Integrand size = 24, antiderivative size = 346 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(11 b c-3 a d) (b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}} \]
-1/6*(-3*a*d+11*b*c)*(-a*d+b*c)*x^(3/2)/c/d^3+2/7*b^2*x^(7/2)/d^2+1/2*(-a* d+b*c)^2*x^(7/2)/c/d^2/(d*x^2+c)-1/8*(-3*a*d+11*b*c)*(-a*d+b*c)*arctan(1-d ^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(1/4)/d^(15/4)*2^(1/2)+1/8*(-3*a*d+11*b* c)*(-a*d+b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(1/4)/d^(15/4)*2 ^(1/2)+1/16*(-3*a*d+11*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))/c^(1/4)/d^(15/4)*2^(1/2)-1/16*(-3*a*d+11*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))/c^(1/4)/d^(15/4)*2 ^(1/2)
Time = 0.66 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.64 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {\frac {4 d^{3/4} x^{3/2} \left (-21 a^2 d^2+14 a b d \left (7 c+4 d x^2\right )+b^2 \left (-77 c^2-44 c d x^2+12 d^2 x^4\right )\right )}{c+d x^2}-\frac {21 \sqrt {2} \left (11 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{c}}-\frac {21 \sqrt {2} \left (11 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{c}}}{168 d^{15/4}} \]
((4*d^(3/4)*x^(3/2)*(-21*a^2*d^2 + 14*a*b*d*(7*c + 4*d*x^2) + b^2*(-77*c^2 - 44*c*d*x^2 + 12*d^2*x^4)))/(c + d*x^2) - (21*Sqrt[2]*(11*b^2*c^2 - 14*a *b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)* Sqrt[x])])/c^(1/4) - (21*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Arc Tanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/c^(1/4))/(1 68*d^(15/4))
Time = 0.53 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {366, 27, 363, 262, 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.
| \(\displaystyle \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle \frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {\int -\frac {x^{5/2} \left (4 a^2 d^2+4 b^2 c x^2 d-7 (b c-a d)^2\right )}{2 \left (d x^2+c\right )}dx}{2 c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^{5/2} \left (4 a^2 d^2+4 b^2 c x^2 d-7 (b c-a d)^2\right )}{d x^2+c}dx}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \int \frac {x^{5/2}}{d x^2+c}dx}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {c \int \frac {\sqrt {x}}{d x^2+c}dx}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \int \frac {x}{d x^2+c}d\sqrt {x}}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {8}{7} b^2 c x^{7/2}-(11 b c-3 a d) (b c-a d) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c d^2}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}\) |
((b*c - a*d)^2*x^(7/2))/(2*c*d^2*(c + d*x^2)) + ((8*b^2*c*x^(7/2))/7 - (11 *b*c - 3*a*d)*(b*c - a*d)*((2*x^(3/2))/(3*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[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])))/d))/(4*c*d^2)
3.5.26.3.1 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[(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 = 2.81 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(\frac {2 b \,x^{\frac {3}{2}} \left (3 b d \,x^{2}+14 a d -14 b c \right )}{21 d^{3}}+\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (-\frac {11 b c}{4}+\frac {3 a d}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{d^{3}}\) | \(177\) |
| derivativedivides | \(\frac {2 b \left (\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -2 b c \right ) x^{\frac {3}{2}}}{3}\right )}{d^{3}}+\frac {\frac {2 \left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (-\frac {7}{2} a b c d +\frac {11}{4} b^{2} c^{2}+\frac {3}{4} a^{2} d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\) | \(200\) |
| default | \(\frac {2 b \left (\frac {b d \,x^{\frac {7}{2}}}{7}+\frac {\left (2 a d -2 b c \right ) x^{\frac {3}{2}}}{3}\right )}{d^{3}}+\frac {\frac {2 \left (-\frac {1}{4} a^{2} d^{2}+\frac {1}{2} a b c d -\frac {1}{4} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (-\frac {7}{2} a b c d +\frac {11}{4} b^{2} c^{2}+\frac {3}{4} a^{2} d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\) | \(200\) |
2/21*b*x^(3/2)*(3*b*d*x^2+14*a*d-14*b*c)/d^3+1/d^3*(2*a*d-2*b*c)*((-1/4*a* d+1/4*b*c)*x^(3/2)/(d*x^2+c)+1/8*(-11/4*b*c+3/4*a*d)/d/(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 = 0.31 (sec) , antiderivative size = 1444, normalized size of antiderivative = 4.17 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
1/168*(21*(d^4*x^2 + c*d^3)*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268* a^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560* a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/( c*d^15))^(1/4)*log(c*d^11*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^ 2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^ 5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c* d^15))^(3/4) + (1331*b^6*c^6 - 5082*a*b^5*c^5*d + 7557*a^2*b^4*c^4*d^2 - 5 516*a^3*b^3*c^3*d^3 + 2061*a^4*b^2*c^2*d^4 - 378*a^5*b*c*d^5 + 27*a^6*d^6) *sqrt(x)) - 21*(I*d^4*x^2 + I*c*d^3)*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a ^8*d^8)/(c*d^15))^(1/4)*log(I*c*d^11*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a ^8*d^8)/(c*d^15))^(3/4) + (1331*b^6*c^6 - 5082*a*b^5*c^5*d + 7557*a^2*b^4* c^4*d^2 - 5516*a^3*b^3*c^3*d^3 + 2061*a^4*b^2*c^2*d^4 - 378*a^5*b*c*d^5 + 27*a^6*d^6)*sqrt(x)) - 21*(-I*d^4*x^2 - I*c*d^3)*(-(14641*b^8*c^8 - 74536* a*b^7*c^7*d + 158268*a^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4 *b^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b* c*d^7 + 81*a^8*d^8)/(c*d^15))^(1/4)*log(-I*c*d^11*(-(14641*b^8*c^8 - 74...
