Integrand size = 24, antiderivative size = 375 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {(13 b c-5 a d) (b c-a d) \sqrt {x}}{2 d^4}-\frac {(13 b c-5 a d) (b c-a d) x^{5/2}}{10 c d^3}+\frac {2 b^2 x^{9/2}}{9 d^2}+\frac {(b c-a d)^2 x^{9/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{17/4}}+\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}}-\frac {\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{17/4}} \]
-1/10*(-5*a*d+13*b*c)*(-a*d+b*c)*x^(5/2)/c/d^3+2/9*b^2*x^(9/2)/d^2+1/2*(-a *d+b*c)^2*x^(9/2)/c/d^2/(d*x^2+c)+1/8*c^(1/4)*(-5*a*d+13*b*c)*(-a*d+b*c)*a rctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(17/4)*2^(1/2)-1/8*c^(1/4)*(-5* a*d+13*b*c)*(-a*d+b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(17/4)* 2^(1/2)+1/16*c^(1/4)*(-5*a*d+13*b*c)*(-a*d+b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/ 4)*d^(1/4)*2^(1/2)*x^(1/2))/d^(17/4)*2^(1/2)-1/16*c^(1/4)*(-5*a*d+13*b*c)* (-a*d+b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/d^(17/4)* 2^(1/2)+1/2*(-5*a*d+13*b*c)*(-a*d+b*c)*x^(1/2)/d^4
Time = 0.75 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.68 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{d} \sqrt {x} \left (45 a^2 d^2 \left (5 c+4 d x^2\right )+18 a b d \left (-45 c^2-36 c d x^2+4 d^2 x^4\right )+b^2 \left (585 c^3+468 c^2 d x^2-52 c d^2 x^4+20 d^3 x^6\right )\right )}{c+d x^2}+45 \sqrt {2} \sqrt [4]{c} \left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-45 \sqrt {2} \sqrt [4]{c} \left (13 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{360 d^{17/4}} \]
((4*d^(1/4)*Sqrt[x]*(45*a^2*d^2*(5*c + 4*d*x^2) + 18*a*b*d*(-45*c^2 - 36*c *d*x^2 + 4*d^2*x^4) + b^2*(585*c^3 + 468*c^2*d*x^2 - 52*c*d^2*x^4 + 20*d^3 *x^6)))/(c + d*x^2) + 45*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2* d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - 45* Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[2]*c^( 1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(360*d^(17/4))
Time = 0.55 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.87, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {366, 27, 363, 262, 262, 266, 755, 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.
| \(\displaystyle \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{7/2} \left ((3 b c-5 a d) (3 b c-a d)-4 b^2 c d x^2\right )}{2 \left (d x^2+c\right )}dx}{2 c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{7/2} \left ((3 b c-5 a d) (3 b c-a d)-4 b^2 c d x^2\right )}{d x^2+c}dx}{4 c d^2}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \int \frac {x^{7/2}}{d x^2+c}dx-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \int \frac {x^{3/2}}{d x^2+c}dx}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {c \int \frac {1}{\sqrt {x} \left (d x^2+c\right )}dx}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \int \frac {1}{d x^2+c}d\sqrt {x}}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(13 b c-5 a d) (b c-a d) \left (\frac {2 x^{5/2}}{5 d}-\frac {c \left (\frac {2 \sqrt {x}}{d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{d}\right )-\frac {8}{9} b^2 c x^{9/2}}{4 c d^2}\) |
((b*c - a*d)^2*x^(9/2))/(2*c*d^2*(c + d*x^2)) - ((-8*b^2*c*x^(9/2))/9 + (1 3*b*c - 5*a*d)*(b*c - a*d)*((2*x^(5/2))/(5*d) - (c*((2*Sqrt[x])/d - (2*c*( (-(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[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr t[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[c])))/d))/d))/(4 *c*d^2)
3.5.25.3.1 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*(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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -1]
