Integrand size = 20, antiderivative size = 493 \[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x \sqrt {a+b x^2+c x^4}}{1155 c^{7/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (8 b^4-21 a b^2 c-30 a^2 c^2+3 b c \left (8 b^2-31 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{1155 c^3}-\frac {x \left (3 \left (2 b^2+a c\right )+14 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{231 c^2}+\frac {x \left (a+b x^2+c x^4\right )^{5/2}}{11 c}+\frac {8 \sqrt [4]{a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{1155 c^{15/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} \left (8 b^4-51 a b^2 c+60 a^2 c^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2310 c^{13/4} \sqrt {a+b x^2+c x^4}} \] Output:
-8/1155*b*(-9*a*c+2*b^2)*(-3*a*c+b^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(7/2)/(a^( 1/2)+c^(1/2)*x^2)+1/1155*x*(8*b^4-21*a*b^2*c-30*c^2*a^2+3*b*c*(-31*a*c+8*b ^2)*x^2)*(c*x^4+b*x^2+a)^(1/2)/c^3-1/231*x*(14*b*c*x^2+3*a*c+6*b^2)*(c*x^4 +b*x^2+a)^(3/2)/c^2+1/11*x*(c*x^4+b*x^2+a)^(5/2)/c+8/1155*a^(1/4)*b*(-9*a* c+2*b^2)*(-3*a*c+b^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)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1 /2)/c^(1/2))^(1/2))/c^(15/4)/(c*x^4+b*x^2+a)^(1/2)-1/2310*a^(1/4)*(8*b*(-9 *a*c+2*b^2)*(-3*a*c+b^2)/c^(1/2)+a^(1/2)*(60*a^2*c^2-51*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)*Invers eJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^( 13/4)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.44 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.33 \[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {2 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (60 a^3 c^2+a^2 c \left (-51 b^2+92 b c x^2+255 c^2 x^4\right )+a \left (8 b^4-57 b^3 c x^2-14 b^2 c^2 x^4+367 b c^3 x^6+300 c^4 x^8\right )+x^2 \left (8 b^5+2 b^4 c x^2-b^3 c^2 x^4+145 b^2 c^3 x^6+245 b c^4 x^8+105 c^5 x^{10}\right )\right )-4 i b \left (2 b^4-15 a b^2 c+27 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-8 b^6+68 a b^4 c-159 a^2 b^2 c^2+60 a^3 c^3+8 b^5 \sqrt {b^2-4 a c}-60 a b^3 c \sqrt {b^2-4 a c}+108 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2310 c^4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[x^4*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(60*a^3*c^2 + a^2*c*(-51*b^2 + 92*b *c*x^2 + 255*c^2*x^4) + a*(8*b^4 - 57*b^3*c*x^2 - 14*b^2*c^2*x^4 + 367*b*c ^3*x^6 + 300*c^4*x^8) + x^2*(8*b^5 + 2*b^4*c*x^2 - b^3*c^2*x^4 + 145*b^2*c ^3*x^6 + 245*b*c^4*x^8 + 105*c^5*x^10)) - (4*I)*b*(2*b^4 - 15*a*b^2*c + 27 *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^6 + 68 *a*b^4*c - 159*a^2*b^2*c^2 + 60*a^3*c^3 + 8*b^5*Sqrt[b^2 - 4*a*c] - 60*a*b ^3*c*Sqrt[b^2 - 4*a*c] + 108*a^2*b*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])])/(2310*c^4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
Time = 1.23 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1436, 27, 1596, 25, 1602, 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.
