Integrand size = 15, antiderivative size = 69 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=-\frac {a \sqrt {a+b x^3}}{6 x^6}-\frac {5 b \sqrt {a+b x^3}}{12 x^3}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}} \] Output:
-1/6*a*(b*x^3+a)^(1/2)/x^6-5/12*b*(b*x^3+a)^(1/2)/x^3-1/4*b^2*arctanh((b*x ^3+a)^(1/2)/a^(1/2))/a^(1/2)
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=\frac {\left (-2 a-5 b x^3\right ) \sqrt {a+b x^3}}{12 x^6}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}} \] Input:
Integrate[(a + b*x^3)^(3/2)/x^7,x]
Output:
((-2*a - 5*b*x^3)*Sqrt[a + b*x^3])/(12*x^6) - (b^2*ArcTanh[Sqrt[a + b*x^3] /Sqrt[a]])/(4*Sqrt[a])
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 51, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^{3/2}}{x^9}dx^3\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} b \int \frac {\sqrt {b x^3+a}}{x^6}dx^3-\frac {\left (a+b x^3\right )^{3/2}}{2 x^6}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3-\frac {\sqrt {a+b x^3}}{x^3}\right )-\frac {\left (a+b x^3\right )^{3/2}}{2 x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}-\frac {\sqrt {a+b x^3}}{x^3}\right )-\frac {\left (a+b x^3\right )^{3/2}}{2 x^6}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^3}}{x^3}\right )-\frac {\left (a+b x^3\right )^{3/2}}{2 x^6}\right )\) |
Input:
Int[(a + b*x^3)^(3/2)/x^7,x]
Output:
(-1/2*(a + b*x^3)^(3/2)/x^6 + (3*b*(-(Sqrt[a + b*x^3]/x^3) - (b*ArcTanh[Sq rt[a + b*x^3]/Sqrt[a]])/Sqrt[a]))/4)/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_)^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_), 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]]
Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (5 b \,x^{3}+2 a \right )}{12 x^{6}}-\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}\) | \(48\) |
default | \(-\frac {a \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {5 b \sqrt {b \,x^{3}+a}}{12 x^{3}}-\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}\) | \(54\) |
elliptic | \(-\frac {a \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {5 b \sqrt {b \,x^{3}+a}}{12 x^{3}}-\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}\) | \(54\) |
pseudoelliptic | \(\frac {-3 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) b^{2} x^{6}-5 b \,x^{3} \sqrt {b \,x^{3}+a}\, \sqrt {a}-2 a^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}{12 x^{6} \sqrt {a}}\) | \(64\) |
Input:
int((b*x^3+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-1/12*(b*x^3+a)^(1/2)*(5*b*x^3+2*a)/x^6-1/4*b^2*arctanh((b*x^3+a)^(1/2)/a^ (1/2))/a^(1/2)
Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} x^{6} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, {\left (5 \, a b x^{3} + 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{24 \, a x^{6}}, \frac {3 \, \sqrt {-a} b^{2} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{3} + a}}\right ) - {\left (5 \, a b x^{3} + 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{12 \, a x^{6}}\right ] \] Input:
integrate((b*x^3+a)^(3/2)/x^7,x, algorithm="fricas")
Output:
[1/24*(3*sqrt(a)*b^2*x^6*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3 ) - 2*(5*a*b*x^3 + 2*a^2)*sqrt(b*x^3 + a))/(a*x^6), 1/12*(3*sqrt(-a)*b^2*x ^6*arctan(sqrt(-a)/sqrt(b*x^3 + a)) - (5*a*b*x^3 + 2*a^2)*sqrt(b*x^3 + a)) /(a*x^6)]
Time = 1.71 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=- \frac {a \sqrt {b} \sqrt {\frac {a}{b x^{3}} + 1}}{6 x^{\frac {9}{2}}} - \frac {5 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}}{12 x^{\frac {3}{2}}} - \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} \] Input:
integrate((b*x**3+a)**(3/2)/x**7,x)
Output:
-a*sqrt(b)*sqrt(a/(b*x**3) + 1)/(6*x**(9/2)) - 5*b**(3/2)*sqrt(a/(b*x**3) + 1)/(12*x**(3/2)) - b**2*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*sqrt(a))
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=\frac {b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{8 \, \sqrt {a}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {b x^{3} + a} a b^{2}}{12 \, {\left ({\left (b x^{3} + a\right )}^{2} - 2 \, {\left (b x^{3} + a\right )} a + a^{2}\right )}} \] Input:
integrate((b*x^3+a)^(3/2)/x^7,x, algorithm="maxima")
Output:
1/8*b^2*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/sqrt( a) - 1/12*(5*(b*x^3 + a)^(3/2)*b^2 - 3*sqrt(b*x^3 + a)*a*b^2)/((b*x^3 + a) ^2 - 2*(b*x^3 + a)*a + a^2)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=\frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{3} - 3 \, \sqrt {b x^{3} + a} a b^{3}}{b^{2} x^{6}}}{12 \, b} \] Input:
integrate((b*x^3+a)^(3/2)/x^7,x, algorithm="giac")
Output:
1/12*(3*b^3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) - (5*(b*x^3 + a)^(3/ 2)*b^3 - 3*sqrt(b*x^3 + a)*a*b^3)/(b^2*x^6))/b
Time = 0.63 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=\frac {b^2\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{8\,\sqrt {a}}-\frac {5\,b\,\sqrt {b\,x^3+a}}{12\,x^3}-\frac {a\,\sqrt {b\,x^3+a}}{6\,x^6} \] Input:
int((a + b*x^3)^(3/2)/x^7,x)
Output:
(b^2*log((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x ^6))/(8*a^(1/2)) - (5*b*(a + b*x^3)^(1/2))/(12*x^3) - (a*(a + b*x^3)^(1/2) )/(6*x^6)
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx=\frac {-4 \sqrt {b \,x^{3}+a}\, a^{2}-10 \sqrt {b \,x^{3}+a}\, a b \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b^{2} x^{6}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b^{2} x^{6}}{24 a \,x^{6}} \] Input:
int((b*x^3+a)^(3/2)/x^7,x)
Output:
( - 4*sqrt(a + b*x**3)*a**2 - 10*sqrt(a + b*x**3)*a*b*x**3 + 3*sqrt(a)*log (sqrt(a + b*x**3) - sqrt(a))*b**2*x**6 - 3*sqrt(a)*log(sqrt(a + b*x**3) + sqrt(a))*b**2*x**6)/(24*a*x**6)