sqrt {x} \left (c+d x^2\right )^2}-\sqrt {2} \left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (3 b^2 c^2+10 a b c d-45 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{64 c^{13/4} d^{7/4}} \]
((-4*c^(1/4)*d^(3/4)*(b^2*c^2*x^2*(c - 3*d*x^2) - 2*a*b*c*d*x^2*(9*c + 5*d *x^2) + a^2*d*(32*c^2 + 81*c*d*x^2 + 45*d^2*x^4)))/(Sqrt[x]*(c + d*x^2)^2) - Sqrt[2]*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d] *x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - Sqrt[2]*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]* x)])/(64*c^(13/4)*d^(7/4))
Time = 0.57 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.85, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {365, 27, 362, 253, 266, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {x} \left (b^2 c x^2+a (2 b c-9 a d)\right )}{2 \left (d x^2+c\right )^3}dx}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {x} \left (b^2 c x^2+a (2 b c-9 a d)\right )}{\left (d x^2+c\right )^3}dx}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \int \frac {\sqrt {x}}{\left (d x^2+c\right )^2}dx+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\int \frac {\sqrt {x}}{d x^2+c}dx}{4 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\int \frac {x}{d x^2+c}d\sqrt {x}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )+\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {x^{3/2} \left (-\frac {9 a^2 d}{c}+2 a b-\frac {b^2 c}{d}\right )}{4 \left (c+d x^2\right )^2}+\frac {1}{8} \left (\frac {5 a (2 b c-9 a d)}{c}+\frac {3 b^2 c}{d}\right ) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}}{2 c}+\frac {x^{3/2}}{2 c \left (c+d x^2\right )}\right )}{c}-\frac {2 a^2}{c \sqrt {x} \left (c+d x^2\right )^2}\) |
(-2*a^2)/(c*Sqrt[x]*(c + d*x^2)^2) + (((2*a*b - (b^2*c)/d - (9*a^2*d)/c)*x ^(3/2))/(4*(c + d*x^2)^2) + (((3*b^2*c)/d + (5*a*(2*b*c - 9*a*d))/c)*(x^(3 /2)/(2*c*(c + d*x^2)) + ((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/ (Sqrt[2]*c^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] /(Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[d]) - (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1 /4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4))) /(2*Sqrt[d]))/(2*c)))/8)/c
3.5.38.3.1 Defintions of rubi rules used
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] :> 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
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[((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 = 2.72 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54
| method | result | size |
| derivativedivides | \(-\frac {2 a^{2}}{c^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {13}{32} a^{2} d^{2}-\frac {5}{16} a b c d -\frac {3}{32} b^{2} c^{2}\right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-18 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (45 a^{2} d^{2}-10 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3}}\) | \(216\) |
| default | \(-\frac {2 a^{2}}{c^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {13}{32} a^{2} d^{2}-\frac {5}{16} a b c d -\frac {3}{32} b^{2} c^{2}\right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-18 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (45 a^{2} d^{2}-10 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3}}\) | \(216\) |
| risch | \(-\frac {2 a^{2}}{c^{3} \sqrt {x}}-\frac {\frac {2 \left (\frac {13}{32} a^{2} d^{2}-\frac {5}{16} a b c d -\frac {3}{32} b^{2} c^{2}\right ) x^{\frac {7}{2}}+\frac {c \left (17 a^{2} d^{2}-18 a b c d +b^{2} c^{2}\right ) x^{\frac {3}{2}}}{16 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (45 a^{2} d^{2}-10 a b c d -3 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{c^{3}}\) | \(217\) |
-2*a^2/c^3/x^(1/2)-2/c^3*(((13/32*a^2*d^2-5/16*a*b*c*d-3/32*b^2*c^2)*x^(7/ 2)+1/32*c*(17*a^2*d^2-18*a*b*c*d+b^2*c^2)/d*x^(3/2))/(d*x^2+c)^2+1/256*(45 *a^2*d^2-10*a*b*c*d-3*b^2*c^2)/d^2/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)* x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+ 2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/ 2)-1)))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1540, normalized size of antiderivative = 3.86 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/64*((c^3*d^3*x^5 + 2*c^4*d^2*x^3 + c^5*d*x)*(-(81*b^8*c^8 + 1080*a*b^7* c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^ 4 + 549000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7))^(1/4)*log(c^10*d^5*(-(81*b^8*c^8 + 1080*a*b ^7*c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4 *d^4 + 549000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d ^7 + 4100625*a^8*d^8)/(c^13*d^7))^(3/4) - (27*b^6*c^6 + 270*a*b^5*c^5*d - 315*a^2*b^4*c^4*d^2 - 7100*a^3*b^3*c^3*d^3 + 4725*a^4*b^2*c^2*d^4 + 60750* a^5*b*c*d^5 - 91125*a^6*d^6)*sqrt(x)) + (-I*c^3*d^3*x^5 - 2*I*c^4*d^2*x^3 - I*c^5*d*x)*(-(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^2*b^6*c^6*d^2 - 3660 0*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c^3*d^5 + 12150 0*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7))^(1/ 4)*log(I*c^10*d^5*(-(81*b^8*c^8 + 1080*a*b^7*c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4*c^4*d^4 + 549000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^7*b*c*d^7 + 4100625*a^8*d^8)/(c^13*d^7) )^(3/4) - (27*b^6*c^6 + 270*a*b^5*c^5*d - 315*a^2*b^4*c^4*d^2 - 7100*a^3*b ^3*c^3*d^3 + 4725*a^4*b^2*c^2*d^4 + 60750*a^5*b*c*d^5 - 91125*a^6*d^6)*sqr t(x)) + (I*c^3*d^3*x^5 + 2*I*c^4*d^2*x^3 + I*c^5*d*x)*(-(81*b^8*c^8 + 1080 *a*b^7*c^7*d + 540*a^2*b^6*c^6*d^2 - 36600*a^3*b^5*c^5*d^3 - 42650*a^4*b^4 *c^4*d^4 + 549000*a^5*b^3*c^3*d^5 + 121500*a^6*b^2*c^2*d^6 - 3645000*a^...