Timed out. \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.79 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=-\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}}}{2 \, {\left (d^{4} x^{2} + c d^{3}\right )}} + \frac {{\left (11 \, b^{2} c^{2} - 14 \, a b c d + 3 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, d^{3}} + \frac {2 \, {\left (3 \, b^{2} d x^{\frac {7}{2}} - 14 \, {\left (b^{2} c - a b d\right )} x^{\frac {3}{2}}\right )}}{21 \, d^{3}} \]
-1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^(3/2)/(d^4*x^2 + c*d^3) + 1/16*(11* b^2*c^2 - 14*a*b*c*d + 3*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)*s qrt(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)*sqrt(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)))/d^3 + 2/21*(3*b^2*d*x^(7/2) - 14*(b^2*c - a* b*d)*x^(3/2))/d^3
Time = 0.32 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.19 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=-\frac {b^{2} c^{2} x^{\frac {3}{2}} - 2 \, a b c d x^{\frac {3}{2}} + a^{2} d^{2} x^{\frac {3}{2}}}{2 \, {\left (d x^{2} + c\right )} d^{3}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c d^{6}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c d^{6}} - \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c d^{6}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c d^{6}} + \frac {2 \, {\left (3 \, b^{2} d^{12} x^{\frac {7}{2}} - 14 \, b^{2} c d^{11} x^{\frac {3}{2}} + 14 \, a b d^{12} x^{\frac {3}{2}}\right )}}{21 \, d^{14}} \]
-1/2*(b^2*c^2*x^(3/2) - 2*a*b*c*d*x^(3/2) + a^2*d^2*x^(3/2))/((d*x^2 + c)* d^3) + 1/8*sqrt(2)*(11*(c*d^3)^(3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*c*d + 3*(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*d^6) + 1/8*sqrt(2)*(11*(c*d^3)^(3/4)*b^2*c^2 - 14*(c*d ^3)^(3/4)*a*b*c*d + 3*(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*d^6) - 1/16*sqrt(2)*(11*(c*d^3)^( 3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*c*d + 3*(c*d^3)^(3/4)*a^2*d^2)*log(sqr t(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^6) + 1/16*sqrt(2)*(11*(c*d^ 3)^(3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*c*d + 3*(c*d^3)^(3/4)*a^2*d^2)*log (-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^6) + 2/21*(3*b^2*d^12* x^(7/2) - 14*b^2*c*d^11*x^(3/2) + 14*a*b*d^12*x^(3/2))/d^14
Time = 5.47 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.46 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx=\frac {2\,b^2\,x^{7/2}}{7\,d^2}-\frac {x^{3/2}\,\left (\frac {a^2\,d^2}{2}-a\,b\,c\,d+\frac {b^2\,c^2}{2}\right )}{d^4\,x^2+c\,d^3}-x^{3/2}\,\left (\frac {4\,b^2\,c}{3\,d^3}-\frac {4\,a\,b}{3\,d^2}\right )+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-11\,b\,c\right )}{4\,{\left (-c\right )}^{1/4}\,d^{15/4}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-11\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{1/4}\,d^{15/4}} \]
(2*b^2*x^(7/2))/(7*d^2) - (x^(3/2)*((a^2*d^2)/2 + (b^2*c^2)/2 - a*b*c*d))/ (c*d^3 + d^4*x^2) - x^(3/2)*((4*b^2*c)/(3*d^3) - (4*a*b)/(3*d^2)) + (atan( (d^(1/4)*x^(1/2))/(-c)^(1/4))*(a*d - b*c)*(3*a*d - 11*b*c))/(4*(-c)^(1/4)* d^(15/4)) + (atan((d^(1/4)*x^(1/2)*1i)/(-c)^(1/4))*(a*d - b*c)*(3*a*d - 11 *b*c)*1i)/(4*(-c)^(1/4)*d^(15/4))