Integrand size = 20, antiderivative size = 204 \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{12 c}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2048 c^{9/2}} \] Output:
-1/1024*(-4*a*c+b^2)*(-4*a*c+7*b^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^4+ 1/384*(-4*a*c+7*b^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(3/2)/c^3-7/120*b*(c*x^4+ b*x^2+a)^(5/2)/c^2+1/12*x^2*(c*x^4+b*x^2+a)^(5/2)/c+1/2048*(-4*a*c+b^2)^2* (-4*a*c+7*b^2)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(9 /2)
Time = 0.53 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95 \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+b x^2+c x^4} \left (-105 b^5+70 b^4 c x^2+8 b^3 c \left (95 a-7 c x^4\right )+48 b^2 c^2 x^2 \left (-9 a+c x^4\right )+160 c^3 x^2 \left (3 a^2+14 a c x^4+8 c^2 x^8\right )+16 b c^2 \left (-81 a^2+18 a c x^4+104 c^2 x^8\right )\right )-15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{30720 c^{9/2}} \] Input:
Integrate[x^5*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(-105*b^5 + 70*b^4*c*x^2 + 8*b^3*c*(95* a - 7*c*x^4) + 48*b^2*c^2*x^2*(-9*a + c*x^4) + 160*c^3*x^2*(3*a^2 + 14*a*c *x^4 + 8*c^2*x^8) + 16*b*c^2*(-81*a^2 + 18*a*c*x^4 + 104*c^2*x^8)) - 15*(b ^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(30720*c^(9/2))
Time = 0.58 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1434, 1166, 27, 1160, 1087, 1087, 1092, 219}
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.
| \(\displaystyle \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int x^4 \left (c x^4+b x^2+a\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {1}{2} \left (7 b x^2+2 a\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2}{6 c}+\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}-\frac {\int \left (7 b x^2+2 a\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2}{12 c}\right )\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}-\frac {\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {\left (7 b^2-4 a c\right ) \int \left (c x^4+b x^2+a\right )^{3/2}dx^2}{2 c}}{12 c}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}-\frac {\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {\left (7 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )}{2 c}}{12 c}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}-\frac {\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {\left (7 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )}{2 c}}{12 c}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}-\frac {\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {\left (7 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )}{2 c}}{12 c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^2+c x^4\right )^{5/2}}{6 c}-\frac {\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {\left (7 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{2 c}}{12 c}\right )\) |
Input:
Int[x^5*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
((x^2*(a + b*x^2 + c*x^4)^(5/2))/(6*c) - ((7*b*(a + b*x^2 + c*x^4)^(5/2))/ (5*c) - ((7*b^2 - 4*a*c)*(((b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^ (3/2))))/(16*c)))/(2*c))/(12*c))/2
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(-\frac {\left (-1280 c^{5} x^{10}-1664 c^{4} x^{8} b -2240 c^{4} a \,x^{6}-48 b^{2} c^{3} x^{6}-288 a b \,c^{3} x^{4}+56 c^{2} x^{4} b^{3}-480 a^{2} c^{3} x^{2}+432 a \,b^{2} c^{2} x^{2}-70 x^{2} c \,b^{4}+1296 a^{2} b \,c^{2}-760 a \,b^{3} c +105 b^{5}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15360 c^{4}}-\frac {\left (64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+60 a \,b^{4} c -7 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2048 c^{\frac {9}{2}}}\) | \(199\) |
| pseudoelliptic | \(-\frac {\left (a c -\frac {b^{2}}{4}\right )^{2} \left (a c -\frac {7 b^{2}}{4}\right ) \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )+\left (\frac {\left (\frac {7}{6} b^{3} x^{4}+9 a \,b^{2} x^{2}+27 a^{2} b \right ) c^{\frac {5}{2}}}{10}+\left (-\frac {1}{10} b^{2} x^{6}-\frac {3}{5} a b \,x^{4}-a^{2} x^{2}\right ) c^{\frac {7}{2}}-\frac {19 b^{3} \left (\frac {7 b \,x^{2}}{76}+a \right ) c^{\frac {3}{2}}}{12}-\frac {14 \left (\frac {26 b \,x^{2}}{35}+a \right ) x^{6} c^{\frac {9}{2}}}{3}-\frac {8 c^{\frac {11}{2}} x^{10}}{3}+\frac {7 \sqrt {c}\, b^{5}}{32}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}-\ln \left (2\right ) \left (a c -\frac {b^{2}}{4}\right )^{2} \left (a c -\frac {7 b^{2}}{4}\right )}{32 c^{\frac {9}{2}}}\) | \(205\) |
| default | \(-\frac {15 b^{4} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}-\frac {9 b^{2} a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 c^{2}}+\frac {3 b a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {27 a^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 c^{2}}+\frac {9 a^{2} b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{128 c^{\frac {5}{2}}}+\frac {a^{2} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 c}+\frac {b^{2} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 c}-\frac {7 b^{3} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1920 c^{2}}+\frac {7 b^{4} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1536 c^{3}}+\frac {7 b^{6} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2048 c^{\frac {9}{2}}}+\frac {19 b^{3} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 c^{3}}+\frac {7 a \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{48}-\frac {a^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {c \,x^{10} \sqrt {c \,x^{4}+b \,x^{2}+a}}{12}+\frac {13 b \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{120}-\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 c^{4}}\) | \(432\) |
| elliptic | \(-\frac {15 b^{4} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}-\frac {9 b^{2} a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 c^{2}}+\frac {3 b a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {27 a^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 c^{2}}+\frac {9 a^{2} b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{128 c^{\frac {5}{2}}}+\frac {a^{2} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 c}+\frac {b^{2} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{320 c}-\frac {7 b^{3} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1920 c^{2}}+\frac {7 b^{4} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1536 c^{3}}+\frac {7 b^{6} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2048 c^{\frac {9}{2}}}+\frac {19 b^{3} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{384 c^{3}}+\frac {7 a \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{48}-\frac {a^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {c \,x^{10} \sqrt {c \,x^{4}+b \,x^{2}+a}}{12}+\frac {13 b \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{120}-\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 c^{4}}\) | \(432\) |
