Integrand size = 20, antiderivative size = 594 \[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {\sqrt {x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (b \left (b^2+20 a c\right )+c \left (b^2+44 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 c^{3/4} \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 a b c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (b^3-68 a b c+\sqrt {b^2-4 a c} \left (b^2+44 a c\right )\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (b^2+44 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 a b c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (b^3-68 a b c+\sqrt {b^2-4 a c} \left (b^2+44 a c\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:
-1/4*x^(1/2)*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/16*x^(1/2)*(b*(2 0*a*c+b^2)+c*(44*a*c+b^2)*x^2)/a/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)-3/64*c^(3/ 4)*(b^2+44*a*c-b^3/(-4*a*c+b^2)^(1/2)+68*a*b*c/(-4*a*c+b^2)^(1/2))*arctan( 2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/a/(-4*a*c+b ^2)^2/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*c^(3/4)*(b^3-68*a*b*c+(-4*a*c+b^2 )^(1/2)*(44*a*c+b^2))*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2 ))^(1/4))*2^(3/4)/a/(-4*a*c+b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)-3/64* c^(3/4)*(b^2+44*a*c-b^3/(-4*a*c+b^2)^(1/2)+68*a*b*c/(-4*a*c+b^2)^(1/2))*ar ctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/a/(-4 *a*c+b^2)^2/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*c^(3/4)*(b^3-68*a*b*c+(-4*a *c+b^2)^(1/2)*(44*a*c+b^2))*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^ 2)^(1/2))^(1/4))*2^(3/4)/a/(-4*a*c+b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4 )
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.36 \[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 \sqrt {x} \left (b^2 x^2 \left (b+c x^2\right )^2+4 a^2 c \left (9 b+19 c x^2\right )+a \left (-3 b^3+13 b^2 c x^2+64 b c^2 x^4+44 c^3 x^6\right )\right )}{\left (a+b x^2+c x^4\right )^2}+3 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-12 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4+44 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{64 a \left (b^2-4 a c\right )^2} \] Input:
Integrate[x^(3/2)/(a + b*x^2 + c*x^4)^3,x]
Output:
((4*Sqrt[x]*(b^2*x^2*(b + c*x^2)^2 + 4*a^2*c*(9*b + 19*c*x^2) + a*(-3*b^3 + 13*b^2*c*x^2 + 64*b*c^2*x^4 + 44*c^3*x^6)))/(a + b*x^2 + c*x^4)^2 + 3*Ro otSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 12*a*b*c*Log[Sqrt[x ] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 44*a*c^2*Log[Sqrt[x] - #1]*#1^4)/ (b*#1^3 + 2*c*#1^7) & ])/(64*a*(b^2 - 4*a*c)^2)
Time = 1.32 (sec) , antiderivative size = 516, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1435, 1700, 1760, 27, 1752, 756, 218, 221}
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.
| \(\displaystyle \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1435 |
| \(\displaystyle 2 \int \frac {x^2}{\left (c x^4+b x^2+a\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 1700 |
| \(\displaystyle 2 \left (\frac {\int \frac {b-22 c x^2}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1760 |
| \(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 \left (c \left (b^2+44 a c\right ) x^2+b \left (b^2-12 a c\right )\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {3 \int \frac {c \left (b^2+44 a c\right ) x^2+b \left (b^2-12 a c\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {1}{2} c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d\sqrt {x}+\frac {1}{2} c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d\sqrt {x}\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {1}{2} c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )+\frac {1}{2} c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {1}{2} c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )+\frac {1}{2} c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {1}{2} c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )+\frac {1}{2} c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+44 a c+b^2\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
Input:
Int[x^(3/2)/(a + b*x^2 + c*x^4)^3,x]
Output:
2*(-1/8*(Sqrt[x]*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (( Sqrt[x]*(b*(b^2 + 20*a*c) + c*(b^2 + 44*a*c)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*((c*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*a*b* c)/Sqrt[b^2 - 4*a*c])*(-(ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTan h[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/ 4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))))/2 + (c*(b^2 + 44*a*c + b^3/Sqrt[b^2 - 4*a*c] - (68*a*b*c)/Sqrt[b^2 - 4*a*c])*(-(ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x] )/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c] )^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/ 4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))))/2))/(4*a*(b^2 - 4*a *c)))/(8*(b^2 - 4*a*c)))
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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/d Subst[Int[x^(k*(m + 1) - 1)*(a + b *(x^(2*k)/d^2) + c*(x^(4*k)/d^4))^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[d^(n - 1)*(d*x)^(m - n + 1)*(b + 2*c*x^n)*((a + b*x^n + c* x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c))), x] - Simp[d^n/(n*(p + 1)*(b^2 - 4*a*c)) Int[(d*x)^(m - n)*(b*(m - n + 1) + 2*c*(m + 2*n*(p + 1) + 1)*x^ n)*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ [n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p _), x_Symbol] :> Simp[(-x)*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)* ((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/ (a*n*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c* d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n ] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.45
| method | result | size |
| derivativedivides | \(\frac {\frac {3 b \left (12 a c -b^{2}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (76 a^{2} c^{2}+13 a \,b^{2} c +b^{4}\right ) x^{\frac {5}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c b \left (32 a c +b^{2}\right ) x^{\frac {9}{2}}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c^{2} \left (44 a c +b^{2}\right ) x^{\frac {13}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (44 a c +b^{2}\right ) \textit {\_R}^{4}-12 a b c +b^{3}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(270\) |
| default | \(\frac {\frac {3 b \left (12 a c -b^{2}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (76 a^{2} c^{2}+13 a \,b^{2} c +b^{4}\right ) x^{\frac {5}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c b \left (32 a c +b^{2}\right ) x^{\frac {9}{2}}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c^{2} \left (44 a c +b^{2}\right ) x^{\frac {13}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (44 a c +b^{2}\right ) \textit {\_R}^{4}-12 a b c +b^{3}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(270\) |
Input:
int(x^(3/2)/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
2*(3/32*b*(12*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)+1/32*(76*a^2*c^2 +13*a*b^2*c+b^4)/a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+1/16*c/a*b*(32*a*c+b ^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(9/2)+1/32*c^2*(44*a*c+b^2)/a/(16*a^2*c^2 -8*a*b^2*c+b^4)*x^(13/2))/(c*x^4+b*x^2+a)^2+3/64/a/(16*a^2*c^2-8*a*b^2*c+b ^4)*sum((c*(44*a*c+b^2)*_R^4-12*a*b*c+b^3)/(2*_R^7*c+_R^3*b)*ln(x^(1/2)-_R ),_R=RootOf(_Z^8*c+_Z^4*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 15770 vs. \(2 (492) = 984\).
