Integrand size = 20, antiderivative size = 533 \[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\sqrt {x} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\sqrt {x} \left (13 b^2-4 a c+24 b c x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {c^{3/4} \left (41 b^2+28 a c+36 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {c^{3/4} \left (41 b^2+28 a c-36 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {c^{3/4} \left (41 b^2+28 a c+36 b \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {c^{3/4} \left (41 b^2+28 a c-36 b \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{16 \sqrt [4]{2} \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:
1/4*x^(1/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2-1/16*x^(1/2)*(24*b* c*x^2-4*a*c+13*b^2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/32*c^(3/4)*(41*b^2+28 *a*c+36*b*(-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)/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4) -1/32*c^(3/4)*(41*b^2+28*a*c-36*b*(-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)/(-4*a*c+b^2)^(5/2)/(-b+( -4*a*c+b^2)^(1/2))^(3/4)+1/32*c^(3/4)*(41*b^2+28*a*c+36*b*(-4*a*c+b^2)^(1/ 2))*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4) /(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-1/32*c^(3/4)*(41*b^2+28* a*c-36*b*(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b ^2)^(1/2))^(1/4))*2^(3/4)/(-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 = 0.61 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.65 \[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{64} \left (-\frac {4 \sqrt {x} \left (28 a^2 c+a \left (5 b^2+36 b c x^2-4 c^2 x^4\right )+b x^2 \left (9 b^2+37 b c x^2+24 c^2 x^4\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2}+\frac {8 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {3 b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-14 a c \log \left (\sqrt {x}-\text {$\#$1}\right )+3 b c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{a b^2 c-4 a^2 c^2}-\frac {3 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {8 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )-71 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+140 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-8 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{a c \left (b^2-4 a c\right )^2}\right ) \] Input:
Integrate[x^(7/2)/(a + b*x^2 + c*x^4)^3,x]
Output:
((-4*Sqrt[x]*(28*a^2*c + a*(5*b^2 + 36*b*c*x^2 - 4*c^2*x^4) + b*x^2*(9*b^2 + 37*b*c*x^2 + 24*c^2*x^4)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + (8 *RootSum[a + b*#1^4 + c*#1^8 & , (3*b^2*Log[Sqrt[x] - #1] - 14*a*c*Log[Sqr t[x] - #1] + 3*b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(a*b^2 *c - 4*a^2*c^2) - (3*RootSum[a + b*#1^4 + c*#1^8 & , (8*b^4*Log[Sqrt[x] - #1] - 71*a*b^2*c*Log[Sqrt[x] - #1] + 140*a^2*c^2*Log[Sqrt[x] - #1] + 8*b^3 *c*Log[Sqrt[x] - #1]*#1^4 - 8*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2* c*#1^7) & ])/(a*c*(b^2 - 4*a*c)^2))/64
Time = 1.25 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1435, 1701, 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.
\(\displaystyle \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1435 |
\(\displaystyle 2 \int \frac {x^4}{\left (c x^4+b x^2+a\right )^3}d\sqrt {x}\) |
\(\Big \downarrow \) 1701 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int \frac {2 a-11 b x^2}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 1760 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\sqrt {x} \left (-4 a c+13 b^2+24 b c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {a \left (5 b^2-72 c x^2 b+28 a c\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\sqrt {x} \left (-4 a c+13 b^2+24 b c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {5 b^2-72 c x^2 b+28 a c}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 1752 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\sqrt {x} \left (-4 a c+13 b^2+24 b c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-c \left (36 b-\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d\sqrt {x}-c \left (\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}+36 b\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d\sqrt {x}}{4 \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\sqrt {x} \left (-4 a c+13 b^2+24 b c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-c \left (\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}+36 b\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )-c \left (36 b-\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )}{4 \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\sqrt {x} \left (-4 a c+13 b^2+24 b c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-c \left (\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}+36 b\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-c \left (36 b-\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 2 \left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\sqrt {x} \left (-4 a c+13 b^2+24 b c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-c \left (\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}+36 b\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-c \left (36 b-\frac {28 a c+41 b^2}{\sqrt {b^2-4 a c}}\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 \left (b^2-4 a c\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
Input:
Int[x^(7/2)/(a + b*x^2 + c*x^4)^3,x]
Output:
2*((Sqrt[x]*(2*a + b*x^2))/(8*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - ((Sqr t[x]*(13*b^2 - 4*a*c + 24*b*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (-(c*(36*b + (41*b^2 + 28*a*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)))) - c*(36 *b - (41*b^2 + 28*a*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))))/(4*(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^(2*n - 1))*(d*x)^(m - 2*n + 1)*(2*a + b*x^n)*((a + b*x ^n + c*x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[d^(2*n)/(n*(p + 1)*(b^2 - 4*a*c)) Int[(d*x)^(m - 2*n)*(2*a*(m - 2*n + 1) + b*(m + n*(2 *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, 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 = 2.48 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.44
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (28 a c +5 b^{2}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 b \left (4 a c +b^{2}\right ) x^{\frac {5}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 c \left (4 a c -37 b^{2}\right ) x^{\frac {9}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}-\frac {3 b \,c^{2} x^{\frac {13}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-72 \textit {\_R}^{4} b c +28 a c +5 b^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{1024 a^{2} c^{2}-512 a \,b^{2} c +64 b^{4}}\) | \(237\) |
default | \(\frac {-\frac {a \left (28 a c +5 b^{2}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 b \left (4 a c +b^{2}\right ) x^{\frac {5}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 c \left (4 a c -37 b^{2}\right ) x^{\frac {9}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}-\frac {3 b \,c^{2} x^{\frac {13}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-72 \textit {\_R}^{4} b c +28 a c +5 b^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{1024 a^{2} c^{2}-512 a \,b^{2} c +64 b^{4}}\) | \(237\) |
Input:
int(x^(7/2)/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
2*(-1/32*a*(28*a*c+5*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)-9/32*b*(4*a*c +b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+1/32*c*(4*a*c-37*b^2)/(16*a^2*c^2 -8*a*b^2*c+b^4)*x^(9/2)-3/4*b*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(13/2))/(c* x^4+b*x^2+a)^2+1/64/(16*a^2*c^2-8*a*b^2*c+b^4)*sum((-72*_R^4*b*c+28*a*c+5* b^2)/(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. 13878 vs. \(2 (429) = 858\).
Time = 6.00 (sec) , antiderivative size = 13878, normalized size of antiderivative = 26.04 \[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^(7/2)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**(7/2)/(c*x**4+b*x**2+a)**3,x)
Output:
Timed out
\[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {7}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(7/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
1/16*((5*b^2*c^2 + 28*a*c^3)*x^(17/2) + 2*(5*b^3*c + 16*a*b*c^2)*x^(13/2) + (5*b^4 + a*b^2*c + 60*a^2*c^2)*x^(9/2) + (a*b^3 + 20*a^2*b*c)*x^(5/2))/( (a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16* a^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2* b^4*c + 32*a^4*c^3)*x^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2) - integrate(1/32*((5*b^2*c + 28*a*c^2)*x^(7/2) + 5*(b^3 + 20*a*b*c)*x^(3/2)) /(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c ^3)*x^4 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2), x)
\[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {7}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(x^(7/2)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
integrate(x^(7/2)/(c*x^4 + b*x^2 + a)^3, x)
Time = 22.48 (sec) , antiderivative size = 47803, normalized size of antiderivative = 89.69 \[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int(x^(7/2)/(a + b*x^2 + c*x^4)^3,x)
Output:
atan(((((171894580*a*b^8*c^7 - 48125*b^10*c^6 - 17210368*a^5*c^11 + 352085 6800*a^2*b^6*c^8 + 3512738432*a^3*b^4*c^9 + 167976704*a^4*b^2*c^10)/(65536 *(b^18 - 262144*a^9*c^9 + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4 *b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 36*a*b^16*c)) + ((((625*b^6*(-(4*a*c - b^2)^25)^(1/2 ) - 625*b^31 + 15192104632320*a^15*b*c^15 + 89000*a^2*b^27*c^2 - 27186416* a^3*b^25*c^3 + 1342297600*a^4*b^23*c^4 - 25492409600*a^5*b^21*c^5 + 265188 833280*a^6*b^19*c^6 - 1688816578560*a^7*b^17*c^7 + 6664504147968*a^8*b^15* c^8 - 14462970429440*a^9*b^13*c^9 + 4163326443520*a^10*b^11*c^10 + 7045524 2260480*a^11*b^9*c^11 - 206669464207360*a^12*b^7*c^12 + 267459844112384*a^ 13*b^5*c^13 - 150009114787840*a^14*b^3*c^14 - 38416*a^3*c^3*(-(4*a*c - b^2 )^25)^(1/2) - 23125*a*b^29*c + 1911000*a^2*b^2*c^2*(-(4*a*c - b^2)^25)^(1/ 2) + 54375*a*b^4*c*(-(4*a*c - b^2)^25)^(1/2))/(33554432*(a^3*b^40 + 109951 1627776*a^23*c^20 - 80*a^4*b^38*c + 3040*a^5*b^36*c^2 - 72960*a^6*b^34*c^3 + 1240320*a^7*b^32*c^4 - 15876096*a^8*b^30*c^5 + 158760960*a^9*b^28*c^6 - 1270087680*a^10*b^26*c^7 + 8255569920*a^11*b^24*c^8 - 44029706240*a^12*b^ 22*c^9 + 193730707456*a^13*b^20*c^10 - 704475299840*a^14*b^18*c^11 + 21134 25899520*a^15*b^16*c^12 - 5202279137280*a^16*b^14*c^13 + 10404558274560*a^ 17*b^12*c^14 - 16647293239296*a^18*b^10*c^15 + 20809116549120*a^19*b^8*c^1 6 - 19585050869760*a^20*b^6*c^17 + 13056700579840*a^21*b^4*c^18 - 54975...
\[ \int \frac {x^{7/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int \frac {x^{\frac {7}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{3}}d x \] Input:
int(x^(7/2)/(c*x^4+b*x^2+a)^3,x)
Output:
int(x^(7/2)/(c*x^4+b*x^2+a)^3,x)