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {32 \, a^{2} c^{2} d - {\left (3 \, b^{2} c^{2} d + 10 \, a b c d^{2} - 45 \, a^{2} d^{3}\right )} x^{4} + {\left (b^{2} c^{3} - 18 \, a b c^{2} d + 81 \, a^{2} c d^{2}\right )} x^{2}}{16 \, {\left (c^{3} d^{3} x^{\frac {9}{2}} + 2 \, c^{4} d^{2} x^{\frac {5}{2}} + c^{5} d \sqrt {x}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 10 \, a b c d - 45 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c^{3} d} \]
-1/16*(32*a^2*c^2*d - (3*b^2*c^2*d + 10*a*b*c*d^2 - 45*a^2*d^3)*x^4 + (b^2 *c^3 - 18*a*b*c^2*d + 81*a^2*c*d^2)*x^2)/(c^3*d^3*x^(9/2) + 2*c^4*d^2*x^(5 /2) + c^5*d*sqrt(x)) + 1/128*(3*b^2*c^2 + 10*a*b*c*d - 45*a^2*d^2)*(2*sqrt (2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt( sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2* sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d) ))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*s qrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^( 1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(c^3*d)
Time = 0.29 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {2 \, a^{2}}{c^{3} \sqrt {x}} + \frac {3 \, b^{2} c^{2} d x^{\frac {7}{2}} + 10 \, a b c d^{2} x^{\frac {7}{2}} - 13 \, a^{2} d^{3} x^{\frac {7}{2}} - b^{2} c^{3} x^{\frac {3}{2}} + 18 \, a b c^{2} d x^{\frac {3}{2}} - 17 \, a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4} d^{4}} - \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{4}} + \frac {\sqrt {2} {\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 45 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{4} d^{4}} \]
-2*a^2/(c^3*sqrt(x)) + 1/16*(3*b^2*c^2*d*x^(7/2) + 10*a*b*c*d^2*x^(7/2) - 13*a^2*d^3*x^(7/2) - b^2*c^3*x^(3/2) + 18*a*b*c^2*d*x^(3/2) - 17*a^2*c*d^2 *x^(3/2))/((d*x^2 + c)^2*c^3*d) + 1/64*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 10*(c*d^3)^(3/4)*a*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(s qrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^4) + 1/64*sqrt(2)*(3*( c*d^3)^(3/4)*b^2*c^2 + 10*(c*d^3)^(3/4)*a*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2 )*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^4* d^4) - 1/128*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 10*(c*d^3)^(3/4)*a*b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d) )/(c^4*d^4) + 1/128*sqrt(2)*(3*(c*d^3)^(3/4)*b^2*c^2 + 10*(c*d^3)^(3/4)*a* b*c*d - 45*(c*d^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + s qrt(c/d))/(c^4*d^4)
Time = 5.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{13/4}\,d^{7/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{13/4}\,d^{7/4}}-\frac {\frac {2\,a^2}{c}-\frac {x^4\,\left (-45\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{16\,c^3}+\frac {x^2\,\left (81\,a^2\,d^2-18\,a\,b\,c\,d+b^2\,c^2\right )}{16\,c^2\,d}}{c^2\,\sqrt {x}+d^2\,x^{9/2}+2\,c\,d\,x^{5/2}} \]
(atanh((d^(1/4)*x^(1/2))/(-c)^(1/4))*(3*b^2*c^2 - 45*a^2*d^2 + 10*a*b*c*d) )/(32*(-c)^(13/4)*d^(7/4)) - (atan((d^(1/4)*x^(1/2))/(-c)^(1/4))*(3*b^2*c^ 2 - 45*a^2*d^2 + 10*a*b*c*d))/(32*(-c)^(13/4)*d^(7/4)) - ((2*a^2)/c - (x^4 *(3*b^2*c^2 - 45*a^2*d^2 + 10*a*b*c*d))/(16*c^3) + (x^2*(81*a^2*d^2 + b^2* c^2 - 18*a*b*c*d))/(16*c^2*d))/(c^2*x^(1/2) + d^2*x^(9/2) + 2*c*d*x^(5/2))