| \(\displaystyle \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1436 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\int 3 x^2 \left (2 \left (b^2-3 a c\right ) x^2+a b\right ) \sqrt {c x^4+b x^2+a}dx}{33 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\int x^2 \left (2 \left (b^2-3 a c\right ) x^2+a b\right ) \sqrt {c x^4+b x^2+a}dx}{11 c}\) |
| \(\Big \downarrow \) 1596 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {\int -\frac {x^2 \left (\left (8 b^4-51 a c b^2+60 a^2 c^2\right ) x^2+2 a b \left (3 b^2-16 a c\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{35 c}+\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}}{11 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}-\frac {\int \frac {x^2 \left (\left (8 b^4-51 a c b^2+60 a^2 c^2\right ) x^2+2 a b \left (3 b^2-16 a c\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{35 c}}{11 c}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}-\frac {\frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\int \frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) x^2+a \left (8 b^4-51 a c b^2+60 a^2 c^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{3 c}}{35 c}}{11 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}-\frac {\frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\sqrt {a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {8 \sqrt {a} b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{35 c}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}-\frac {\frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\sqrt {a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{35 c}}{11 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}-\frac {\frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{35 c}}{11 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {x^3 \left (b+3 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{33 c}-\frac {\frac {x^3 \left (10 c x^2 \left (b^2-3 a c\right )+b \left (a c+2 b^2\right )\right ) \sqrt {a+b x^2+c x^4}}{35 c}-\frac {\frac {x \left (60 a^2 c^2-51 a b^2 c+8 b^4\right ) \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a} \left (60 a^2 c^2-51 a b^2 c+8 b^4\right )+\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {8 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{3 c}}{35 c}}{11 c}\) |
Input:
Int[x^4*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(x^3*(b + 3*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(33*c) - ((x^3*(b*(2*b^2 + a *c) + 10*c*(b^2 - 3*a*c)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(35*c) - (((8*b^4 - 51*a*b^2*c + 60*a^2*c^2)*x*Sqrt[a + b*x^2 + c*x^4])/(3*c) - ((-8*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]* x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a ]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c] + (a^(1/4)*((8 *b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c))/Sqrt[c] + Sqrt[a]*(8*b^4 - 51*a*b^2*c + 60*a^2*c^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + S qrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sq rt[c]))/4])/(2*c^(1/4)*Sqrt[a + b*x^2 + c*x^4]))/(3*c))/(35*c))/(11*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)/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]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^p*((b*e*2 *p + c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2)/(c*f*(4*p + m + 1)*(m + 4*p + 3))), x] + Simp[2*(p/(c*(4*p + m + 1)*(m + 4*p + 3))) Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p - 1)*Simp[2*a*c*d*(m + 4*p + 3) - a*b*e*(m + 1) + (2*a*c *e*(4*p + m + 1) + b*c*d*(m + 4*p + 3) - b^2*e*(m + 2*p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] & & NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[ p] || IntegerQ[m])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
Time = 3.70 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(\frac {c \,x^{9} \sqrt {c \,x^{4}+b \,x^{2}+a}}{11}+\frac {4 b \,x^{7} \sqrt {c \,x^{4}+b \,x^{2}+a}}{33}+\frac {\left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 c}+\frac {\left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) b}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}-\frac {\left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\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 (-\frac {3 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) a}{5 c}-\frac {2 \left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\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 )}\) | \(674\) |
| elliptic | \(\frac {c \,x^{9} \sqrt {c \,x^{4}+b \,x^{2}+a}}{11}+\frac {4 b \,x^{7} \sqrt {c \,x^{4}+b \,x^{2}+a}}{33}+\frac {\left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 c}+\frac {\left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) b}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}-\frac {\left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\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 (-\frac {3 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\right ) a}{5 c}-\frac {2 \left (a^{2}-\frac {5 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) a}{7 c}-\frac {4 \left (\frac {38 a b}{33}-\frac {6 \left (\frac {13 a c}{11}+\frac {b^{2}}{33}\right ) b}{7 c}\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 )}\) | \(674\) |
| risch | \(\frac {x \left (105 c^{4} x^{8}+140 b \,c^{3} x^{6}+195 a \,c^{3} x^{4}+5 b^{2} c^{2} x^{4}+32 a b \,c^{2} x^{2}-6 x^{2} b^{3} c +60 a^{2} c^{2}-51 a \,b^{2} c +8 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{1155 c^{3}}-\frac {-\frac {4 b \left (27 a^{2} c^{2}-15 a \,b^{2} c +2 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 )}{\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 {15 a^{3} c^{2} \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 {2 b^{4} 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 {51 a^{2} b^{2} 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 )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{1155 c^{3}}\) | \(788\) |
