Integrand size = 20, antiderivative size = 223 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {3 b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{2048 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (21 b^2-16 a c-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{560 c^3}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4096 c^{11/2}} \] Output:
3/2048*b*(-4*a*c+b^2)*(-4*a*c+3*b^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^5 -1/256*b*(-4*a*c+3*b^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(3/2)/c^4+1/14*x^4*(c* x^4+b*x^2+a)^(5/2)/c+1/560*(-30*b*c*x^2-16*a*c+21*b^2)*(c*x^4+b*x^2+a)^(5/ 2)/c^3-3/4096*b*(-4*a*c+b^2)^2*(-4*a*c+3*b^2)*arctanh(1/2*(2*c*x^2+b)/c^(1 /2)/(c*x^4+b*x^2+a)^(1/2))/c^(11/2)
Time = 0.69 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.99 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (315 b^6-210 b^5 c x^2+16 b^3 c^2 x^2 \left (91 a-9 c x^4\right )+168 b^4 c \left (-15 a+c x^4\right )+1024 c^3 \left (a+c x^4\right )^2 \left (-2 a+5 c x^4\right )+16 b^2 c^2 \left (343 a^2-62 a c x^4+8 c^2 x^8\right )+32 b c^3 x^2 \left (-73 a^2+22 a c x^4+200 c^2 x^8\right )\right )}{71680 c^5}+\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{4096 c^{11/2}} \] Input:
Integrate[x^7*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(Sqrt[a + b*x^2 + c*x^4]*(315*b^6 - 210*b^5*c*x^2 + 16*b^3*c^2*x^2*(91*a - 9*c*x^4) + 168*b^4*c*(-15*a + c*x^4) + 1024*c^3*(a + c*x^4)^2*(-2*a + 5*c *x^4) + 16*b^2*c^2*(343*a^2 - 62*a*c*x^4 + 8*c^2*x^8) + 32*b*c^3*x^2*(-73* a^2 + 22*a*c*x^4 + 200*c^2*x^8)))/(71680*c^5) + (3*b*(b^2 - 4*a*c)^2*(3*b^ 2 - 4*a*c)*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(4096*c^( 11/2))
Time = 0.61 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.02, 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, 1225, 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.
| \(\displaystyle \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int x^6 \left (c x^4+b x^2+a\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {1}{2} x^2 \left (9 b x^2+4 a\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2}{7 c}+\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}-\frac {\int x^2 \left (9 b x^2+4 a\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2}{14 c}\right )\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}-\frac {\frac {7 b \left (3 b^2-4 a c\right ) \int \left (c x^4+b x^2+a\right )^{3/2}dx^2}{8 c^2}-\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{20 c^2}}{14 c}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}-\frac {\frac {7 b \left (3 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )}{8 c^2}-\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{20 c^2}}{14 c}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}-\frac {\frac {7 b \left (3 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{20 c^2}}{14 c}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}-\frac {\frac {7 b \left (3 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{20 c^2}}{14 c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^4 \left (a+b x^2+c x^4\right )^{5/2}}{7 c}-\frac {\frac {7 b \left (3 b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{8 c^2}-\frac {\left (-16 a c+21 b^2-30 b c x^2\right ) \left (a+b x^2+c x^4\right )^{5/2}}{20 c^2}}{14 c}\right )\) |
Input:
Int[x^7*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
((x^4*(a + b*x^2 + c*x^4)^(5/2))/(7*c) - (-1/20*((21*b^2 - 16*a*c - 30*b*c *x^2)*(a + b*x^2 + c*x^4)^(5/2))/c^2 + (7*b*(3*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)))/(8*c^2))/(14*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_))^(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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
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 = 0.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {105 \left (a c -\frac {3 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )^{2} b \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{64}+\left (\left (\frac {9}{128} b^{3} x^{6}+\frac {31}{64} b^{2} x^{4} a +\frac {73}{64} a^{2} b \,x^{2}+a^{3}\right ) c^{\frac {7}{2}}-\frac {343 \left (\frac {3}{98} b^{2} x^{4}+\frac {13}{49} a b \,x^{2}+a^{2}\right ) b^{2} c^{\frac {5}{2}}}{128}+\left (-\frac {1}{16} b^{2} x^{8}-\frac {11}{32} a b \,x^{6}-\frac {1}{2} a^{2} x^{4}\right ) c^{\frac {9}{2}}+\left (-\frac {25}{8} b \,x^{10}-4 a \,x^{8}\right ) c^{\frac {11}{2}}+\frac {315 b^{4} \left (\frac {b \,x^{2}}{12}+a \right ) c^{\frac {3}{2}}}{256}-\frac {5 c^{\frac {13}{2}} x^{12}}{2}-\frac {315 \sqrt {c}\, b^{6}}{2048}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {105 \ln \left (2\right ) \left (a c -\frac {3 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )^{2} b}{64}}{35 c^{\frac {11}{2}}}\) | \(241\) |
| risch | \(-\frac {\left (-5120 c^{6} x^{12}-6400 b \,c^{5} x^{10}-8192 a \,c^{5} x^{8}-128 b^{2} c^{4} x^{8}-704 a b \,c^{4} x^{6}+144 b^{3} c^{3} x^{6}-1024 a^{2} c^{4} x^{4}+992 a \,b^{2} c^{3} x^{4}-168 b^{4} c^{2} x^{4}+2336 a^{2} b \,c^{3} x^{2}-1456 a \,b^{3} c^{2} x^{2}+210 b^{5} c \,x^{2}+2048 a^{3} c^{3}-5488 a^{2} b^{2} c^{2}+2520 a \,b^{4} c -315 b^{6}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{71680 c^{5}}+\frac {3 b \left (64 a^{3} c^{3}-80 a^{2} b^{2} c^{2}+28 a \,b^{4} c -3 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}\) | \(245\) |
| default | \(\frac {13 b^{3} a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}-\frac {31 b^{2} a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {11 b a \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1120 c}+\frac {21 b^{5} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{1024 c^{\frac {9}{2}}}-\frac {73 a^{2} b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {b^{2} x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{560 c}-\frac {9 b^{3} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4480 c^{2}}+\frac {3 b^{4} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1280 c^{3}}-\frac {3 b^{5} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 c^{4}}-\frac {9 b^{7} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}-\frac {9 b^{4} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{4}}+\frac {a^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{70 c}+\frac {49 a^{2} b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}-\frac {15 a^{2} b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 a^{3} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}-\frac {a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 c^{2}}+\frac {9 b^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2048 c^{5}}+\frac {4 a \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {c \,x^{12} \sqrt {c \,x^{4}+b \,x^{2}+a}}{14}+\frac {5 b \,x^{10} \sqrt {c \,x^{4}+b \,x^{2}+a}}{56}\) | \(534\) |
| elliptic | \(\frac {13 b^{3} a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}-\frac {31 b^{2} a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {11 b a \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1120 c}+\frac {21 b^{5} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{1024 c^{\frac {9}{2}}}-\frac {73 a^{2} b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2240 c^{2}}+\frac {b^{2} x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{560 c}-\frac {9 b^{3} x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4480 c^{2}}+\frac {3 b^{4} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1280 c^{3}}-\frac {3 b^{5} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{1024 c^{4}}-\frac {9 b^{7} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4096 c^{\frac {11}{2}}}-\frac {9 b^{4} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{4}}+\frac {a^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{70 c}+\frac {49 a^{2} b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{640 c^{3}}-\frac {15 a^{2} b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 a^{3} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}-\frac {a^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 c^{2}}+\frac {9 b^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{2048 c^{5}}+\frac {4 a \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {c \,x^{12} \sqrt {c \,x^{4}+b \,x^{2}+a}}{14}+\frac {5 b \,x^{10} \sqrt {c \,x^{4}+b \,x^{2}+a}}{56}\) | \(534\) |
Input:
int(x^7*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/35*(-105/64*(a*c-3/4*b^2)*(a*c-1/4*b^2)^2*b*ln((2*c*x^2+2*(c*x^4+b*x^2+ a)^(1/2)*c^(1/2)+b)/c^(1/2))+((9/128*b^3*x^6+31/64*b^2*x^4*a+73/64*a^2*b*x ^2+a^3)*c^(7/2)-343/128*(3/98*b^2*x^4+13/49*a*b*x^2+a^2)*b^2*c^(5/2)+(-1/1 6*b^2*x^8-11/32*a*b*x^6-1/2*a^2*x^4)*c^(9/2)+(-25/8*b*x^10-4*a*x^8)*c^(11/ 2)+315/256*b^4*(1/12*b*x^2+a)*c^(3/2)-5/2*c^(13/2)*x^12-315/2048*c^(1/2)*b ^6)*(c*x^4+b*x^2+a)^(1/2)+105/64*ln(2)*(a*c-3/4*b^2)*(a*c-1/4*b^2)^2*b)/c^ (11/2)
