Integrand size = 24, antiderivative size = 332 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}-\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) \sqrt {x}}{6 c^2 d \left (c+d x^2\right )}-\frac {(b c-a d) (b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{11/4} d^{5/4}}-\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(b c-a d) (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{11/4} d^{5/4}} \]
-2/3*a^2/c/x^(3/2)/(d*x^2+c)-1/8*(-a*d+b*c)*(7*a*d+b*c)*arctan(1-d^(1/4)*2 ^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/d^(5/4)*2^(1/2)+1/8*(-a*d+b*c)*(7*a*d+b*c )*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/d^(5/4)*2^(1/2)-1/16* (-a*d+b*c)*(7*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^(11/4)/d^(5/4)*2^(1/2)+1/16*(-a*d+b*c)*(7*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^(11/4)/d^(5/4)*2^(1/2)-1/6*(7*a^2*d^ 2-6*a*b*c*d+3*b^2*c^2)*x^(1/2)/c^2/d/(d*x^2+c)
Time = 0.74 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.63 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\frac {-\frac {4 c^{3/4} \sqrt [4]{d} \left (3 b^2 c^2 x^2-6 a b c d x^2+a^2 d \left (4 c+7 d x^2\right )\right )}{x^{3/2} \left (c+d x^2\right )}-3 \sqrt {2} \left (b^2 c^2+6 a b c d-7 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+3 \sqrt {2} \left (b^2 c^2+6 a b c d-7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{24 c^{11/4} d^{5/4}} \]
((-4*c^(3/4)*d^(1/4)*(3*b^2*c^2*x^2 - 6*a*b*c*d*x^2 + a^2*d*(4*c + 7*d*x^2 )))/(x^(3/2)*(c + d*x^2)) - 3*Sqrt[2]*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*Ar cTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] + 3*Sqrt[2]* (b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] )/(Sqrt[c] + Sqrt[d]*x)])/(24*c^(11/4)*d^(5/4))
Time = 0.55 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {365, 27, 362, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {3 b^2 c x^2+a (6 b c-7 a d)}{2 \sqrt {x} \left (d x^2+c\right )^2}dx}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 b^2 c x^2+a (6 b c-7 a d)}{\sqrt {x} \left (d x^2+c\right )^2}dx}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \int \frac {1}{\sqrt {x} \left (d x^2+c\right )}dx}{4 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \int \frac {1}{d x^2+c}d\sqrt {x}}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c d}+\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {x} \left (-\frac {7 a^2 d}{c}+6 a b-\frac {3 b^2 c}{d}\right )}{2 \left (c+d x^2\right )}+\frac {3 (b c-a d) (7 a d+b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{2 c d}}{3 c}-\frac {2 a^2}{3 c x^{3/2} \left (c+d x^2\right )}\) |
(-2*a^2)/(3*c*x^(3/2)*(c + d*x^2)) + (((6*a*b - (3*b^2*c)/d - (7*a^2*d)/c) *Sqrt[x])/(2*(c + d*x^2)) + (3*(b*c - a*d)*(b*c + 7*a*d)*((-(ArcTan[1 - (S qrt[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] + 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[c])))/(2*c*d))/(3*c)
3.5.31.3.1 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] :> 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[((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 = 2.76 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(-\frac {2 a^{2}}{3 c^{2} x^{\frac {3}{2}}}-\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a d +b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d c}\right )}{c^{2}}\) | \(169\) |
| derivativedivides | \(-\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a^{2} d^{2}-6 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d c}\right )}{c^{2}}-\frac {2 a^{2}}{3 c^{2} x^{\frac {3}{2}}}\) | \(188\) |
| default | \(-\frac {2 \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a^{2} d^{2}-6 a b c d -b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d c}\right )}{c^{2}}-\frac {2 a^{2}}{3 c^{2} x^{\frac {3}{2}}}\) | \(188\) |
-2/3*a^2/c^2/x^(3/2)-1/c^2*(2*a*d-2*b*c)*(1/4/d*(a*d-b*c)*x^(1/2)/(d*x^2+c )+1/32*(7*a*d+b*c)/d*(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*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 = 0.29 (sec) , antiderivative size = 1219, normalized size of antiderivative = 3.67 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
-1/24*(3*(c^2*d^2*x^4 + c^3*d*x^2)*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b ^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3 *d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5)) ^(1/4)*log(c^3*d*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a ^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^ 2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4) - (b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*sqrt(x)) + 3*(I*c^2*d^2*x^4 + I*c^3*d*x^2)*(-(b^8 *c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a ^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b* c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4)*log(I*c^3*d*(-(b^8*c^8 + 24*a*b^7* c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8 *d^8)/(c^11*d^5))^(1/4) - (b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*sqrt(x)) + 3*( -I*c^2*d^2*x^4 - I*c^3*d*x^2)*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^ 6*d^2 + 360*a^3*b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2*c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4 )*log(-I*c^3*d*(-(b^8*c^8 + 24*a*b^7*c^7*d + 188*a^2*b^6*c^6*d^2 + 360*a^3 *b^5*c^5*d^3 - 1434*a^4*b^4*c^4*d^4 - 2520*a^5*b^3*c^3*d^5 + 9212*a^6*b^2* c^2*d^6 - 8232*a^7*b*c*d^7 + 2401*a^8*d^8)/(c^11*d^5))^(1/4) - (b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*sqrt(x)) - 3*(c^2*d^2*x^4 + c^3*d*x^2)*(-(b^8*c^...
Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (313) = 626\).
