Integrand size = 20, antiderivative size = 445 \[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {\left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) x \sqrt {a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {x \left (b \left (4 b^2-9 a c\right )+6 c \left (2 b^2-7 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\sqrt [4]{a} \left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (4 \sqrt {a} b \left (b^2-6 a c\right )+\frac {8 b^4-57 a b^2 c+84 a^2 c^2}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{9/4} \sqrt {a+b x^2+c x^4}} \] Output:
1/315*(84*a^2*c^2-57*a*b^2*c+8*b^4)*x*(c*x^4+b*x^2+a)^(1/2)/c^(5/2)/(a^(1/ 2)+c^(1/2)*x^2)-1/315*x*(b*(-9*a*c+4*b^2)+6*c*(-7*a*c+2*b^2)*x^2)*(c*x^4+b *x^2+a)^(1/2)/c^2+1/63*x*(7*c*x^2+3*b)*(c*x^4+b*x^2+a)^(3/2)/c-1/315*a^(1/ 4)*(84*a^2*c^2-57*a*b^2*c+8*b^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a ^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/ 2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)+1/630*a^(1/4 )*(4*a^(1/2)*b*(-6*a*c+b^2)+(84*a^2*c^2-57*a*b^2*c+8*b^4)/c^(1/2))*(a^(1/2 )+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJaco biAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(9/4)/ (c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.97 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.35 \[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}+a^2 c \left (24 b+77 c x^2\right )+a \left (-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right )\right )+i \left (8 b^4-57 a b^2 c+84 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-8 b^5+65 a b^3 c-132 a^2 b c^2+8 b^4 \sqrt {b^2-4 a c}-57 a b^2 c \sqrt {b^2-4 a c}+84 a^2 c^2 \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{1260 c^3 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[x^2*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b^4*x^2 - b^3*c*x^4 + 53*b^2*c^ 2*x^6 + 85*b*c^3*x^8 + 35*c^4*x^10 + a^2*c*(24*b + 77*c*x^2) + a*(-4*b^3 + 27*b^2*c*x^2 + 151*b*c^2*x^4 + 112*c^3*x^6)) + I*(8*b^4 - 57*a*b^2*c + 84 *a^2*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/ (b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - S qrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a* c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-8*b^5 + 65 *a*b^3*c - 132*a^2*b*c^2 + 8*b^4*Sqrt[b^2 - 4*a*c] - 57*a*b^2*c*Sqrt[b^2 - 4*a*c] + 84*a^2*c^2*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c* x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/( b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(1260*c^3* Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
Time = 0.97 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1436, 1490, 25, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1436 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\int \left (2 \left (2 b^2-7 a c\right ) x^2+a b\right ) \sqrt {c x^4+b x^2+a}dx}{21 c}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\frac {\int -\frac {\left (8 b^4-57 a c b^2+84 a^2 c^2\right ) x^2+4 a b \left (b^2-6 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right )}{15 c}}{21 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\int \frac {\left (8 b^4-57 a c b^2+84 a^2 c^2\right ) x^2+4 a b \left (b^2-6 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}}{21 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt {a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt {a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}-\frac {\frac {x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt {a} b \sqrt {c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}\) |
Input:
Int[x^2*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(x*(3*b + 7*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c) - ((x*(b*(4*b^2 - 9*a *c) + 6*c*(2*b^2 - 7*a*c)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) - (-(((8*b^ 4 - 57*a*b^2*c + 84*a^2*c^2)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqr t[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sq rt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(S qrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/ 4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2 + 4*Sqrt[a]*b*Sqrt[c]*(b^2 - 6*a*c))*( Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2] *EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2 *c^(3/4)*Sqrt[a + b*x^2 + c*x^4]))/(15*c))/(21*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(d*x)^(m - 1)*(a + b*x^2 + c*x^4)^p*((2*b*p + c*(m + 4*p - 1)*x^2 )/(c*(m + 4*p + 1)*(m + 4*p - 1))), x] - Simp[2*p*(d^2/(c*(m + 4*p + 1)*(m + 4*p - 1))) Int[(d*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p - 1)*Simp[a*b*(m - 1) - (2*a*c*(m + 4*p - 1) - b^2*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && GtQ[m, 1] && IntegerQ[ 2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Time = 4.05 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {c \,x^{7} \sqrt {c \,x^{4}+b \,x^{2}+a}}{9}+\frac {10 b \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{63}+\frac {\left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}-\frac {\left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (a^{2}-\frac {3 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) a}{5 c}-\frac {2 \left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) b}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(545\) |