Time = 0.10 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.40 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (5120 \, c^{7} x^{12} + 6400 \, b c^{6} x^{10} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{8} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{6} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{4} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{286720 \, c^{6}}, \frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (5120 \, c^{7} x^{12} + 6400 \, b c^{6} x^{10} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{8} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{6} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{4} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{143360 \, c^{6}}\right ] \] Input:
integrate(x^7*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[-1/286720*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*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*(5120*c^7*x^12 + 6400*b*c^6*x^10 + 128*(b^2*c^5 + 64*a*c^6)*x^8 + 315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3* c^4 - 16*(9*b^3*c^4 - 44*a*b*c^5)*x^6 + 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 12 8*a^2*c^5)*x^4 - 2*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*x^2)*sqr t(c*x^4 + b*x^2 + a))/c^6, 1/143360*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3* c^2 - 64*a^3*b*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*(5120*c^7*x^12 + 6400*b*c^6*x^ 10 + 128*(b^2*c^5 + 64*a*c^6)*x^8 + 315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2* b^2*c^3 - 2048*a^3*c^4 - 16*(9*b^3*c^4 - 44*a*b*c^5)*x^6 + 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*x^4 - 2*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168* a^2*b*c^4)*x^2)*sqrt(c*x^4 + b*x^2 + a))/c^6]
\[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{7} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**7*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(x**7*(a + b*x**2 + c*x**4)**(3/2), x)
Exception generated. \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^7*(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. 663 vs. \(2 (197) = 394\).
Time = 0.17 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.97 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate(x^7*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
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(a bs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) + b))/c^(9/2))*a + 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 - (105*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))* b + 1/430080*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(2*(8*(10*(12*x^2 + b/c)*x^2 - (11*b^2*c^4 - 24*a*c^5)/c^6)*x^2 + (99*b^3*c^3 - 316*a*b*c^4)/c^6)*x^2 - (231*b^4*c^2 - 972*a*b^2*c^3 + 512*a^2*c^4)/c^6)*x^2 + (1155*b^5*c - 604 8*a*b^3*c^2 + 6352*a^2*b*c^3)/c^6)*x^2 - (3465*b^6 - 21840*a*b^4*c + 34608 *a^2*b^2*c^2 - 8192*a^3*c^3)/c^6) - 105*(33*b^7 - 252*a*b^5*c + 560*a^2*b^ 3*c^2 - 320*a^3*b*c^3)*log(abs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*s qrt(c) + b))/c^(13/2))*c
Timed out. \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^7\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int(x^7*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^7*(a + b*x^2 + c*x^4)^(3/2), x)
Time = 0.61 (sec) , antiderivative size = 7599, normalized size of antiderivative = 34.08 \[ \int x^7 \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int(x^7*(c*x^4+b*x^2+a)^(3/2),x)
Output:
(860160*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**6*b*c**6 + 3440640*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**3*c**5 + 20643840*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**5*b**2*c**6*x**2 + 20643840*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**7*x**4 - 3386880*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**5*c**4 - 1720320*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**4 *c**5*x**2 + 67092480*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqr t(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**4*b**3*c**6* x**4 + 137625600*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**7*x**6 + 68812800*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**8*x**8 - 322560* 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**7*c**3 - 18063360*sqrt(c)*sqr t(a + b*x**2 + c*x**4)*log((2*sqrt(c)*sqrt(a + b*x**2 + c*x**4) + b + 2...