Integrand size = 26, antiderivative size = 365 \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {5 x^2}{243 a^2 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^5}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 x^2}{108 b^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{324 a b^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{7/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{7/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{7/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \] Output:
5/243*x^2/a^2/b^2/((b*x^3+a)^2)^(1/2)-1/12*x^5/b/(b*x^3+a)^3/((b*x^3+a)^2) ^(1/2)-5/108*x^2/b^2/(b*x^3+a)^2/((b*x^3+a)^2)^(1/2)+5/324*x^2/a/b^2/(b*x^ 3+a)/((b*x^3+a)^2)^(1/2)-5/729*(b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)* 3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)/b^(8/3)/((b*x^3+a)^2)^(1/2)-5/729*(b*x^3+ a)*ln(a^(1/3)+b^(1/3)*x)/a^(7/3)/b^(8/3)/((b*x^3+a)^2)^(1/2)+5/1458*(b*x^3 +a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(7/3)/b^(8/3)/((b*x^3+a)^2 )^(1/2)
Time = 1.16 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.62 \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (243 a b^{2/3} x^2-378 b^{2/3} x^2 \left (a+b x^3\right )+\frac {45 b^{2/3} x^2 \left (a+b x^3\right )^2}{a}+\frac {60 b^{2/3} x^2 \left (a+b x^3\right )^3}{a^2}+\frac {20 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{7/3}}-\frac {20 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3}}+\frac {10 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3}}\right )}{2916 b^{8/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \] Input:
Integrate[x^7/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
Output:
((a + b*x^3)*(243*a*b^(2/3)*x^2 - 378*b^(2/3)*x^2*(a + b*x^3) + (45*b^(2/3 )*x^2*(a + b*x^3)^2)/a + (60*b^(2/3)*x^2*(a + b*x^3)^3)/a^2 + (20*Sqrt[3]* (a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(7/3) - (20*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x])/a^(7/3) + (10*(a + b*x^3)^4* Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(7/3)))/(2916*b^(8/3)*(( a + b*x^3)^2)^(5/2))
Time = 0.45 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.70, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1384, 27, 817, 817, 819, 819, 821, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \int \frac {x^7}{b^5 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {x^7}{\left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \int \frac {x^4}{\left (b x^3+a\right )^4}dx}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \int \frac {x}{\left (b x^3+a\right )^3}dx}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \int \frac {x}{\left (b x^3+a\right )^2}dx}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\int \frac {x}{b x^3+a}dx}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 b}-\frac {x^2}{9 b \left (a+b x^3\right )^3}\right )}{12 b}-\frac {x^5}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
Input:
Int[x^7/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
Output:
((a + b*x^3)*(-1/12*x^5/(b*(a + b*x^3)^4) + (5*(-1/9*x^2/(b*(a + b*x^3)^3) + (2*(x^2/(6*a*(a + b*x^3)^2) + (2*(x^2/(3*a*(a + b*x^3)) + (-1/3*Log[a^( 1/3) + b^(1/3)*x]/(a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x )/a^(1/3))/Sqrt[3]])/b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)* x^2]/(2*b^(1/3)))/(3*a^(1/3)*b^(1/3)))/(3*a)))/(3*a)))/(9*b)))/(12*b)))/Sq rt[a^2 + 2*a*b*x^3 + b^2*x^6]
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.62 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.29
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {5 b \,x^{11}}{243 a^{2}}+\frac {25 x^{8}}{324 a}-\frac {x^{5}}{27 b}-\frac {5 a \,x^{2}}{486 b^{2}}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {5 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{729 \left (b \,x^{3}+a \right ) a^{2} b^{3}}\) | \(107\) |
| default | \(-\frac {\left (20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{4} x^{12}+20 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}-10 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{4} x^{12}-60 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4} x^{11}+80 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a \,b^{3} x^{9}+80 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}-40 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a \,b^{3} x^{9}-225 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{3} x^{8}+120 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} b^{2} x^{6}+120 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}-60 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} b^{2} x^{6}+108 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b^{2} x^{5}+80 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{3} b \,x^{3}+80 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}-40 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{3} b \,x^{3}+30 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b \,x^{2}+20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{4}+20 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}-10 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3} a^{2} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(521\) |
Input:
int(x^7/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
((b*x^3+a)^2)^(1/2)/(b*x^3+a)^5*(5/243/a^2*b*x^11+25/324/a*x^8-1/27/b*x^5- 5/486*a/b^2*x^2)+5/729*((b*x^3+a)^2)^(1/2)/(b*x^3+a)/a^2/b^3*sum(1/_R*ln(x -_R),_R=RootOf(_Z^3*b+a))
Time = 0.08 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.01 \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(x^7/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")
Output:
[1/2916*(60*a*b^5*x^11 + 225*a^2*b^4*x^8 - 108*a^3*b^3*x^5 - 30*a^4*b^2*x^ 2 + 30*sqrt(1/3)*(a*b^5*x^12 + 4*a^2*b^4*x^9 + 6*a^3*b^3*x^6 + 4*a^4*b^2*x ^3 + a^5*b)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b *x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*( -a*b^2)^(2/3)*x)/(b*x^3 + a)) + 10*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (- a*b^2)^(2/3)) - 20*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^3*b^8*x^12 + 4*a^4*b^7* x^9 + 6*a^5*b^6*x^6 + 4*a^6*b^5*x^3 + a^7*b^4), 1/2916*(60*a*b^5*x^11 + 22 5*a^2*b^4*x^8 - 108*a^3*b^3*x^5 - 30*a^4*b^2*x^2 + 60*sqrt(1/3)*(a*b^5*x^1 2 + 4*a^2*b^4*x^9 + 6*a^3*b^3*x^6 + 4*a^4*b^2*x^3 + a^5*b)*sqrt(-(-a*b^2)^ (1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a) /b) + 10*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a* b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 20*(b^4*x^ 12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(2/3)*log(b *x - (-a*b^2)^(1/3)))/(a^3*b^8*x^12 + 4*a^4*b^7*x^9 + 6*a^5*b^6*x^6 + 4*a^ 6*b^5*x^3 + a^7*b^4)]
\[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^{7}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**7/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
Output:
Integral(x**7/((a + b*x**3)**2)**(5/2), x)
Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.54 \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {20 \, b^{3} x^{11} + 75 \, a b^{2} x^{8} - 36 \, a^{2} b x^{5} - 10 \, a^{3} x^{2}}{972 \, {\left (a^{2} b^{6} x^{12} + 4 \, a^{3} b^{5} x^{9} + 6 \, a^{4} b^{4} x^{6} + 4 \, a^{5} b^{3} x^{3} + a^{6} b^{2}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x^7/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")
Output:
1/972*(20*b^3*x^11 + 75*a*b^2*x^8 - 36*a^2*b*x^5 - 10*a^3*x^2)/(a^2*b^6*x^ 12 + 4*a^3*b^5*x^9 + 6*a^4*b^4*x^6 + 4*a^5*b^3*x^3 + a^6*b^2) + 5/729*sqrt (3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^3*(a/b)^(1/ 3)) + 5/1458*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b^3*(a/b)^(1/3)) - 5/729*log(x + (a/b)^(1/3))/(a^2*b^3*(a/b)^(1/3))
Time = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.57 \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {5 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {5 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{3} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {20 \, b^{3} x^{11} + 75 \, a b^{2} x^{8} - 36 \, a^{2} b x^{5} - 10 \, a^{3} x^{2}}{972 \, {\left (b x^{3} + a\right )}^{4} a^{2} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} \] Input:
integrate(x^7/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")
Output:
5/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b ^2)^(1/3)*a^2*b^2*sgn(b*x^3 + a)) - 5/1458*log(x^2 + x*(-a/b)^(1/3) + (-a/ b)^(2/3))/((-a*b^2)^(1/3)*a^2*b^2*sgn(b*x^3 + a)) - 5/729*(-a/b)^(2/3)*log (abs(x - (-a/b)^(1/3)))/(a^3*b^2*sgn(b*x^3 + a)) + 1/972*(20*b^3*x^11 + 75 *a*b^2*x^8 - 36*a^2*b*x^5 - 10*a^3*x^2)/((b*x^3 + a)^4*a^2*b^2*sgn(b*x^3 + a))
Timed out. \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^7}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \] Input:
int(x^7/(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2),x)
Output:
int(x^7/(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2), x)
Time = 0.18 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.32 \[ \int \frac {x^7}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {-20 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{4}-80 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{3} b \,x^{3}-120 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b^{2} x^{6}-80 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{3} x^{9}-20 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{4} x^{12}-30 b^{\frac {2}{3}} a^{\frac {10}{3}} x^{2}-108 b^{\frac {5}{3}} a^{\frac {7}{3}} x^{5}+225 b^{\frac {8}{3}} a^{\frac {4}{3}} x^{8}+60 b^{\frac {11}{3}} a^{\frac {1}{3}} x^{11}+10 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{4}+40 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} b \,x^{3}+60 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b^{2} x^{6}+40 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{3} x^{9}+10 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{4} x^{12}-20 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{4}-80 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} b \,x^{3}-120 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b^{2} x^{6}-80 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{3} x^{9}-20 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{4} x^{12}}{2916 b^{\frac {8}{3}} a^{\frac {7}{3}} \left (b^{4} x^{12}+4 a \,b^{3} x^{9}+6 a^{2} b^{2} x^{6}+4 a^{3} b \,x^{3}+a^{4}\right )} \] Input:
int(x^7/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)
Output:
( - 20*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4 - 8 0*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b*x**3 - 120*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2* x**6 - 80*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b** 3*x**9 - 20*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b** 4*x**12 - 30*b**(2/3)*a**(1/3)*a**3*x**2 - 108*b**(2/3)*a**(1/3)*a**2*b*x* *5 + 225*b**(2/3)*a**(1/3)*a*b**2*x**8 + 60*b**(2/3)*a**(1/3)*b**3*x**11 + 10*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4 + 40*log(a**( 2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*x**3 + 60*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**2*x**6 + 40*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**3*x**9 + 10*log(a**(2/3) - b**(1 /3)*a**(1/3)*x + b**(2/3)*x**2)*b**4*x**12 - 20*log(a**(1/3) + b**(1/3)*x) *a**4 - 80*log(a**(1/3) + b**(1/3)*x)*a**3*b*x**3 - 120*log(a**(1/3) + b** (1/3)*x)*a**2*b**2*x**6 - 80*log(a**(1/3) + b**(1/3)*x)*a*b**3*x**9 - 20*l og(a**(1/3) + b**(1/3)*x)*b**4*x**12)/(2916*b**(2/3)*a**(1/3)*a**2*b**2*(a **4 + 4*a**3*b*x**3 + 6*a**2*b**2*x**6 + 4*a*b**3*x**9 + b**4*x**12))