Integrand size = 24, antiderivative size = 59 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\frac {\left (-2 b-3 a x^3\right ) \sqrt {b+a x^3}}{6 x^6}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{6 \sqrt {b}} \] Output:
1/6*(-3*a*x^3-2*b)*(a*x^3+b)^(1/2)/x^6-1/6*a^2*arctanh((a*x^3+b)^(1/2)/b^( 1/2))/b^(1/2)
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\frac {\left (-2 b-3 a x^3\right ) \sqrt {b+a x^3}}{6 x^6}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{6 \sqrt {b}} \] Input:
Integrate[(Sqrt[b + a*x^3]*(2*b + a*x^3))/x^7,x]
Output:
((-2*b - 3*a*x^3)*Sqrt[b + a*x^3])/(6*x^6) - (a^2*ArcTanh[Sqrt[b + a*x^3]/ Sqrt[b]])/(6*Sqrt[b])
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {948, 87, 51, 73, 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.
\(\displaystyle \int \frac {\sqrt {a x^3+b} \left (a x^3+2 b\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {a x^3+b} \left (a x^3+2 b\right )}{x^9}dx^3\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} a \int \frac {\sqrt {a x^3+b}}{x^6}dx^3-\frac {\left (a x^3+b\right )^{3/2}}{x^6}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} a \left (\frac {1}{2} a \int \frac {1}{x^3 \sqrt {a x^3+b}}dx^3-\frac {\sqrt {a x^3+b}}{x^3}\right )-\frac {\left (a x^3+b\right )^{3/2}}{x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} a \left (\int \frac {1}{\frac {x^6}{a}-\frac {b}{a}}d\sqrt {a x^3+b}-\frac {\sqrt {a x^3+b}}{x^3}\right )-\frac {\left (a x^3+b\right )^{3/2}}{x^6}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} a \left (-\frac {a \text {arctanh}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {\sqrt {a x^3+b}}{x^3}\right )-\frac {\left (a x^3+b\right )^{3/2}}{x^6}\right )\) |
Input:
Int[(Sqrt[b + a*x^3]*(2*b + a*x^3))/x^7,x]
Output:
(-((b + a*x^3)^(3/2)/x^6) + (a*(-(Sqrt[b + a*x^3]/x^3) - (a*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/Sqrt[b]))/2)/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\sqrt {a \,x^{3}+b}\, \left (3 a \,x^{3}+2 b \right )}{6 x^{6}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{6 \sqrt {b}}\) | \(48\) |
elliptic | \(-\frac {b \sqrt {a \,x^{3}+b}}{3 x^{6}}-\frac {a \sqrt {a \,x^{3}+b}}{2 x^{3}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{6 \sqrt {b}}\) | \(54\) |
pseudoelliptic | \(\frac {-\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right ) a^{2} x^{6}-3 \sqrt {a \,x^{3}+b}\, a \,x^{3} \sqrt {b}-2 b^{\frac {3}{2}} \sqrt {a \,x^{3}+b}}{6 x^{6} \sqrt {b}}\) | \(64\) |
default | \(a \left (-\frac {\sqrt {a \,x^{3}+b}}{3 x^{3}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\right )+2 b \left (-\frac {\sqrt {a \,x^{3}+b}}{6 x^{6}}-\frac {a \sqrt {a \,x^{3}+b}}{12 b \,x^{3}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{12 b^{\frac {3}{2}}}\right )\) | \(97\) |
Input:
int((a*x^3+b)^(1/2)*(a*x^3+2*b)/x^7,x,method=_RETURNVERBOSE)
Output:
-1/6*(a*x^3+b)^(1/2)*(3*a*x^3+2*b)/x^6-1/6*a^2*arctanh((a*x^3+b)^(1/2)/b^( 1/2))/b^(1/2)
Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\left [\frac {a^{2} \sqrt {b} x^{6} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right ) - 2 \, {\left (3 \, a b x^{3} + 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{12 \, b x^{6}}, \frac {a^{2} \sqrt {-b} x^{6} \arctan \left (\frac {\sqrt {-b}}{\sqrt {a x^{3} + b}}\right ) - {\left (3 \, a b x^{3} + 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{6 \, b x^{6}}\right ] \] Input:
integrate((a*x^3+b)^(1/2)*(a*x^3+2*b)/x^7,x, algorithm="fricas")
Output:
[1/12*(a^2*sqrt(b)*x^6*log((a*x^3 - 2*sqrt(a*x^3 + b)*sqrt(b) + 2*b)/x^3) - 2*(3*a*b*x^3 + 2*b^2)*sqrt(a*x^3 + b))/(b*x^6), 1/6*(a^2*sqrt(-b)*x^6*ar ctan(sqrt(-b)/sqrt(a*x^3 + b)) - (3*a*b*x^3 + 2*b^2)*sqrt(a*x^3 + b))/(b*x ^6)]
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (51) = 102\).