Input:
int(x^4*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/11*c*x^9*(c*x^4+b*x^2+a)^(1/2)+4/33*b*x^7*(c*x^4+b*x^2+a)^(1/2)+1/7*(13/ 11*a*c+1/33*b^2)/c*x^5*(c*x^4+b*x^2+a)^(1/2)+1/5*(38/33*a*b-6/7*(13/11*a*c +1/33*b^2)/c*b)/c*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(a^2-5/7*(13/11*a*c+1/33*b ^2)/c*a-4/5*(38/33*a*b-6/7*(13/11*a*c+1/33*b^2)/c*b)/c*b)/c*x*(c*x^4+b*x^2 +a)^(1/2)-1/12*(a^2-5/7*(13/11*a*c+1/33*b^2)/c*a-4/5*(38/33*a*b-6/7*(13/11 *a*c+1/33*b^2)/c*b)/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*(-3/5*(38/3 3*a*b-6/7*(13/11*a*c+1/33*b^2)/c*b)/c*a-2/3*(a^2-5/7*(13/11*a*c+1/33*b^2)/ c*a-4/5*(38/33*a*b-6/7*(13/11*a*c+1/33*b^2)/c*b)/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 = 528, normalized size of antiderivative = 1.07 \[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {8 \, \sqrt {\frac {1}{2}} {\left ({\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 27 \, a^{2} b c^{3}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (2 \, b^{6} - 15 \, a b^{4} c + 27 \, a^{2} b^{2} c^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (16 \, b^{5} c - 60 \, a^{2} c^{4} + 3 \, {\left (72 \, a^{2} b + 17 \, a b^{2}\right )} c^{3} - 8 \, {\left (15 \, a b^{3} + b^{4}\right )} c^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (16 \, b^{6} + 60 \, a^{2} b c^{3} + 3 \, {\left (72 \, a^{2} b^{2} - 17 \, a b^{3}\right )} c^{2} - 8 \, {\left (15 \, a b^{4} - b^{5}\right )} c\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, {\left (105 \, c^{6} x^{10} + 140 \, b c^{5} x^{8} + 5 \, {\left (b^{2} c^{4} + 39 \, a c^{5}\right )} x^{6} - 16 \, b^{5} c + 120 \, a b^{3} c^{2} - 216 \, a^{2} b c^{3} - 2 \, {\left (3 \, b^{3} c^{3} - 16 \, a b c^{4}\right )} x^{4} + {\left (8 \, b^{4} c^{2} - 51 \, a b^{2} c^{3} + 60 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2310 \, c^{5} x} \] Input:
integrate(x^4*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/2310*(8*sqrt(1/2)*((2*b^5*c - 15*a*b^3*c^2 + 27*a^2*b*c^3)*x*sqrt((b^2 - 4*a*c)/c^2) - (2*b^6 - 15*a*b^4*c + 27*a^2*b^2*c^2)*x)*sqrt(c)*sqrt((c*s qrt((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)*((16*b^5*c - 60*a^2*c^4 + 3*(72*a^2*b + 17*a*b^2)*c^ 3 - 8*(15*a*b^3 + b^4)*c^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (16*b^6 + 60*a^2*b *c^3 + 3*(72*a^2*b^2 - 17*a*b^3)*c^2 - 8*(15*a*b^4 - b^5)*c)*x)*sqrt(c)*sq rt((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*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - 2*(105*c^6*x^10 + 140*b*c^5*x^8 + 5*(b^2*c^4 + 39*a*c^5 )*x^6 - 16*b^5*c + 120*a*b^3*c^2 - 216*a^2*b*c^3 - 2*(3*b^3*c^3 - 16*a*b*c ^4)*x^4 + (8*b^4*c^2 - 51*a*b^2*c^3 + 60*a^2*c^4)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(c^5*x)
\[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{4} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**4*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(x**4*(a + b*x**2 + c*x**4)**(3/2), x)
\[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{4} \,d x } \] Input:
integrate(x^4*(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*x^4, x)
\[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{4} \,d x } \] Input:
integrate(x^4*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*x^4, x)
Timed out. \[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^4\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int(x^4*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^4*(a + b*x^2 + c*x^4)^(3/2), x)
\[ \int x^4 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {60 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} c^{2} x -51 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{2} c x +32 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,c^{2} x^{3}+195 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,c^{3} x^{5}+8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4} x -6 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} c \,x^{3}+5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c^{2} x^{5}+140 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{3} x^{7}+105 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{4} x^{9}-60 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{3} c^{2}+51 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} b^{2} c -8 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{4}-216 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} b \,c^{2}+120 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{3} c -16 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{5}}{1155 c^{3}} \] Input:
int(x^4*(c*x^4+b*x^2+a)^(3/2),x)
Output:
(60*sqrt(a + b*x**2 + c*x**4)*a**2*c**2*x - 51*sqrt(a + b*x**2 + c*x**4)*a *b**2*c*x + 32*sqrt(a + b*x**2 + c*x**4)*a*b*c**2*x**3 + 195*sqrt(a + b*x* *2 + c*x**4)*a*c**3*x**5 + 8*sqrt(a + b*x**2 + c*x**4)*b**4*x - 6*sqrt(a + b*x**2 + c*x**4)*b**3*c*x**3 + 5*sqrt(a + b*x**2 + c*x**4)*b**2*c**2*x**5 + 140*sqrt(a + b*x**2 + c*x**4)*b*c**3*x**7 + 105*sqrt(a + b*x**2 + c*x** 4)*c**4*x**9 - 60*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a **3*c**2 + 51*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a**2* b**2*c - 8*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a*b**4 - 216*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*a**2*b* c**2 + 120*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*a *b**3*c - 16*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x) *b**5)/(1155*c**3)