Integrand size = 26, antiderivative size = 161 \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=-\frac {(5 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{8 b d^2}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 d^2}+\frac {(b c-a d) (3 b c+a d) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{8 b^{3/2} d^{5/2}} \]
1/8*(-a*d+b*c)*(a*d+3*b*c)*arctanh(d^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b ^(1/2)/e^(1/2))*e^(1/2)/b^(3/2)/d^(5/2)-1/8*(-a*d+5*b*c)*(d*x^2+c)*(e*(b*x ^2+a)/(d*x^2+c))^(1/2)/b/d^2+1/4*(d*x^2+c)^2*(e*(b*x^2+a)/(d*x^2+c))^(1/2) /d^2
Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00 \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {b} \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-3 b c+a d+2 b d x^2\right )+\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )\right )}{8 b^{3/2} d^{5/2} \sqrt {a+b x^2}} \]
(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*(c + d *x^2)*(-3*b*c + a*d + 2*b*d*x^2) + (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*Sqrt[ c + d*x^2]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])]))/ (8*b^(3/2)*d^(5/2)*Sqrt[a + b*x^2])
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2053, 2052, 25, 360, 25, 298, 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.
\(\displaystyle \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int x^2 \sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}dx^2\) |
\(\Big \downarrow \) 2052 |
\(\displaystyle e (b c-a d) \int -\frac {x^4 \left (a e-c x^4\right )}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\left (e (b c-a d) \int \frac {x^4 \left (a e-c x^4\right )}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\right )\) |
\(\Big \downarrow \) 360 |
\(\displaystyle e (b c-a d) \left (\frac {\int -\frac {4 c d x^4+(b c-a d) e}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 d^2}+\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle e (b c-a d) \left (\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\int \frac {4 c d x^4+(b c-a d) e}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 d^2}\right )\) |
\(\Big \downarrow \) 298 |
\(\displaystyle e (b c-a d) \left (\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\frac {(5 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b \left (b e-d x^4\right )}-\frac {(a d+3 b c) \int \frac {1}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 b}}{4 d^2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle e (b c-a d) \left (\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 d^2 \left (b e-d x^4\right )^2}-\frac {\frac {(5 b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b \left (b e-d x^4\right )}-\frac {(a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 b^{3/2} \sqrt {d} \sqrt {e}}}{4 d^2}\right )\) |
(b*c - a*d)*e*(((b*c - a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*d^2*(b *e - d*x^4)^2) - (((5*b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(2*b*( b*e - d*x^4)) - ((3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b]*Sqrt[e])])/(2*b^(3/2)*Sqrt[d]*Sqrt[e]))/(4*d^2))
3.3.64.3.1 Defintions of rubi rules used
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_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 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 = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {\left (2 b d \,x^{2}+d a -3 b c \right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{8 b \,d^{2}}-\frac {\left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) \ln \left (\frac {\frac {1}{2} e d a +\frac {1}{2} e b c +b d e \,x^{2}}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{16 b \,d^{2} \sqrt {b d e}\, \left (b \,x^{2}+a \right )}\) | \(189\) |
default | \(\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (4 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b d \,x^{2}-\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+d a +b c}{2 \sqrt {b d}}\right ) a^{2} d^{2}-2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+d a +b c}{2 \sqrt {b d}}\right ) a b c d +3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+d a +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2}+2 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a d -6 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b c \right )}{16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, d^{2} b \sqrt {b d}}\) | \(342\) |
1/8*(2*b*d*x^2+a*d-3*b*c)*(d*x^2+c)/b/d^2*(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/ 16*(a^2*d^2+2*a*b*c*d-3*b^2*c^2)/b/d^2*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)/(b*x^2+a)
Time = 0.34 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.53 \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {\frac {e}{b d}} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e - 4 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d + a b c d^{2} + {\left (3 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{b d}}\right ) - 4 \, {\left (2 \, b d^{2} x^{4} - 3 \, b c^{2} + a c d - {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, b d^{2}}, -\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-\frac {e}{b d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{b d}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) - 2 \, {\left (2 \, b d^{2} x^{4} - 3 \, b c^{2} + a c d - {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, b d^{2}}\right ] \]
[-1/32*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(e/(b*d))*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^2*d^3*x^4 + b^2*c^2*d + 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))*sqrt(e/(b*d))) - 4*(2*b*d^2*x^4 - 3*b*c^2 + a* c*d - (b*c*d - a*d^2)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b*d^2), -1/ 16*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(-e/(b*d))*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(-e/(b*d))/(b*e*x^2 + a*e)) - 2*(2*b*d^2*x^4 - 3*b*c^2 + a*c*d - (b*c*d - a*d^2)*x^2)*sqrt((b*e* x^2 + a*e)/(d*x^2 + c)))/(b*d^2)]
Timed out. \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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 = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.05 \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {1}{16} \, {\left (2 \, \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e} {\left (\frac {2 \, x^{2}}{d} - \frac {3 \, b c - a d}{b d^{2}}\right )} - \frac {{\left (3 \, b^{2} c^{2} e - 2 \, a b c d e - a^{2} d^{2} e\right )} \log \left ({\left | -b c e - a d e - 2 \, \sqrt {b d e} {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} \right |}\right )}{\sqrt {b d e} b d^{2}}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \]
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/d - (3*b*c - a*d)/(b*d^2)) - (3*b^2*c^2*e - 2*a*b*c*d*e - a^2*d^2*e)*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*d^2))*sgn(d*x^2 + c)
Timed out. \[ \int x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\int x^3\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}} \,d x \]