Time = 31.72 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.17 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=- \frac {a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} - \frac {a^{\frac {3}{2}}}{6 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {\sqrt {a} b}{2 x^{\frac {9}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{6 \sqrt {b}} - \frac {b^{2}}{3 \sqrt {a} x^{\frac {15}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} \] Input:
integrate((a*x**3+b)**(1/2)*(a*x**3+2*b)/x**7,x)
Output:
-a**(3/2)*sqrt(1 + b/(a*x**3))/(3*x**(3/2)) - a**(3/2)/(6*x**(3/2)*sqrt(1 + b/(a*x**3))) - sqrt(a)*b/(2*x**(9/2)*sqrt(1 + b/(a*x**3))) - a**2*asinh( sqrt(b)/(sqrt(a)*x**(3/2)))/(6*sqrt(b)) - b**2/(3*sqrt(a)*x**(15/2)*sqrt(1 + b/(a*x**3)))
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (47) = 94\).
Time = 0.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.68 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\frac {1}{6} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} a - \frac {1}{12} \, {\left (\frac {a^{2} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (a x^{3} + b\right )}^{\frac {3}{2}} a^{2} + \sqrt {a x^{3} + b} a^{2} b\right )}}{{\left (a x^{3} + b\right )}^{2} b - 2 \, {\left (a x^{3} + b\right )} b^{2} + b^{3}}\right )} b \] Input:
integrate((a*x^3+b)^(1/2)*(a*x^3+2*b)/x^7,x, algorithm="maxima")
Output:
1/6*(a*log((sqrt(a*x^3 + b) - sqrt(b))/(sqrt(a*x^3 + b) + sqrt(b)))/sqrt(b ) - 2*sqrt(a*x^3 + b)/x^3)*a - 1/12*(a^2*log((sqrt(a*x^3 + b) - sqrt(b))/( sqrt(a*x^3 + b) + sqrt(b)))/b^(3/2) + 2*((a*x^3 + b)^(3/2)*a^2 + sqrt(a*x^ 3 + b)*a^2*b)/((a*x^3 + b)^2*b - 2*(a*x^3 + b)*b^2 + b^3))*b
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\frac {\frac {a^{3} \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {3 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} a^{3} - \sqrt {a x^{3} + b} a^{3} b}{a^{2} x^{6}}}{6 \, a} \] Input:
integrate((a*x^3+b)^(1/2)*(a*x^3+2*b)/x^7,x, algorithm="giac")
Output:
1/6*(a^3*arctan(sqrt(a*x^3 + b)/sqrt(-b))/sqrt(-b) - (3*(a*x^3 + b)^(3/2)* a^3 - sqrt(a*x^3 + b)*a^3*b)/(a^2*x^6))/a
Time = 8.44 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\frac {a^2\,\ln \left (\frac {{\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )}^3\,\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}{x^6}\right )}{12\,\sqrt {b}}-\frac {b\,\sqrt {a\,x^3+b}}{3\,x^6}-\frac {a\,\sqrt {a\,x^3+b}}{2\,x^3} \] Input:
int(((b + a*x^3)^(1/2)*(2*b + a*x^3))/x^7,x)
Output:
(a^2*log((((b + a*x^3)^(1/2) - b^(1/2))^3*((b + a*x^3)^(1/2) + b^(1/2)))/x ^6))/(12*b^(1/2)) - (b*(b + a*x^3)^(1/2))/(3*x^6) - (a*(b + a*x^3)^(1/2))/ (2*x^3)
Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx=\frac {-6 \sqrt {a \,x^{3}+b}\, a b \,x^{3}-4 \sqrt {a \,x^{3}+b}\, b^{2}+\sqrt {b}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}-\sqrt {b}\right ) a^{2} x^{6}-\sqrt {b}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}+\sqrt {b}\right ) a^{2} x^{6}}{12 b \,x^{6}} \] Input:
int((a*x^3+b)^(1/2)*(a*x^3+2*b)/x^7,x)
Output:
( - 6*sqrt(a*x**3 + b)*a*b*x**3 - 4*sqrt(a*x**3 + b)*b**2 + sqrt(b)*log(sq rt(a*x**3 + b) - sqrt(b))*a**2*x**6 - sqrt(b)*log(sqrt(a*x**3 + b) + sqrt( b))*a**2*x**6)/(12*b*x**6)