Integrand size = 20, antiderivative size = 150 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c}-\frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2}} \] Output:
3/256*b*(-4*a*c+b^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^3-1/32*b*(2*c*x^2 +b)*(c*x^4+b*x^2+a)^(3/2)/c^2+1/10*(c*x^4+b*x^2+a)^(5/2)/c-3/512*b*(-4*a*c +b^2)^2*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(7/2)
Time = 0.36 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (15 b^4-10 b^3 c x^2+128 c^2 \left (a+c x^4\right )^2+4 b^2 c \left (-25 a+2 c x^4\right )+8 b c^2 x^2 \left (7 a+22 c x^4\right )\right )}{1280 c^3}+\frac {3 b \left (b^2-4 a c\right )^2 \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{512 c^{7/2}} \] Input:
Integrate[x^3*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(Sqrt[a + b*x^2 + c*x^4]*(15*b^4 - 10*b^3*c*x^2 + 128*c^2*(a + c*x^4)^2 + 4*b^2*c*(-25*a + 2*c*x^4) + 8*b*c^2*x^2*(7*a + 22*c*x^4)))/(1280*c^3) + (3 *b*(b^2 - 4*a*c)^2*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/( 512*c^(7/2))
Time = 0.48 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1434, 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.
| \(\displaystyle \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int x^2 \left (c x^4+b x^2+a\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \int \left (c x^4+b x^2+a\right )^{3/2}dx^2}{2 c}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )}{2 c}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )}{2 c}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )}{2 c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{2 c}\right )\) |
Input:
Int[x^3*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
((a + b*x^2 + c*x^4)^(5/2)/(5*c) - (b*(((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))/2
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[(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 = 0.24 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(\frac {\left (128 c^{4} x^{8}+176 b \,c^{3} x^{6}+256 a \,c^{3} x^{4}+8 b^{2} c^{2} x^{4}+56 a b \,c^{2} x^{2}-10 x^{2} b^{3} c +128 a^{2} c^{2}-100 a \,b^{2} c +15 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{1280 c^{3}}-\frac {3 b \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}\) | \(152\) |
| pseudoelliptic | \(\frac {-\frac {15 \left (a c -\frac {b^{2}}{4}\right )^{2} b \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{16}+\left (\left (\frac {1}{16} b^{2} x^{4}+\frac {7}{16} a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}-\frac {25 \left (\frac {b \,x^{2}}{10}+a \right ) b^{2} c^{\frac {3}{2}}}{32}+\left (\frac {11}{8} b \,x^{6}+2 a \,x^{4}\right ) c^{\frac {7}{2}}+c^{\frac {9}{2}} x^{8}+\frac {15 \sqrt {c}\, b^{4}}{128}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {15 \ln \left (2\right ) \left (a c -\frac {b^{2}}{4}\right )^{2} b}{16}}{10 c^{\frac {7}{2}}}\) | \(156\) |
| default | \(\frac {3 b^{3} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}+\frac {7 b a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {b^{3} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 c^{2}}-\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}-\frac {5 b^{2} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c^{2}}+\frac {b^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {3 a^{2} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 c}+\frac {11 b \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{80}+\frac {3 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{3}}+\frac {c \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10}\) | \(316\) |
| elliptic | \(\frac {3 b^{3} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{64 c^{\frac {5}{2}}}+\frac {7 b a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {b^{3} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 c^{2}}-\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {7}{2}}}-\frac {5 b^{2} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c^{2}}+\frac {b^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{160 c}-\frac {3 a^{2} b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {3}{2}}}+\frac {a \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {a^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10 c}+\frac {11 b \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}+a}}{80}+\frac {3 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{3}}+\frac {c \,x^{8} \sqrt {c \,x^{4}+b \,x^{2}+a}}{10}\) | \(316\) |
Input:
int(x^3*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/1280*(128*c^4*x^8+176*b*c^3*x^6+256*a*c^3*x^4+8*b^2*c^2*x^4+56*a*b*c^2*x ^2-10*b^3*c*x^2+128*a^2*c^2-100*a*b^2*c+15*b^4)*(c*x^4+b*x^2+a)^(1/2)/c^3- 3/512*b*(16*a^2*c^2-8*a*b^2*c+b^4)/c^(7/2)*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 = 361, normalized size of antiderivative = 2.41 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{5120 \, c^{4}}, \frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2560 \, c^{4}}\right ] \] Input:
integrate(x^3*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/5120*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*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 *(128*c^5*x^8 + 176*b*c^4*x^6 + 15*b^4*c - 100*a*b^2*c^2 + 128*a^2*c^3 + 8 *(b^2*c^3 + 32*a*c^4)*x^4 - 2*(5*b^3*c^2 - 28*a*b*c^3)*x^2)*sqrt(c*x^4 + b *x^2 + a))/c^4, 1/2560*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*sqrt(-c)*arcta n(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*(128*c^5*x^8 + 176*b*c^4*x^6 + 15*b^4*c - 100*a*b^2*c^2 + 128*a^ 2*c^3 + 8*(b^2*c^3 + 32*a*c^4)*x^4 - 2*(5*b^3*c^2 - 28*a*b*c^3)*x^2)*sqrt( c*x^4 + b*x^2 + a))/c^4]
\[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int x^{3} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**3*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(x**3*(a + b*x**2 + c*x**4)**(3/2), x)
Exception generated. \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(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. 408 vs. \(2 (128) = 256\).
Time = 0.16 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.72 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {1}{96} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {5}{2}}}\right )} a + \frac {1}{768} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {5 \, b^{2} c - 12 \, a c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, b^{3} - 52 \, a b c}{c^{3}}\right )} + \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {7}{2}}}\right )} b + \frac {1}{7680} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {9}{2}}}\right )} c \] Input:
integrate(x^3*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
1/96*(2*sqrt(c*x^4 + b*x^2 + a)*(2*(4*x^2 + b/c)*x^2 - (3*b^2 - 8*a*c)/c^2 ) - 3*(b^3 - 4*a*b*c)*log(abs(2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sq rt(c) + b))/c^(5/2))*a + 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))*b + 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))*c
Time = 17.78 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.49 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {{\left (c\,x^4+b\,x^2+a\right )}^{5/2}}{10\,c}-\frac {b\,\left (\frac {3\,a\,\left (\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (2\,c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2+a}}{4\,c}\right )}{4}+\frac {x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{4}-\frac {3\,b^2\,\left (\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (2\,c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2+a}}{4\,c}\right )}{16\,c}+\frac {b\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{8\,c}\right )}{4\,c} \] Input:
int(x^3*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
(a + b*x^2 + c*x^4)^(5/2)/(10*c) - (b*((3*a*(log((a + b*x^2 + c*x^4)^(1/2) + (b/2 + c*x^2)/c^(1/2))*(a/(2*c^(1/2)) - b^2/(8*c^(3/2))) + ((b + 2*c*x^ 2)*(a + b*x^2 + c*x^4)^(1/2))/(4*c)))/4 + (x^2*(a + b*x^2 + c*x^4)^(3/2))/ 4 - (3*b^2*(log((a + b*x^2 + c*x^4)^(1/2) + (b/2 + c*x^2)/c^(1/2))*(a/(2*c ^(1/2)) - b^2/(8*c^(3/2))) + ((b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(4* c)))/(16*c) + (b*(a + b*x^2 + c*x^4)^(3/2))/(8*c)))/(4*c)
Time = 0.22 (sec) , antiderivative size = 3844, normalized size of antiderivative = 25.63 \[ \int x^3 \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int(x^3*(c*x^4+b*x^2+a)^(3/2),x)
Output:
( - 7680*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**4 - 15360*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**3 - 92160*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**2*c**4*x**2 - 92160*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**5*x**4 + 6720*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*log((2*sqr t(c)*sqrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a**2*b* *5*c**2 + 7680*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**2*b**4*c**3*x**2 - 115200*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**2*b**3*c**4*x**4 - 245760*s qrt(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**2*b**2*c**5*x**6 - 122880*sqrt(c)*s qrt(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**2*b*c**6*x**8 + 13440*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*b**6*c**2*x**2 + 74880*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 - ...