Input:
int(x^5*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/15360*(-1280*c^5*x^10-1664*b*c^4*x^8-2240*a*c^4*x^6-48*b^2*c^3*x^6-288* a*b*c^3*x^4+56*b^3*c^2*x^4-480*a^2*c^3*x^2+432*a*b^2*c^2*x^2-70*b^4*c*x^2+ 1296*a^2*b*c^2-760*a*b^3*c+105*b^5)*(c*x^4+b*x^2+a)^(1/2)/c^4-1/2048*(64*a ^3*c^3-144*a^2*b^2*c^2+60*a*b^4*c-7*b^6)/c^(9/2)*ln((1/2*b+c*x^2)/c^(1/2)+ (c*x^4+b*x^2+a)^(1/2))
Time = 0.09 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.21 \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (1280 \, c^{6} x^{10} + 1664 \, b c^{5} x^{8} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{61440 \, c^{5}}, -\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \, {\left (1280 \, c^{6} x^{10} + 1664 \, b c^{5} x^{8} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30720 \, c^{5}}\right ] \] Input:
integrate(x^5*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[-1/61440*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c)* log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b) *sqrt(c) - 4*a*c) - 4*(1280*c^6*x^10 + 1664*b*c^5*x^8 + 16*(3*b^2*c^4 + 14 0*a*c^5)*x^6 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^4 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*x^2)*sqrt( c*x^4 + b*x^2 + a))/c^5, -1/30720*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^ 2 - 64*a^3*c^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)* sqrt(-c)/(c^2*x^4 + b*c*x^2 + a*c)) - 2*(1280*c^6*x^10 + 1664*b*c^5*x^8 + 16*(3*b^2*c^4 + 140*a*c^5)*x^6 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^ 3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^4 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a ^2*c^4)*x^2)*sqrt(c*x^4 + b*x^2 + a))/c^5]
\[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**5*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(x**5*(a + b*x**2 + c*x**4)**(3/2), x)
Exception generated. \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^5*(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (178) = 356\).
Time = 0.16 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.59 \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {1}{768} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {5 \, b^{2} c - 12 \, a c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, b^{3} - 52 \, a b c}{c^{3}}\right )} + \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {7}{2}}}\right )} a + \frac {1}{7680} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {9}{2}}}\right )} b + \frac {1}{30720} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {9 \, b^{2} c^{3} - 20 \, a c^{4}}{c^{5}}\right )} x^{2} + \frac {21 \, b^{3} c^{2} - 68 \, a b c^{3}}{c^{5}}\right )} x^{2} - \frac {105 \, b^{4} c - 448 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}}{c^{5}}\right )} x^{2} + \frac {315 \, b^{5} - 1680 \, a b^{3} c + 1808 \, a^{2} b c^{2}}{c^{5}}\right )} + \frac {15 \, {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {11}{2}}}\right )} c \] Input:
integrate(x^5*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
1/768*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(6*x^2 + b/c)*x^2 - (5*b^2*c - 12*a *c^2)/c^3)*x^2 + (15*b^3 - 52*a*b*c)/c^3) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2 *c^2)*log(abs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) + b))/c^(7 /2))*a + 1/7680*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(6*(8*x^2 + b/c)*x^2 - (7 *b^2*c^2 - 16*a*c^3)/c^4)*x^2 + (35*b^3*c - 116*a*b*c^2)/c^4)*x^2 - (105*b ^4 - 460*a*b^2*c + 256*a^2*c^2)/c^4) - 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c ^2)*log(abs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) + b))/c^(9/2 ))*b + 1/30720*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(2*(8*(10*x^2 + b/c)*x^2 - (9*b^2*c^3 - 20*a*c^4)/c^5)*x^2 + (21*b^3*c^2 - 68*a*b*c^3)/c^5)*x^2 - (1 05*b^4*c - 448*a*b^2*c^2 + 240*a^2*c^3)/c^5)*x^2 + (315*b^5 - 1680*a*b^3*c + 1808*a^2*b*c^2)/c^5) + 15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64* a^3*c^3)*log(abs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) + b))/c ^(11/2))*c
Timed out. \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int(x^5*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^5*(a + b*x^2 + c*x^4)^(3/2), x)
Time = 0.32 (sec) , antiderivative size = 5920, normalized size of antiderivative = 29.02 \[ \int x^5 \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int(x^5*(c*x^4+b*x^2+a)^(3/2),x)
Output:
( - 184320*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x** 2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**5*b*c**5 - 368640*sqrt( c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**5*c**6*x**2 + 261120*sqrt(c)*sqrt(a + b *x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/s qrt(4*a*c - b**2))*a**4*b**3*c**4 - 460800*sqrt(c)*sqrt(a + b*x**2 + c*x** 4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b **2))*a**4*b**2*c**5*x**2 - 2949120*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log( (2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a **4*b*c**6*x**4 - 1966080*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c) *sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**4*c**7*x **6 + 161280*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x **2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**3*b**5*c**3 + 2288640 *sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4 ) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**3*b**4*c**4*x**2 + 4915200*sqrt(c )*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**3*b**3*c**5*x**4 - 4915200*sqrt(c)*sqrt( a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x* *2)/sqrt(4*a*c - b**2))*a**3*b*c**7*x**8 - 1966080*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqr...