Integrand size = 15, antiderivative size = 62 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\frac {2}{3} b \sqrt {a+b x^3}-\frac {a \sqrt {a+b x^3}}{3 x^3}-\sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \] Output:
2/3*b*(b*x^3+a)^(1/2)-1/3*a*(b*x^3+a)^(1/2)/x^3-a^(1/2)*b*arctanh((b*x^3+a )^(1/2)/a^(1/2))
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {a+b x^3} \left (-a+2 b x^3\right )}{3 x^3}-\sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \] Input:
Integrate[(a + b*x^3)^(3/2)/x^4,x]
Output:
(Sqrt[a + b*x^3]*(-a + 2*b*x^3))/(3*x^3) - Sqrt[a]*b*ArcTanh[Sqrt[a + b*x^ 3]/Sqrt[a]]
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 51, 60, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\left (b x^3+a\right )^{3/2}}{x^6}dx^3\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} b \int \frac {\sqrt {b x^3+a}}{x^3}dx^3-\frac {\left (a+b x^3\right )^{3/2}}{x^3}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} b \left (a \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3+2 \sqrt {a+b x^3}\right )-\frac {\left (a+b x^3\right )^{3/2}}{x^3}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} b \left (\frac {2 a \int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{b}+2 \sqrt {a+b x^3}\right )-\frac {\left (a+b x^3\right )^{3/2}}{x^3}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} b \left (2 \sqrt {a+b x^3}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\right )-\frac {\left (a+b x^3\right )^{3/2}}{x^3}\right )\) |
Input:
Int[(a + b*x^3)^(3/2)/x^4,x]
Output:
(-((a + b*x^3)^(3/2)/x^3) + (3*b*(2*Sqrt[a + b*x^3] - 2*Sqrt[a]*ArcTanh[Sq rt[a + b*x^3]/Sqrt[a]]))/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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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 = 0.52 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {2 b \sqrt {b \,x^{3}+a}}{3}-\frac {a \sqrt {b \,x^{3}+a}}{3 x^{3}}-\sqrt {a}\, b \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )\) | \(49\) |
elliptic | \(\frac {2 b \sqrt {b \,x^{3}+a}}{3}-\frac {a \sqrt {b \,x^{3}+a}}{3 x^{3}}-\sqrt {a}\, b \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )\) | \(49\) |
risch | \(-\frac {a \sqrt {b \,x^{3}+a}}{3 x^{3}}+\frac {b \left (\frac {4 \sqrt {b \,x^{3}+a}}{3}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )\right )}{2}\) | \(51\) |
pseudoelliptic | \(-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) a b \,x^{3}+\sqrt {a}\, \left (-2 b \,x^{3}+a \right ) \sqrt {b \,x^{3}+a}}{3 \sqrt {a}\, x^{3}}\) | \(52\) |
Input:
int((b*x^3+a)^(3/2)/x^4,x,method=_RETURNVERBOSE)
Output:
2/3*b*(b*x^3+a)^(1/2)-1/3*a*(b*x^3+a)^(1/2)/x^3-a^(1/2)*b*arctanh((b*x^3+a )^(1/2)/a^(1/2))
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\left [\frac {3 \, \sqrt {a} b x^{3} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (2 \, b x^{3} - a\right )} \sqrt {b x^{3} + a}}{6 \, x^{3}}, \frac {3 \, \sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{3} + a}}\right ) + {\left (2 \, b x^{3} - a\right )} \sqrt {b x^{3} + a}}{3 \, x^{3}}\right ] \] Input:
integrate((b*x^3+a)^(3/2)/x^4,x, algorithm="fricas")
Output:
[1/6*(3*sqrt(a)*b*x^3*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2*(2*b*x^3 - a)*sqrt(b*x^3 + a))/x^3, 1/3*(3*sqrt(-a)*b*x^3*arctan(sqrt(- a)/sqrt(b*x^3 + a)) + (2*b*x^3 - a)*sqrt(b*x^3 + a))/x^3]
Time = 1.51 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=- \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} - \frac {a^{2}}{3 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {a \sqrt {b}}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} \] Input:
integrate((b*x**3+a)**(3/2)/x**4,x)
Output:
-sqrt(a)*b*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) - a**2/(3*sqrt(b)*x**(9/2)*sq rt(a/(b*x**3) + 1)) + a*sqrt(b)/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*b**( 3/2)*x**(3/2)/(3*sqrt(a/(b*x**3) + 1))
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\frac {1}{2} \, \sqrt {a} b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {b x^{3} + a} b - \frac {\sqrt {b x^{3} + a} a}{3 \, x^{3}} \] Input:
integrate((b*x^3+a)^(3/2)/x^4,x, algorithm="maxima")
Output:
1/2*sqrt(a)*b*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a))) + 2/3*sqrt(b*x^3 + a)*b - 1/3*sqrt(b*x^3 + a)*a/x^3
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\frac {1}{3} \, {\left (\frac {3 \, a \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {b x^{3} + a} - \frac {\sqrt {b x^{3} + a} a}{b x^{3}}\right )} b \] Input:
integrate((b*x^3+a)^(3/2)/x^4,x, algorithm="giac")
Output:
1/3*(3*a*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + 2*sqrt(b*x^3 + a) - s qrt(b*x^3 + a)*a/(b*x^3))*b
Time = 0.51 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\frac {2\,b\,\sqrt {b\,x^3+a}}{3}-\frac {a\,\sqrt {b\,x^3+a}}{3\,x^3}+\frac {\sqrt {a}\,b\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{2} \] Input:
int((a + b*x^3)^(3/2)/x^4,x)
Output:
(2*b*(a + b*x^3)^(1/2))/3 - (a*(a + b*x^3)^(1/2))/(3*x^3) + (a^(1/2)*b*log ((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x^6))/2
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^3\right )^{3/2}}{x^4} \, dx=\frac {-2 \sqrt {b \,x^{3}+a}\, a +4 \sqrt {b \,x^{3}+a}\, b \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b \,x^{3}}{6 x^{3}} \] Input:
int((b*x^3+a)^(3/2)/x^4,x)
Output:
( - 2*sqrt(a + b*x**3)*a + 4*sqrt(a + b*x**3)*b*x**3 + 3*sqrt(a)*log(sqrt( a + b*x**3) - sqrt(a))*b*x**3 - 3*sqrt(a)*log(sqrt(a + b*x**3) + sqrt(a))* b*x**3)/(6*x**3)