Time = 112.77 (sec) , antiderivative size = 1418, normalized size of antiderivative = 4.27 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*a**2/(11*x**(11/2)) - 4*a*b/(7*x**(7/2)) - 2*b**2/(3*x* *(3/2))), Eq(c, 0) & Eq(d, 0)), ((-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2 *b**2*x**(5/2)/5)/c**2, Eq(d, 0)), ((-2*a**2/(11*x**(11/2)) - 4*a*b/(7*x** (7/2)) - 2*b**2/(3*x**(3/2)))/d**2, Eq(c, 0)), (-16*a**2*c**2*d/(24*c**4*d *x**(3/2) + 24*c**3*d**2*x**(7/2)) + 21*a**2*c*d**2*x**(3/2)*(-c/d)**(1/4) *log(sqrt(x) - (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 21*a**2*c*d**2*x**(3/2)*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(24* c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 42*a**2*c*d**2*x**(3/2)*(-c/d)* *(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**( 7/2)) - 28*a**2*c*d**2*x**2/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 21*a**2*d**3*x**(7/2)*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(24*c**4 *d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 21*a**2*d**3*x**(7/2)*(-c/d)**(1/4) *log(sqrt(x) + (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 42*a**2*d**3*x**(7/2)*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c** 4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) - 18*a*b*c**2*d*x**(3/2)*(-c/d)**(1/ 4)*log(sqrt(x) - (-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2 )) + 18*a*b*c**2*d*x**(3/2)*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(24 *c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2)) + 36*a*b*c**2*d*x**(3/2)*(-c/d)* *(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**( 7/2)) + 24*a*b*c**2*d*x**2/(24*c**4*d*x**(3/2) + 24*c**3*d**2*x**(7/2))...
Time = 0.39 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=-\frac {4 \, a^{2} c d + {\left (3 \, b^{2} c^{2} - 6 \, a b c d + 7 \, a^{2} d^{2}\right )} x^{2}}{6 \, {\left (c^{2} d^{2} x^{\frac {7}{2}} + c^{3} d x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{2} c^{2} + 6 \, a b c d - 7 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, c^{2} d} \]
-1/6*(4*a^2*c*d + (3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x^2)/(c^2*d^2*x^(7/2 ) + c^3*d*x^(3/2)) + 1/16*(2*sqrt(2)*(b^2*c^2 + 6*a*b*c*d - 7*a^2*d^2)*arc tan(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))) + 2*sqrt(2)*(b^2*c^2 + 6*a*b*c* d - 7*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sq rt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(b ^2*c^2 + 6*a*b*c*d - 7*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)*(b^2*c^2 + 6*a*b*c*d - 7*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)))/(c^2*d)
Time = 0.29 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=-\frac {2 \, a^{2}}{3 \, c^{2} x^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{3} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{3} d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{3} d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{3} d^{2}} - \frac {b^{2} c^{2} \sqrt {x} - 2 \, a b c d \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} c^{2} d} \]
-2/3*a^2/(c^2*x^(3/2)) + 1/8*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1 /4)*a*b*c*d - 7*(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))/(c^3*d^2) + 1/8*sqrt(2)*((c*d^3)^(1/4)*b^2* c^2 + 6*(c*d^3)^(1/4)*a*b*c*d - 7*(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))/(c^3*d^2) + 1/16*sqrt(2) *((c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d - 7*(c*d^3)^(1/4)*a^2*d^ 2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^3*d^2) - 1/16*sqrt( 2)*((c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d - 7*(c*d^3)^(1/4)*a^2* d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^3*d^2) - 1/2*(b^ 2*c^2*sqrt(x) - 2*a*b*c*d*sqrt(x) + a^2*d^2*sqrt(x))/((d*x^2 + c)*c^2*d)
Time = 5.39 (sec) , antiderivative size = 1340, normalized size of antiderivative = 4.04 \[ \int \frac {\left (a+b x^2\right )^2}{x^{5/2} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
(atan((((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) - ((a*d - b*c)*(7*a*d + b*c)*( 256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8))/(8*(-c)^(11/4)*d ^(5/4)))*(a*d - b*c)*(7*a*d + b*c)*1i)/(8*(-c)^(11/4)*d^(5/4)) + ((x^(1/2) *(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7 *d^9 + 704*a^2*b^2*c^8*d^8) + ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8))/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c)*1i)/(8*(-c)^(11/4)*d^(5/4)))/(((x^(1/2)*(1568*a^4*c^6*d ^10 + 32*b^4*c^10*d^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b ^2*c^8*d^8) - ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9* d^9 + 1536*a*b*c^10*d^8))/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b* c))/(8*(-c)^(11/4)*d^(5/4)) - ((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d ^6 + 384*a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) + ((a*d - b*c)*(7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10 *d^8))/(8*(-c)^(11/4)*d^(5/4)))*(a*d - b*c)*(7*a*d + b*c))/(8*(-c)^(11/4)* d^(5/4))))*(a*d - b*c)*(7*a*d + b*c)*1i)/(4*(-c)^(11/4)*d^(5/4)) - ((2*a^2 )/(3*c) + (x^2*(7*a^2*d^2 + 3*b^2*c^2 - 6*a*b*c*d))/(6*c^2*d))/(c*x^(3/2) + d*x^(7/2)) + (atan((((x^(1/2)*(1568*a^4*c^6*d^10 + 32*b^4*c^10*d^6 + 384 *a*b^3*c^9*d^7 - 2688*a^3*b*c^7*d^9 + 704*a^2*b^2*c^8*d^8) - ((a*d - b*c)* (7*a*d + b*c)*(256*b^2*c^11*d^7 - 1792*a^2*c^9*d^9 + 1536*a*b*c^10*d^8)...