| elliptic | \(\frac {c \,x^{7} \sqrt {c \,x^{4}+b \,x^{2}+a}}{9}+\frac {10 b \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{63}+\frac {\left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}-\frac {\left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (a^{2}-\frac {3 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) a}{5 c}-\frac {2 \left (\frac {76 a b}{63}-\frac {4 \left (\frac {11 a c}{9}+\frac {b^{2}}{21}\right ) b}{5 c}\right ) b}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(545\) |
| risch | \(\frac {x \left (35 c^{3} x^{6}+50 b \,c^{2} x^{4}+77 a \,c^{2} x^{2}+3 b^{2} c \,x^{2}+24 a b c -4 b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{315 c^{2}}-\frac {\frac {\left (84 a^{2} c^{2}-57 a \,b^{2} c +8 b^{4}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}-\frac {b^{3} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {6 a^{2} b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{315 c^{2}}\) | \(605\) |
Input:
int(x^2*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/9*c*x^7*(c*x^4+b*x^2+a)^(1/2)+10/63*b*x^5*(c*x^4+b*x^2+a)^(1/2)+1/5*(11/ 9*a*c+1/21*b^2)/c*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(76/63*a*b-4/5*(11/9*a*c+1 /21*b^2)/c*b)/c*x*(c*x^4+b*x^2+a)^(1/2)-1/12*(76/63*a*b-4/5*(11/9*a*c+1/21 *b^2)/c*b)/c*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+ b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b *x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/ 2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*(a^2-3/5*(11/9*a*c+1/21*b ^2)/c*a-2/3*(76/63*a*b-4/5*(11/9*a*c+1/21*b^2)/c*b)/c*b)*a*2^(1/2)/((-b+(- 4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2* (b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^ (1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+ 2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a* c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.06 \[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, b^{5} - 57 \, a b^{3} c + 84 \, a^{2} b c^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (8 \, b^{4} c + 12 \, {\left (7 \, a^{2} + 2 \, a b\right )} c^{3} - {\left (57 \, a b^{2} + 4 \, b^{3}\right )} c^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, b^{5} + 12 \, {\left (7 \, a^{2} b - 2 \, a b^{2}\right )} c^{2} - {\left (57 \, a b^{3} - 4 \, b^{4}\right )} c\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (35 \, c^{5} x^{8} + 50 \, b c^{4} x^{6} + 8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 77 \, a c^{4}\right )} x^{4} - 4 \, {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{630 \, c^{4} x} \] Input:
integrate(x^2*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/630*(sqrt(1/2)*((8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3)*x*sqrt((b^2 - 4*a* c)/c^2) - (8*b^5 - 57*a*b^3*c + 84*a^2*b*c^2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a *c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c) ) - sqrt(1/2)*((8*b^4*c + 12*(7*a^2 + 2*a*b)*c^3 - (57*a*b^2 + 4*b^3)*c^2) *x*sqrt((b^2 - 4*a*c)/c^2) - (8*b^5 + 12*(7*a^2*b - 2*a*b^2)*c^2 - (57*a*b ^3 - 4*b^4)*c)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic _f(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*s qrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 2*(35*c^5*x^8 + 50*b*c^4*x^ 6 + 8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3 + (3*b^2*c^3 + 77*a*c^4)*x^4 - 4*( b^3*c^2 - 6*a*b*c^3)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(c^4*x)
\[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{2} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(x**2*(a + b*x**2 + c*x**4)**(3/2), x)
\[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*x^2, x)
\[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*x^2, x)
Timed out. \[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int(x^2*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^2*(a + b*x^2 + c*x^4)^(3/2), x)
\[ \int x^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {24 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b c x +77 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,c^{2} x^{3}-4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} x +3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c \,x^{3}+50 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{2} x^{5}+35 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{3} x^{7}-24 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} b c +4 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{3}+84 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} c^{2}-57 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{2} c +8 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{4}}{315 c^{2}} \] Input:
int(x^2*(c*x^4+b*x^2+a)^(3/2),x)
Output:
(24*sqrt(a + b*x**2 + c*x**4)*a*b*c*x + 77*sqrt(a + b*x**2 + c*x**4)*a*c** 2*x**3 - 4*sqrt(a + b*x**2 + c*x**4)*b**3*x + 3*sqrt(a + b*x**2 + c*x**4)* b**2*c*x**3 + 50*sqrt(a + b*x**2 + c*x**4)*b*c**2*x**5 + 35*sqrt(a + b*x** 2 + c*x**4)*c**3*x**7 - 24*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x **4),x)*a**2*b*c + 4*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x )*a*b**3 + 84*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x )*a**2*c**2 - 57*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4 ),x)*a*b**2*c + 8*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x** 4),x)*b**4)/(315*c**2)