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[((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 = 2.81 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {2 \left (5 b^{2} d^{2} x^{4}+18 x^{2} a b \,d^{2}-18 x^{2} b^{2} c d +45 a^{2} d^{2}-180 a b c d +135 b^{2} c^{2}\right ) \sqrt {x}}{45 d^{4}}-\frac {c \left (2 a d -2 b c \right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a d -13 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(216\) |
| derivativedivides | \(\frac {\frac {2 b^{2} d^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {4 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-8 a b c d \sqrt {x}+6 b^{2} c^{2} \sqrt {x}}{d^{4}}-\frac {2 c \left (\frac {\left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(240\) |
| default | \(\frac {\frac {2 b^{2} d^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} x^{\frac {5}{2}}}{5}-\frac {4 b^{2} c d \,x^{\frac {5}{2}}}{5}+2 a^{2} d^{2} \sqrt {x}-8 a b c d \sqrt {x}+6 b^{2} c^{2} \sqrt {x}}{d^{4}}-\frac {2 c \left (\frac {\left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{d^{4}}\) | \(240\) |
2/45*(5*b^2*d^2*x^4+18*a*b*d^2*x^2-18*b^2*c*d*x^2+45*a^2*d^2-180*a*b*c*d+1 35*b^2*c^2)*x^(1/2)/d^4-c/d^4*(2*a*d-2*b*c)*((-1/4*a*d+1/4*b*c)*x^(1/2)/(d *x^2+c)+1/32*(5*a*d-13*b*c)*(c/d)^(1/4)/c*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*arc tan(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 = 0.30 (sec) , antiderivative size = 1248, normalized size of antiderivative = 3.33 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
-1/360*(45*(d^5*x^2 + c*d^4)*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 37247 6*a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 1868 40*a^5*b^3*c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8* c*d^8)/d^17)^(1/4)*log(d^4*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476* a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840 *a^5*b^3*c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c* d^8)/d^17)^(1/4) + (13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*sqrt(x)) + 45*(I* d^5*x^2 + I*c*d^4)*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^6* c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3 *c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^1 7)^(1/4)*log(I*d^4*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^6* c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3 *c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^1 7)^(1/4) + (13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*sqrt(x)) + 45*(-I*d^5*x^2 - I*c*d^4)*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3*c^4*d^ 5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^17)^(1/4 )*log(-I*d^4*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^6*c^7*d^ 2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3*c^4*d ^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^17)^...
Timed out. \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.01 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {x}}{2 \, {\left (d^{5} x^{2} + c d^{4}\right )}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (13 \, b^{2} c^{2} - 18 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )} c}{16 \, d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{2} x^{\frac {9}{2}} - 18 \, {\left (b^{2} c d - a b d^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}\right )}}{45 \, d^{4}} \]
