Integrand size = 27, antiderivative size = 151 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=\frac {2}{7} \left (-b+a x^2\right )^{3/4} \left (5 b+2 a x^2\right )+\frac {3 b^{7/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{-\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2}}+\frac {3 b^{7/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}} \]
2/7*(a*x^2-b)^(3/4)*(2*a*x^2+5*b)+3/2*b^(7/4)*arctan(2^(1/2)*b^(1/4)*(a*x^ 2-b)^(1/4)/(-b^(1/2)+(a*x^2-b)^(1/2)))*2^(1/2)+3/2*b^(7/4)*arctanh((1/2*b^ (1/4)*2^(1/2)+1/2*(a*x^2-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x^2-b)^(1/4))*2^(1/2 )
Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=\frac {2}{7} \left (-b+a x^2\right )^{3/4} \left (5 b+2 a x^2\right )-\frac {3 b^{7/4} \arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}}+\frac {3 b^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \]
(2*(-b + a*x^2)^(3/4)*(5*b + 2*a*x^2))/7 - (3*b^(7/4)*ArcTan[(-Sqrt[b] + S qrt[-b + a*x^2])/(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))])/Sqrt[2] + (3*b^(7/ 4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2 ])])/Sqrt[2]
Time = 0.40 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.54, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {354, 90, 60, 73, 27, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a x^2-b\right )^{3/4} \left (2 a x^2+3 b\right )}{x} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (a x^2-b\right )^{3/4} \left (2 a x^2+3 b\right )}{x^2}dx^2\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} \left (3 b \int \frac {\left (a x^2-b\right )^{3/4}}{x^2}dx^2+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-b \int \frac {1}{x^2 \sqrt [4]{a x^2-b}}dx^2\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-\frac {4 b \int \frac {a x^4}{x^8+b}d\sqrt [4]{a x^2-b}}{a}\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \int \frac {x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \int \frac {x^4+\sqrt {b}}{x^8+b}d\sqrt [4]{a x^2-b}-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^4-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^4}{x^8+b}d\sqrt [4]{a x^2-b}\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}\right )}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}\right )}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^2-b}}{x^4+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}d\sqrt [4]{a x^2-b}}{2 \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (3 b \left (\frac {4}{3} \left (a x^2-b\right )^{3/4}-4 b \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {b}+x^4\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {b}+x^4\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\right )\right )+\frac {8}{7} \left (a x^2-b\right )^{7/4}\right )\) |
((8*(-b + a*x^2)^(7/4))/7 + 3*b*((4*(-b + a*x^2)^(3/4))/3 - 4*b*((-(ArcTan [1 - (Sqrt[2]*(-b + a*x^2)^(1/4))/b^(1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*(-b + a*x^2)^(1/4))/b^(1/4)]/(Sqrt[2]*b^(1/4)))/2 + (Log[Sqrt[b] + x^4 - Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[Sqr t[b] + x^4 + Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4)))/2))) /2
3.21.85.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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 = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05
| method | result | size |
| pseudoelliptic | \(-\frac {3 \sqrt {2}\, \left (\ln \left (\frac {\sqrt {a \,x^{2}-b}-b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{2}-b}+b^{\frac {1}{4}} \left (a \,x^{2}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{2}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {7}{4}}}{4}+\frac {2 \left (a \,x^{2}-b \right )^{\frac {3}{4}} \left (2 a \,x^{2}+5 b \right )}{7}\) | \(158\) |
