Integrand size = 15, antiderivative size = 135 \[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=\left (-\frac {1}{2 a x^2}-\frac {b}{4 a^2 x}\right ) \sqrt {(a+b x)^5}+\frac {3 b^2 \left (\frac {b \sqrt {a+b x}}{4 a x}-\frac {\sqrt {(a+b x)^3}}{2 a x^2}-\frac {b^2 \log \left (\frac {-\sqrt {a}+\sqrt {a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )}{8 a^{3/2}}\right )}{8 a^2} \]
-(1/2/a/x^2+1/4*b/a^2/x)*((b*x+a)^5)^(1/2)+3/8*b^2/a^2*(-1/2*((b*x+a)^3)^( 1/2)/a/x^2+1/4*b*(b*x+a)^(1/2)/a/x-1/8*b^2/a^(3/2)*ln(((b*x+a)^(1/2)-a^(1/ 2))/((b*x+a)^(1/2)+a^(1/2))))
Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=-\frac {\sqrt {(a+b x)^3} \left (\sqrt {a} \sqrt {a+b x} (2 a+5 b x)+3 b^2 x^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{4 \sqrt {a} x^2 (a+b x)^{3/2}} \]
-1/4*(Sqrt[(a + b*x)^3]*(Sqrt[a]*Sqrt[a + b*x]*(2*a + 5*b*x) + 3*b^2*x^2*A rcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(Sqrt[a]*x^2*(a + b*x)^(3/2))
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.61, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2008, 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.
\(\displaystyle \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx\) |
\(\Big \downarrow \) 2008 |
\(\displaystyle \frac {\sqrt {(a+b x)^3} \int \frac {(a+b x)^{3/2}}{x^3}dx}{(a+b x)^{3/2}}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\sqrt {(a+b x)^3} \left (\frac {3}{4} b \int \frac {\sqrt {a+b x}}{x^2}dx-\frac {(a+b x)^{3/2}}{2 x^2}\right )}{(a+b x)^{3/2}}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\sqrt {(a+b x)^3} \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{x}\right )-\frac {(a+b x)^{3/2}}{2 x^2}\right )}{(a+b x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {(a+b x)^3} \left (\frac {3}{4} b \left (\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}-\frac {\sqrt {a+b x}}{x}\right )-\frac {(a+b x)^{3/2}}{2 x^2}\right )}{(a+b x)^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {(a+b x)^3} \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x}}{x}\right )-\frac {(a+b x)^{3/2}}{2 x^2}\right )}{(a+b x)^{3/2}}\) |
(Sqrt[(a + b*x)^3]*(-1/2*(a + b*x)^(3/2)/x^2 + (3*b*(-(Sqrt[a + b*x]/x) - (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a]))/4))/(a + b*x)^(3/2)
3.2.45.3.1 Defintions of rubi rules used
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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Simp[((a + b*x)^Exp on[Px, x])^p/(a + b*x)^(Expon[Px, x]*p) Int[u*(a + b*x)^(Expon[Px, x]*p), x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; !IntegerQ[p] && PolyQ[Px, x ] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {\left (5 b x +2 a \right ) \sqrt {\left (b x +a \right )^{3}}}{4 \left (b x +a \right ) x^{2}}-\frac {3 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {\left (b x +a \right )^{3}}}{4 \sqrt {a}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(67\) |
default | \(-\frac {\sqrt {\left (b x +a \right )^{3}}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{2} x^{2}+5 \left (b x +a \right )^{\frac {3}{2}} \sqrt {a}-3 \sqrt {b x +a}\, a^{\frac {3}{2}}\right )}{4 \left (b x +a \right )^{\frac {3}{2}} x^{2} \sqrt {a}}\) | \(70\) |
-1/4/(b*x+a)*(5*b*x+2*a)/x^2*((b*x+a)^3)^(1/2)-3/4*b^2/a^(1/2)*arctanh((b* x+a)^(1/2)/a^(1/2))*((b*x+a)^3)^(1/2)/(b*x+a)^(3/2)
Time = 0.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=\left [\frac {3 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt {a} \log \left (\frac {b^{2} x^{2} + 3 \, a b x + 2 \, a^{2} - 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {a}}{b x^{2} + a x}\right ) - 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (5 \, a b x + 2 \, a^{2}\right )}}{8 \, {\left (a b x^{3} + a^{2} x^{2}\right )}}, \frac {3 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {-a}}{a b x + a^{2}}\right ) - \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (5 \, a b x + 2 \, a^{2}\right )}}{4 \, {\left (a b x^{3} + a^{2} x^{2}\right )}}\right ] \]
[1/8*(3*(b^3*x^3 + a*b^2*x^2)*sqrt(a)*log((b^2*x^2 + 3*a*b*x + 2*a^2 - 2*s qrt(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(a))/(b*x^2 + a*x)) - 2*s qrt(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*(5*a*b*x + 2*a^2))/(a*b*x^3 + a^2*x^2), 1/4*(3*(b^3*x^3 + a*b^2*x^2)*sqrt(-a)*arctan(sqrt(b^3*x^3 + 3*a *b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(-a)/(a*b*x + a^2)) - sqrt(b^3*x^3 + 3*a*b ^2*x^2 + 3*a^2*b*x + a^3)*(5*a*b*x + 2*a^2))/(a*b*x^3 + a^2*x^2)]
\[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=\int \frac {\sqrt {\left (a + b x\right )^{3}}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=\int { \frac {\sqrt {{\left (b x + a\right )}^{3}}}{x^{3}} \,d x } \]
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=\frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} - 3 \, \sqrt {b x + a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \]
1/4*(3*b^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - (5*(b*x + a)^(3/2)*b^ 3 - 3*sqrt(b*x + a)*a*b^3)/(b^2*x^2))/b
Timed out. \[ \int \frac {\sqrt {(a+b x)^3}}{x^3} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^3}}{x^3} \,d x \]