1/2*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(x)/(d^5*x^2 + c*d^4) - 1/16*( 2*sqrt(2)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^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(c)*sqr t(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*arct an(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c) *sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(13*b^2*c^2 - 18*a*b* c*d + 5*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c) )/(c^(3/4)*d^(1/4)) - sqrt(2)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*log(-s qrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))*c /d^4 + 2/45*(5*b^2*d^2*x^(9/2) - 18*(b^2*c*d - a*b*d^2)*x^(5/2) + 45*(3*b^ 2*c^2 - 4*a*b*c*d + a^2*d^2)*sqrt(x))/d^4
Time = 0.31 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.17 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=-\frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, d^{5}} - \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {\sqrt {2} {\left (13 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, d^{5}} + \frac {b^{2} c^{3} \sqrt {x} - 2 \, a b c^{2} d \sqrt {x} + a^{2} c d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} d^{4}} + \frac {2 \, {\left (5 \, b^{2} d^{16} x^{\frac {9}{2}} - 18 \, b^{2} c d^{15} x^{\frac {5}{2}} + 18 \, a b d^{16} x^{\frac {5}{2}} + 135 \, b^{2} c^{2} d^{14} \sqrt {x} - 180 \, a b c d^{15} \sqrt {x} + 45 \, a^{2} d^{16} \sqrt {x}\right )}}{45 \, d^{18}} \]
-1/8*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d ^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c /d)^(1/4))/d^5 - 1/8*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)* a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4 ) - 2*sqrt(x))/(c/d)^(1/4))/d^5 - 1/16*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*( c/d)^(1/4) + x + sqrt(c/d))/d^5 + 1/16*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)* (c/d)^(1/4) + x + sqrt(c/d))/d^5 + 1/2*(b^2*c^3*sqrt(x) - 2*a*b*c^2*d*sqrt (x) + a^2*c*d^2*sqrt(x))/((d*x^2 + c)*d^4) + 2/45*(5*b^2*d^16*x^(9/2) - 18 *b^2*c*d^15*x^(5/2) + 18*a*b*d^16*x^(5/2) + 135*b^2*c^2*d^14*sqrt(x) - 180 *a*b*c*d^15*sqrt(x) + 45*a^2*d^16*sqrt(x))/d^18
Time = 4.94 (sec) , antiderivative size = 1367, normalized size of antiderivative = 3.65 \[ \int \frac {x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
x^(1/2)*((2*a^2)/d^2 + (2*c*((4*b^2*c)/d^3 - (4*a*b)/d^2))/d - (2*b^2*c^2) /d^4) - x^(5/2)*((4*b^2*c)/(5*d^3) - (4*a*b)/(5*d^2)) + (x^(1/2)*((b^2*c^3 )/2 + (a^2*c*d^2)/2 - a*b*c^2*d))/(c*d^4 + d^5*x^2) + (2*b^2*x^(9/2))/(9*d ^2) + ((-c)^(1/4)*atan((((-c)^(1/4)*((x^(1/2)*(169*b^4*c^6 + 25*a^4*c^2*d^ 4 - 180*a^3*b*c^3*d^3 + 454*a^2*b^2*c^4*d^2 - 468*a*b^3*c^5*d))/d^5 + ((-c )^(1/4)*(a*d - b*c)*(5*a*d - 13*b*c)*(13*b^2*c^4 + 5*a^2*c^2*d^2 - 18*a*b* c^3*d))/d^(21/4))*(a*d - b*c)*(5*a*d - 13*b*c)*1i)/(8*d^(17/4)) + ((-c)^(1 /4)*((x^(1/2)*(169*b^4*c^6 + 25*a^4*c^2*d^4 - 180*a^3*b*c^3*d^3 + 454*a^2* b^2*c^4*d^2 - 468*a*b^3*c^5*d))/d^5 - ((-c)^(1/4)*(a*d - b*c)*(5*a*d - 13* b*c)*(13*b^2*c^4 + 5*a^2*c^2*d^2 - 18*a*b*c^3*d))/d^(21/4))*(a*d - b*c)*(5 *a*d - 13*b*c)*1i)/(8*d^(17/4)))/(((-c)^(1/4)*((x^(1/2)*(169*b^4*c^6 + 25* a^4*c^2*d^4 - 180*a^3*b*c^3*d^3 + 454*a^2*b^2*c^4*d^2 - 468*a*b^3*c^5*d))/ d^5 + ((-c)^(1/4)*(a*d - b*c)*(5*a*d - 13*b*c)*(13*b^2*c^4 + 5*a^2*c^2*d^2 - 18*a*b*c^3*d))/d^(21/4))*(a*d - b*c)*(5*a*d - 13*b*c))/(8*d^(17/4)) - ( (-c)^(1/4)*((x^(1/2)*(169*b^4*c^6 + 25*a^4*c^2*d^4 - 180*a^3*b*c^3*d^3 + 4 54*a^2*b^2*c^4*d^2 - 468*a*b^3*c^5*d))/d^5 - ((-c)^(1/4)*(a*d - b*c)*(5*a* d - 13*b*c)*(13*b^2*c^4 + 5*a^2*c^2*d^2 - 18*a*b*c^3*d))/d^(21/4))*(a*d - b*c)*(5*a*d - 13*b*c))/(8*d^(17/4))))*(a*d - b*c)*(5*a*d - 13*b*c)*1i)/(4* d^(17/4)) - ((-c)^(1/4)*atan((((-c)^(1/4)*((x^(1/2)*(169*b^4*c^6 + 25*a^4* c^2*d^4 - 180*a^3*b*c^3*d^3 + 454*a^2*b^2*c^4*d^2 - 468*a*b^3*c^5*d))/d...