Time = 15.98 (sec) , antiderivative size = 15770, normalized size of antiderivative = 26.55 \[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^(3/2)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**(3/2)/(c*x**4+b*x**2+a)**3,x)
Output:
Timed out
\[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(3/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
1/16*(3*(b^3*c^2 - 12*a*b*c^3)*x^(17/2) + (6*b^4*c - 71*a*b^2*c^2 + 44*a^2 *c^3)*x^(13/2) + (3*b^5 - 28*a*b^3*c - 8*a^2*b*c^2)*x^(9/2) + (7*a*b^4 - 5 9*a^2*b^2*c + 76*a^3*c^2)*x^(5/2))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4* c^4)*x^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c ^2 + 16*a^4*b*c^3)*x^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3 *b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) + integrate(-3/32*((b^3*c - 12*a*b *c^2)*x^(7/2) + (b^4 - 13*a*b^2*c - 44*a^2*c^2)*x^(3/2))/(a^3*b^4 - 8*a^4* b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b ^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2), x)
\[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(3/2)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
integrate(x^(3/2)/(c*x^4 + b*x^2 + a)^3, x)
Time = 23.63 (sec) , antiderivative size = 54027, normalized size of antiderivative = 90.95 \[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int(x^(3/2)/(a + b*x^2 + c*x^4)^3,x)
Output:
atan(((((3*(230850*a*b^11*c^8 - 4455*b^13*c^7 + 24287662080*a^6*b*c^13 - 3 679344*a^2*b^9*c^9 + 8309952*a^3*b^7*c^10 - 548653824*a^4*b^5*c^11 + 97602 27840*a^5*b^3*c^12))/(65536*(a^4*b^18 - 262144*a^13*c^9 - 36*a^5*b^16*c + 576*a^6*b^14*c^2 - 5376*a^7*b^12*c^3 + 32256*a^8*b^10*c^4 - 129024*a^9*b^8 *c^5 + 344064*a^10*b^6*c^6 - 589824*a^11*b^4*c^7 + 589824*a^12*b^2*c^8)) + ((3*(-(81*(b^35 + b^10*(-(4*a*c - b^2)^25)^(1/2) + 12505065717760*a^17*b* c^17 + 3910*a^2*b^31*c^2 - 91335*a^3*b^29*c^3 + 1329320*a^4*b^27*c^4 - 123 56816*a^5*b^25*c^5 + 70316800*a^6*b^23*c^6 - 181190400*a^7*b^21*c^7 - 6687 23200*a^8*b^19*c^8 + 10912870400*a^9*b^17*c^9 - 83490242560*a^10*b^15*c^10 + 502626713600*a^11*b^13*c^11 - 2379389337600*a^12*b^11*c^12 + 8291284418 560*a^13*b^9*c^13 - 20114959237120*a^14*b^7*c^14 + 31974471237632*a^15*b^5 *c^15 - 29919144837120*a^16*b^3*c^16 - 234256*a^5*c^5*(-(4*a*c - b^2)^25)^ (1/2) - 95*a*b^33*c + 510*a^2*b^6*c^2*(-(4*a*c - b^2)^25)^(1/2) + 2015*a^3 *b^4*c^3*(-(4*a*c - b^2)^25)^(1/2) - 33880*a^4*b^2*c^4*(-(4*a*c - b^2)^25) ^(1/2) - 45*a*b^8*c*(-(4*a*c - b^2)^25)^(1/2)))/(33554432*(a^7*b^40 + 1099 511627776*a^27*c^20 - 80*a^8*b^38*c + 3040*a^9*b^36*c^2 - 72960*a^10*b^34* c^3 + 1240320*a^11*b^32*c^4 - 15876096*a^12*b^30*c^5 + 158760960*a^13*b^28 *c^6 - 1270087680*a^14*b^26*c^7 + 8255569920*a^15*b^24*c^8 - 44029706240*a ^16*b^22*c^9 + 193730707456*a^17*b^20*c^10 - 704475299840*a^18*b^18*c^11 + 2113425899520*a^19*b^16*c^12 - 5202279137280*a^20*b^14*c^13 + 10404558...
\[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{3}}d x \] Input:
int(x^(3/2)/(c*x^4+b*x^2+a)^3,x)
Output:
int(x^(3/2)/(c*x^4+b*x^2+a)^3,x)