Integrand size = 24, antiderivative size = 288 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^2}{9 a x^{9/2}}+\frac {2 c (b c-2 a d)}{5 a^2 x^{5/2}}-\frac {2 (b c-a d)^2}{a^3 \sqrt {x}}+\frac {\sqrt [4]{b} (b c-a d)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (b c-a d)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{13/4}} \]
-2/9*c^2/a/x^(9/2)+2/5*c*(-2*a*d+b*c)/a^2/x^(5/2)+1/2*b^(1/4)*(-a*d+b*c)^2 *arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)*2^(1/2)-1/2*b^(1/4)*(- a*d+b*c)^2*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)*2^(1/2)-1/4* b^(1/4)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2)) /a^(13/4)*2^(1/2)+1/4*b^(1/4)*(-a*d+b*c)^2*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^ (1/4)*2^(1/2)*x^(1/2))/a^(13/4)*2^(1/2)-2*(-a*d+b*c)^2/a^3/x^(1/2)
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (45 b^2 c^2 x^4-9 a b c x^2 \left (c+10 d x^2\right )+a^2 \left (5 c^2+18 c d x^2+45 d^2 x^4\right )\right )}{x^{9/2}}+45 \sqrt {2} \sqrt [4]{b} (b c-a d)^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{90 a^{13/4}} \]
((-4*a^(1/4)*(45*b^2*c^2*x^4 - 9*a*b*c*x^2*(c + 10*d*x^2) + a^2*(5*c^2 + 1 8*c*d*x^2 + 45*d^2*x^4)))/x^(9/2) + 45*Sqrt[2]*b^(1/4)*(b*c - a*d)^2*ArcTa n[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 45*Sqrt[2]*b^ (1/4)*(b*c - a*d)^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + S qrt[b]*x)])/(90*a^(13/4))
Time = 0.51 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {365, 27, 359, 264, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int -\frac {9 \left (c (b c-2 a d)-a d^2 x^2\right )}{2 x^{7/2} \left (b x^2+a\right )}dx}{9 a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {c (b c-2 a d)-a d^2 x^2}{x^{7/2} \left (b x^2+a\right )}dx}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \int \frac {1}{x^{3/2} \left (b x^2+a\right )}dx}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {b \int \frac {\sqrt {x}}{b x^2+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \int \frac {x}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {-\frac {(b c-a d)^2 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2 c (b c-2 a d)}{5 a x^{5/2}}}{a}-\frac {2 c^2}{9 a x^{9/2}}\) |
(-2*c^2)/(9*a*x^(9/2)) - ((-2*c*(b*c - 2*a*d))/(5*a*x^(5/2)) - ((b*c - a*d )^2*(-2/(a*Sqrt[x]) - (2*b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4 )]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/ 4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a ^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a ] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4 )))/(2*Sqrt[b])))/a))/a)/a
3.5.24.3.1 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)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[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 = 2.84 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65
| method | result | size |
| derivativedivides | \(-\frac {2 c^{2}}{9 a \,x^{\frac {9}{2}}}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{3} \sqrt {x}}-\frac {2 c \left (2 a d -b c \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(186\) |
| default | \(-\frac {2 c^{2}}{9 a \,x^{\frac {9}{2}}}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{3} \sqrt {x}}-\frac {2 c \left (2 a d -b c \right )}{5 a^{2} x^{\frac {5}{2}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(186\) |
| risch | \(-\frac {2 \left (45 a^{2} d^{2} x^{4}-90 x^{4} a b c d +45 b^{2} c^{2} x^{4}+18 a^{2} c d \,x^{2}-9 x^{2} b \,c^{2} a +5 a^{2} c^{2}\right )}{45 a^{3} x^{\frac {9}{2}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(196\) |
-2/9*c^2/a/x^(9/2)-2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^3/x^(1/2)-2/5*c*(2*a*d- b*c)/a^2/x^(5/2)-1/4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^3/(a/b)^(1/4)*2^(1/2)*( ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1 /2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2) /(a/b)^(1/4)*x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1383, normalized size of antiderivative = 4.80 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \]
-1/90*(45*a^3*x^5*(-(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3 *b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^ 6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8)/a^13)^(1/4)*log(a^10*(-(b^9*c^8 - 8*a*b^8 *c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56 *a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8)/a^13) ^(3/4) + (b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^ 3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*sqrt(x)) - 45*I*a^3* x^5*(-(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2* c*d^7 + a^8*b*d^8)/a^13)^(1/4)*log(I*a^10*(-(b^9*c^8 - 8*a*b^8*c^7*d + 28* a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3 *d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8)/a^13)^(3/4) + (b^ 7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b ^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*sqrt(x)) + 45*I*a^3*x^5*(-(b^9*c ^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5* c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8* b*d^8)/a^13)^(1/4)*log(-I*a^10*(-(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6 *d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4 - 56*a^5*b^4*c^3*d^5 + 28*a ^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8)/a^13)^(3/4) + (b^7*c^6 - 6*a *b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d...
Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, a^{3}} - \frac {2 \, {\left (45 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 5 \, a^{2} c^{2} - 9 \, {\left (a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )}}{45 \, a^{3} x^{\frac {9}{2}}} \]
-1/4*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sq rt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sq rt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b ^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))* sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt( a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqr t(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/a^3 - 2/45*(45*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4 + 5*a^2*c^2 - 9*(a*b*c^2 - 2*a^2*c*d)*x^2)/(a^3*x^(9/2))
Time = 0.31 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (a b^{3}\right )^{\frac {3}{4}} a b c d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a^{4} b^{2}} - \frac {2 \, {\left (45 \, b^{2} c^{2} x^{4} - 90 \, a b c d x^{4} + 45 \, a^{2} d^{2} x^{4} - 9 \, a b c^{2} x^{2} + 18 \, a^{2} c d x^{2} + 5 \, a^{2} c^{2}\right )}}{45 \, a^{3} x^{\frac {9}{2}}} \]
-1/2*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a*b^3)^(3/4)*a*b*c*d + (a*b^3)^(3 /4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1 /4))/(a^4*b^2) - 1/2*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a*b^3)^(3/4)*a*b* c*d + (a*b^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2* sqrt(x))/(a/b)^(1/4))/(a^4*b^2) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*( a*b^3)^(3/4)*a*b*c*d + (a*b^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(a/b)^(1 /4) + x + sqrt(a/b))/(a^4*b^2) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^2*c^2 - 2*(a *b^3)^(3/4)*a*b*c*d + (a*b^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(a/b)^(1 /4) + x + sqrt(a/b))/(a^4*b^2) - 2/45*(45*b^2*c^2*x^4 - 90*a*b*c*d*x^4 + 4 5*a^2*d^2*x^4 - 9*a*b*c^2*x^2 + 18*a^2*c*d*x^2 + 5*a^2*c^2)/(a^3*x^(9/2))
Time = 0.20 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.57 \[ \int \frac {\left (c+d x^2\right )^2}{x^{11/2} \left (a+b x^2\right )} \, dx=\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (16\,a^{14}\,b^4\,d^4-64\,a^{13}\,b^5\,c\,d^3+96\,a^{12}\,b^6\,c^2\,d^2-64\,a^{11}\,b^7\,c^3\,d+16\,a^{10}\,b^8\,c^4\right )}{a^{13/4}\,\left (16\,a^{13}\,b^4\,d^6-96\,a^{12}\,b^5\,c\,d^5+240\,a^{11}\,b^6\,c^2\,d^4-320\,a^{10}\,b^7\,c^3\,d^3+240\,a^9\,b^8\,c^4\,d^2-96\,a^8\,b^9\,c^5\,d+16\,a^7\,b^{10}\,c^6\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{13/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (16\,a^{14}\,b^4\,d^4-64\,a^{13}\,b^5\,c\,d^3+96\,a^{12}\,b^6\,c^2\,d^2-64\,a^{11}\,b^7\,c^3\,d+16\,a^{10}\,b^8\,c^4\right )}{a^{13/4}\,\left (16\,a^{13}\,b^4\,d^6-96\,a^{12}\,b^5\,c\,d^5+240\,a^{11}\,b^6\,c^2\,d^4-320\,a^{10}\,b^7\,c^3\,d^3+240\,a^9\,b^8\,c^4\,d^2-96\,a^8\,b^9\,c^5\,d+16\,a^7\,b^{10}\,c^6\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{13/4}}-\frac {\frac {2\,c^2}{9\,a}+\frac {2\,x^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}+\frac {2\,c\,x^2\,\left (2\,a\,d-b\,c\right )}{5\,a^2}}{x^{9/2}} \]
((-b)^(1/4)*atanh(((-b)^(1/4)*x^(1/2)*(a*d - b*c)^2*(16*a^10*b^8*c^4 + 16* a^14*b^4*d^4 - 64*a^11*b^7*c^3*d - 64*a^13*b^5*c*d^3 + 96*a^12*b^6*c^2*d^2 ))/(a^(13/4)*(16*a^7*b^10*c^6 + 16*a^13*b^4*d^6 - 96*a^8*b^9*c^5*d - 96*a^ 12*b^5*c*d^5 + 240*a^9*b^8*c^4*d^2 - 320*a^10*b^7*c^3*d^3 + 240*a^11*b^6*c ^2*d^4)))*(a*d - b*c)^2)/a^(13/4) - ((-b)^(1/4)*atan(((-b)^(1/4)*x^(1/2)*( a*d - b*c)^2*(16*a^10*b^8*c^4 + 16*a^14*b^4*d^4 - 64*a^11*b^7*c^3*d - 64*a ^13*b^5*c*d^3 + 96*a^12*b^6*c^2*d^2))/(a^(13/4)*(16*a^7*b^10*c^6 + 16*a^13 *b^4*d^6 - 96*a^8*b^9*c^5*d - 96*a^12*b^5*c*d^5 + 240*a^9*b^8*c^4*d^2 - 32 0*a^10*b^7*c^3*d^3 + 240*a^11*b^6*c^2*d^4)))*(a*d - b*c)^2)/a^(13/4) - ((2 *c^2)/(9*a) + (2*x^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/a^3 + (2*c*x^2*(2*a* d - b*c))/(5*a^2))/x^(9/2)