-3/4*2^(1/2)*(ln(((a*x^2-b)^(1/2)-b^(1/4)*(a*x^2-b)^(1/4)*2^(1/2)+b^(1/2)) /((a*x^2-b)^(1/2)+b^(1/4)*(a*x^2-b)^(1/4)*2^(1/2)+b^(1/2)))-2*arctan((-2^( 1/2)*(a*x^2-b)^(1/4)+b^(1/4))/b^(1/4))+2*arctan((2^(1/2)*(a*x^2-b)^(1/4)+b ^(1/4))/b^(1/4)))*b^(7/4)+2/7*(a*x^2-b)^(3/4)*(2*a*x^2+5*b)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=\frac {2}{7} \, {\left (2 \, a x^{2} + 5 \, b\right )} {\left (a x^{2} - b\right )}^{\frac {3}{4}} - \frac {3}{2} \, \left (-b^{7}\right )^{\frac {1}{4}} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} + 27 \, \left (-b^{7}\right )^{\frac {3}{4}}\right ) + \frac {3}{2} i \, \left (-b^{7}\right )^{\frac {1}{4}} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} + 27 i \, \left (-b^{7}\right )^{\frac {3}{4}}\right ) - \frac {3}{2} i \, \left (-b^{7}\right )^{\frac {1}{4}} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} - 27 i \, \left (-b^{7}\right )^{\frac {3}{4}}\right ) + \frac {3}{2} \, \left (-b^{7}\right )^{\frac {1}{4}} \log \left (27 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{5} - 27 \, \left (-b^{7}\right )^{\frac {3}{4}}\right ) \]
2/7*(2*a*x^2 + 5*b)*(a*x^2 - b)^(3/4) - 3/2*(-b^7)^(1/4)*log(27*(a*x^2 - b )^(1/4)*b^5 + 27*(-b^7)^(3/4)) + 3/2*I*(-b^7)^(1/4)*log(27*(a*x^2 - b)^(1/ 4)*b^5 + 27*I*(-b^7)^(3/4)) - 3/2*I*(-b^7)^(1/4)*log(27*(a*x^2 - b)^(1/4)* b^5 - 27*I*(-b^7)^(3/4)) + 3/2*(-b^7)^(1/4)*log(27*(a*x^2 - b)^(1/4)*b^5 - 27*(-b^7)^(3/4))
Time = 10.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.53 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=- \frac {3 a^{\frac {3}{4}} b x^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{4}\right )} + 2 a \left (\begin {cases} \frac {x^{2} \left (- b\right )^{\frac {3}{4}}}{2} & \text {for}\: a = 0 \\\frac {2 \left (a x^{2} - b\right )^{\frac {7}{4}}}{7 a} & \text {otherwise} \end {cases}\right ) \]
-3*a**(3/4)*b*x**(3/2)*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), b*exp_polar (2*I*pi)/(a*x**2))/(2*gamma(1/4)) + 2*a*Piecewise((x**2*(-b)**(3/4)/2, Eq( a, 0)), (2*(a*x**2 - b)**(7/4)/(7*a), True))
Time = 0.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=-\frac {1}{4} \, {\left (3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} b - 8 \, {\left (a x^{2} - b\right )}^{\frac {3}{4}}\right )} b + \frac {4}{7} \, {\left (a x^{2} - b\right )}^{\frac {7}{4}} \]
-1/4*(3*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^2 - b)^(1/ 4))/b^(1/4))/b^(1/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2* (a*x^2 - b)^(1/4))/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(2)*(a*x^2 - b)^(1/4 )*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b))/b^(1/4) + sqrt(2)*log(-sqrt(2)*(a*x ^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b))/b^(1/4))*b - 8*(a*x^2 - b)^(3/4))*b + 4/7*(a*x^2 - b)^(7/4)
Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.25 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=-\frac {3}{2} \, \sqrt {2} b^{\frac {7}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {3}{2} \, \sqrt {2} b^{\frac {7}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {3}{4} \, \sqrt {2} b^{\frac {7}{4}} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) - \frac {3}{4} \, \sqrt {2} b^{\frac {7}{4}} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + \frac {4}{7} \, {\left (a x^{2} - b\right )}^{\frac {7}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {3}{4}} b \]
-3/2*sqrt(2)*b^(7/4)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^2 - b)^( 1/4))/b^(1/4)) - 3/2*sqrt(2)*b^(7/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^2 - b)^(1/4))/b^(1/4)) + 3/4*sqrt(2)*b^(7/4)*log(sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b)) - 3/4*sqrt(2)*b^(7/4)*log(- sqrt(2)*(a*x^2 - b)^(1/4)*b^(1/4) + sqrt(a*x^2 - b) + sqrt(b)) + 4/7*(a*x^ 2 - b)^(7/4) + 2*(a*x^2 - b)^(3/4)*b
Time = 6.69 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.52 \[ \int \frac {\left (-b+a x^2\right )^{3/4} \left (3 b+2 a x^2\right )}{x} \, dx=\frac {4\,{\left (a\,x^2-b\right )}^{7/4}}{7}-3\,{\left (-b\right )}^{7/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )+3\,{\left (-b\right )}^{7/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )+2\,b\,{\left (a\,x^2-b\right )}^{3/4} \]