Integrand size = 26, antiderivative size = 169 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {(b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{8 b^2 d e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 b d e}-\frac {(b c-a d) (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{8 b^{5/2} d^{3/2} \sqrt {e}} \]
-1/8*(-a*d+b*c)*(3*a*d+b*c)*arctanh(d^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/ b^(1/2)/e^(1/2))/b^(5/2)/d^(3/2)/e^(1/2)-1/8*(3*a*d+b*c)*(d*x^2+c)*(e*(b*x ^2+a)/(d*x^2+c))^(1/2)/b^2/d/e+1/4*(d*x^2+c)^2*(e*(b*x^2+a)/(d*x^2+c))^(1/ 2)/b/d/e
Time = 0.89 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {b} \sqrt {d} \left (a+b x^2\right ) \sqrt {c+d x^2} \left (-3 a d+b \left (c+2 d x^2\right )\right )-\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{5/2} d^{3/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \]
(Sqrt[b]*Sqrt[d]*(a + b*x^2)*Sqrt[c + d*x^2]*(-3*a*d + b*(c + 2*d*x^2)) - (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(8*b^(5/2)*d^(3/2)*Sqrt[(e*(a + b*x^ 2))/(c + d*x^2)]*Sqrt[c + d*x^2])
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2053, 2052, 25, 298, 215, 221}
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.
| \(\displaystyle \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle e (b c-a d) \int -\frac {a e-c x^4}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\left (e (b c-a d) \int \frac {a e-c x^4}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle e (b c-a d) \left (\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b d \left (b e-d x^4\right )^2}-\frac {(3 a d+b c) \int \frac {1}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 b d}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle e (b c-a d) \left (\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b d \left (b e-d x^4\right )^2}-\frac {(3 a d+b c) \left (\frac {\int \frac {1}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b d}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle e (b c-a d) \left (\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b d \left (b e-d x^4\right )^2}-\frac {(3 a d+b c) \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 b^{3/2} \sqrt {d} e^{3/2}}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b d}\right )\) |
(b*c - a*d)*e*(((b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*b*d*(b*e - d*x^4)^2) - ((b*c + 3*a*d)*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/(2*b*e*(b *e - d*x^4)) + ArcTanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b ]*Sqrt[e])]/(2*b^(3/2)*Sqrt[d]*e^(3/2))))/(4*b*d))
3.3.97.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(-\frac {\left (-2 b d \,x^{2}+3 a d -b c \right ) \left (b \,x^{2}+a \right )}{8 b^{2} d \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) \ln \left (\frac {\frac {1}{2} e d a +\frac {1}{2} e b c +b d e \,x^{2}}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{16 b^{2} d \sqrt {b d e}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(191\) |
| default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-4 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b d \,x^{2}-3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{2}+2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c d +\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2}+6 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a d -2 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b c \right )}{16 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} d \sqrt {b d}}\) | \(341\) |
-1/8*(-2*b*d*x^2+3*a*d-b*c)*(b*x^2+a)/b^2/d/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+ 1/16*(3*a^2*d^2-2*a*b*c*d-b^2*c^2)/b^2/d*ln((1/2*e*d*a+1/2*e*b*c+b*d*e*x^2 )/(b*d*e)^(1/2)+(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^(1/2))/(b*d*e)^(1/2)/( e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2)/(d*x^2+c)
Time = 0.35 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.44 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\left [-\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {b d e} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e + 4 \, {\left (2 \, b d^{2} x^{4} + b c^{2} + a c d + {\left (3 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}\right ) - 4 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d - 3 \, a b c d^{2} + 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, b^{3} d^{2} e}, \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-b d e} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {-b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (b^{2} d e x^{2} + a b d e\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d - 3 \, a b c d^{2} + 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, b^{3} d^{2} e}\right ] \]
[-1/32*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*sqrt(b*d*e)*log(8*b^2*d^2*e*x^4 + 8*(b^2*c*d + a*b*d^2)*e*x^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e + 4*(2*b *d^2*x^4 + b*c^2 + a*c*d + (3*b*c*d + a*d^2)*x^2)*sqrt(b*d*e)*sqrt((b*e*x^ 2 + a*e)/(d*x^2 + c))) - 4*(2*b^2*d^3*x^4 + b^2*c^2*d - 3*a*b*c*d^2 + 3*(b ^2*c*d^2 - a*b*d^3)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^3*d^2*e), 1 /16*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*sqrt(-b*d*e)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(b^2*d*e*x^2 + a*b*d*e)) + 2*(2*b^2*d^3*x^4 + b^2*c^2*d - 3*a*b*c*d^2 + 3*(b^2*c*d^2 - a *b*d^3)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^3*d^2*e)]
Timed out. \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {2 \, \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e} {\left (\frac {2 \, x^{2}}{b e} + \frac {b c e - 3 \, a d e}{b^{2} d e^{2}}\right )} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left ({\left | -b c e - a d e - 2 \, \sqrt {b d e} {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} \right |}\right )}{\sqrt {b d e} b^{2} d}}{16 \, \mathrm {sgn}\left (d x^{2} + c\right )} \]
1/16*(2*sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e)*(2*x^2/(b*e) + (b* c*e - 3*a*d*e)/(b^2*d*e^2)) + (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*log(abs(-b *c*e - a*d*e - 2*sqrt(b*d*e)*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))))/(sqrt(b*d*e)*b^2*d))/sgn(d*x^2 + c)
Timed out. \[ \int \frac {x^3}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {x^3}{\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \]