Integrand size = 26, antiderivative size = 165 \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {a+b x^3}{6 a x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right )}{3 a^2 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^2 \left (a+b x^3\right ) \log (x)}{a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^2 \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \] Output:
-1/6*(b*x^3+a)/a/x^6/((b*x^3+a)^2)^(1/2)+1/3*b*(b*x^3+a)/a^2/x^3/((b*x^3+a )^2)^(1/2)+b^2*(b*x^3+a)*ln(x)/a^3/((b*x^3+a)^2)^(1/2)-1/3*b^2*(b*x^3+a)*l n(b*x^3+a)/a^3/((b*x^3+a)^2)^(1/2)
Time = 0.59 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {\frac {a^3 \left (a-2 b x^3\right )}{\sqrt {a^2} x^6}-\frac {a \left (a-3 b x^3\right ) \sqrt {\left (a+b x^3\right )^2}}{x^6}-4 \sqrt {a^2} b^2 \log \left (x^3\right )+2 \left (-a+\sqrt {a^2}\right ) b^2 \log \left (\sqrt {a^2}-b x^3-\sqrt {\left (a+b x^3\right )^2}\right )+2 \left (a+\sqrt {a^2}\right ) b^2 \log \left (\sqrt {a^2}+b x^3-\sqrt {\left (a+b x^3\right )^2}\right )}{12 a^4} \] Input:
Integrate[1/(x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
Output:
((a^3*(a - 2*b*x^3))/(Sqrt[a^2]*x^6) - (a*(a - 3*b*x^3)*Sqrt[(a + b*x^3)^2 ])/x^6 - 4*Sqrt[a^2]*b^2*Log[x^3] + 2*(-a + Sqrt[a^2])*b^2*Log[Sqrt[a^2] - b*x^3 - Sqrt[(a + b*x^3)^2]] + 2*(a + Sqrt[a^2])*b^2*Log[Sqrt[a^2] + b*x^ 3 - Sqrt[(a + b*x^3)^2]])/(12*a^4)
Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1384, 27, 798, 54, 2009}
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.
\(\displaystyle \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {b \left (a+b x^3\right ) \int \frac {1}{b x^7 \left (b x^3+a\right )}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {1}{x^7 \left (b x^3+a\right )}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {1}{x^9 \left (b x^3+a\right )}dx^3}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\left (a+b x^3\right ) \int \left (-\frac {b^3}{a^3 \left (b x^3+a\right )}+\frac {b^2}{a^3 x^3}-\frac {b}{a^2 x^6}+\frac {1}{a x^9}\right )dx^3}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {b^2 \log \left (x^3\right )}{a^3}-\frac {b^2 \log \left (a+b x^3\right )}{a^3}+\frac {b}{a^2 x^3}-\frac {1}{2 a x^6}\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
Input:
Int[1/(x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]
Output:
((a + b*x^3)*(-1/2*1/(a*x^6) + b/(a^2*x^3) + (b^2*Log[x^3])/a^3 - (b^2*Log [a + b*x^3])/a^3))/(3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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 = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.35
method | result | size |
pseudoelliptic | \(-\frac {\left (2 b^{2} \ln \left (b \,x^{3}+a \right ) x^{6}-2 \ln \left (b \,x^{3}\right ) b^{2} x^{6}-2 a \,x^{3} b +a^{2}\right ) \operatorname {csgn}\left (b \,x^{3}+a \right )}{6 a^{3} x^{6}}\) | \(58\) |
default | \(-\frac {\left (b \,x^{3}+a \right ) \left (2 b^{2} \ln \left (b \,x^{3}+a \right ) x^{6}-6 b^{2} \ln \left (x \right ) x^{6}-2 a \,x^{3} b +a^{2}\right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{3} x^{6}}\) | \(64\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {b \,x^{3}}{3 a^{2}}-\frac {1}{6 a}\right )}{\left (b \,x^{3}+a \right ) x^{6}}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{2} \ln \left (x \right )}{\left (b \,x^{3}+a \right ) a^{3}}-\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{2} \ln \left (b \,x^{3}+a \right )}{3 \left (b \,x^{3}+a \right ) a^{3}}\) | \(106\) |
Input:
int(1/x^7/((b*x^3+a)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(2*b^2*ln(b*x^3+a)*x^6-2*ln(b*x^3)*b^2*x^6-2*a*x^3*b+a^2)*csgn(b*x^3+ a)/a^3/x^6
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {2 \, b^{2} x^{6} \log \left (b x^{3} + a\right ) - 6 \, b^{2} x^{6} \log \left (x\right ) - 2 \, a b x^{3} + a^{2}}{6 \, a^{3} x^{6}} \] Input:
integrate(1/x^7/((b*x^3+a)^2)^(1/2),x, algorithm="fricas")
Output:
-1/6*(2*b^2*x^6*log(b*x^3 + a) - 6*b^2*x^6*log(x) - 2*a*b*x^3 + a^2)/(a^3* x^6)
\[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\int \frac {1}{x^{7} \sqrt {\left (a + b x^{3}\right )^{2}}}\, dx \] Input:
integrate(1/x**7/((b*x**3+a)**2)**(1/2),x)
Output:
Integral(1/(x**7*sqrt((a + b*x**3)**2)), x)
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {\left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right )}{3 \, a^{3}} + \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b}{2 \, a^{3} x^{3}} - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}{6 \, a^{2} x^{6}} \] Input:
integrate(1/x^7/((b*x^3+a)^2)^(1/2),x, algorithm="maxima")
Output:
-1/3*(-1)^(2*a*b*x^3 + 2*a^2)*b^2*log(2*a*b*x/abs(x) + 2*a^2/(x^2*abs(x))) /a^3 + 1/2*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b/(a^3*x^3) - 1/6*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)/(a^2*x^6)
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=-\frac {1}{6} \, {\left (\frac {2 \, b^{2} \log \left ({\left | b x^{3} + a \right |}\right )}{a^{3}} - \frac {6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {3 \, b^{2} x^{6} - 2 \, a b x^{3} + a^{2}}{a^{3} x^{6}}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \] Input:
integrate(1/x^7/((b*x^3+a)^2)^(1/2),x, algorithm="giac")
Output:
-1/6*(2*b^2*log(abs(b*x^3 + a))/a^3 - 6*b^2*log(abs(x))/a^3 + (3*b^2*x^6 - 2*a*b*x^3 + a^2)/(a^3*x^6))*sgn(b*x^3 + a)
Timed out. \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\int \frac {1}{x^7\,\sqrt {{\left (b\,x^3+a\right )}^2}} \,d x \] Input:
int(1/(x^7*((a + b*x^3)^2)^(1/2)),x)
Output:
int(1/(x^7*((a + b*x^3)^2)^(1/2)), x)
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx=\frac {-2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{6}-2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{6}+6 \,\mathrm {log}\left (x \right ) b^{2} x^{6}-a^{2}+2 a b \,x^{3}}{6 a^{3} x^{6}} \] Input:
int(1/x^7/((b*x^3+a)^2)^(1/2),x)
Output:
( - 2*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**2*x**6 - 2*lo g(a**(1/3) + b**(1/3)*x)*b**2*x**6 + 6*log(x)*b**2*x**6 - a**2 + 2*a*b*x** 3)/(6*a**3*x**6)