Integrand size = 33, antiderivative size = 145 \[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {(b d-a e) x \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e x^3 \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {a} (b d-a e) \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
(-a*e+b*d)*x*(b*x^2+a)/b^2/((b*x^2+a)^2)^(1/2)+1/3*e*x^3*(b*x^2+a)/b/((b*x ^2+a)^2)^(1/2)-(-a*e+b*d)*(b*x^2+a)*arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/b^(5 /2)/((b*x^2+a)^2)^(1/2)
Time = 1.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.55 \[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\left (a+b x^2\right ) \left (\sqrt {b} x \left (3 b d-3 a e+b e x^2\right )+3 \sqrt {a} (-b d+a e) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{3 b^{5/2} \sqrt {\left (a+b x^2\right )^2}} \]
((a + b*x^2)*(Sqrt[b]*x*(3*b*d - 3*a*e + b*e*x^2) + 3*Sqrt[a]*(-(b*d) + a* e)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(3*b^(5/2)*Sqrt[(a + b*x^2)^2])
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1384, 27, 363, 262, 218}
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.
\(\displaystyle \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {b \left (a+b x^2\right ) \int \frac {x^2 \left (e x^2+d\right )}{b \left (b x^2+a\right )}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (a+b x^2\right ) \int \frac {x^2 \left (e x^2+d\right )}{b x^2+a}dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
\(\Big \downarrow \) 363 |
\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {(b d-a e) \int \frac {x^2}{b x^2+a}dx}{b}+\frac {e x^3}{3 b}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {(b d-a e) \left (\frac {x}{b}-\frac {a \int \frac {1}{b x^2+a}dx}{b}\right )}{b}+\frac {e x^3}{3 b}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\left (a+b x^2\right ) \left (\frac {\left (\frac {x}{b}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}\right ) (b d-a e)}{b}+\frac {e x^3}{3 b}\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\) |
((a + b*x^2)*((e*x^3)/(3*b) + ((b*d - a*e)*(x/b - (Sqrt[a]*ArcTan[(Sqrt[b] *x)/Sqrt[a]])/b^(3/2)))/b))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
3.1.75.3.1 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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 2.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-\sqrt {a b}\, b e \,x^{3}+3 \sqrt {a b}\, a e x -3 \sqrt {a b}\, b d x -3 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{2} e +3 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a b d \right )}{3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b^{2} \sqrt {a b}}\) | \(91\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {1}{3} e \,x^{3} b -a e x +b d x \right )}{\left (b \,x^{2}+a \right ) b^{2}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {-a b}\, \left (a e -b d \right ) \ln \left (-\sqrt {-a b}\, x +a \right )}{2 \left (b \,x^{2}+a \right ) b^{3}}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {-a b}\, \left (a e -b d \right ) \ln \left (\sqrt {-a b}\, x +a \right )}{2 \left (b \,x^{2}+a \right ) b^{3}}\) | \(144\) |
-1/3*(b*x^2+a)*(-(a*b)^(1/2)*b*e*x^3+3*(a*b)^(1/2)*a*e*x-3*(a*b)^(1/2)*b*d *x-3*arctan(b*x/(a*b)^(1/2))*a^2*e+3*arctan(b*x/(a*b)^(1/2))*a*b*d)/((b*x^ 2+a)^2)^(1/2)/b^2/(a*b)^(1/2)
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\left [\frac {2 \, b e x^{3} - 3 \, {\left (b d - a e\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (b d - a e\right )} x}{6 \, b^{2}}, \frac {b e x^{3} - 3 \, {\left (b d - a e\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, {\left (b d - a e\right )} x}{3 \, b^{2}}\right ] \]
[1/6*(2*b*e*x^3 - 3*(b*d - a*e)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 6*(b*d - a*e)*x)/b^2, 1/3*(b*e*x^3 - 3*(b*d - a*e)*sqrt (a/b)*arctan(b*x*sqrt(a/b)/a) + 3*(b*d - a*e)*x)/b^2]
\[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {x^{2} \left (d + e x^{2}\right )}{\sqrt {\left (a + b x^{2}\right )^{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.37 \[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {{\left (a b d - a^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b e x^{3} + 3 \, {\left (b d - a e\right )} x}{3 \, b^{2}} \]
-(a*b*d - a^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/3*(b*e*x^3 + 3* (b*d - a*e)*x)/b^2
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {{\left (a b d \mathrm {sgn}\left (b x^{2} + a\right ) - a^{2} e \mathrm {sgn}\left (b x^{2} + a\right )\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} e x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, b^{2} d x \mathrm {sgn}\left (b x^{2} + a\right ) - 3 \, a b e x \mathrm {sgn}\left (b x^{2} + a\right )}{3 \, b^{3}} \]
-(a*b*d*sgn(b*x^2 + a) - a^2*e*sgn(b*x^2 + a))*arctan(b*x/sqrt(a*b))/(sqrt (a*b)*b^2) + 1/3*(b^2*e*x^3*sgn(b*x^2 + a) + 3*b^2*d*x*sgn(b*x^2 + a) - 3* a*b*e*x*sgn(b*x^2 + a))/b^3
Timed out. \[ \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {x^2\,\left (e\,x^2+d\right )}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]