Optimal. Leaf size=218 \[ -\frac{3 a b^3 x^{-2 (1-n)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (1-n) \left (a b+b^2 x^n\right )}-\frac{3 a^2 b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{b^4 x^{3 n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-3 n) \left (a b+b^2 x^n\right )}-\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0698201, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1355, 270} \[ -\frac{3 a b^3 x^{-2 (1-n)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (1-n) \left (a b+b^2 x^n\right )}-\frac{3 a^2 b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{b^4 x^{3 n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-3 n) \left (a b+b^2 x^n\right )}-\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1355
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x^3} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \frac{\left (a b+b^2 x^n\right )^3}{x^3} \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \left (\frac{a^3 b^3}{x^3}+3 a^2 b^4 x^{-3+n}+b^6 x^{3 (-1+n)}+3 a b^5 x^{-3+2 n}\right ) \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )}-\frac{3 a b^3 x^{-2 (1-n)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (1-n) \left (a b+b^2 x^n\right )}-\frac{3 a^2 b^2 x^{-2+n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{b^4 x^{-2+3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-3 n) \left (a b+b^2 x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.102713, size = 124, normalized size = 0.57 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (6 a^2 b \left (3 n^2-5 n+2\right ) x^n+a^3 \left (-3 n^3+11 n^2-12 n+4\right )+3 a b^2 \left (3 n^2-8 n+4\right ) x^{2 n}+2 b^3 \left (n^2-3 n+2\right ) x^{3 n}\right )}{2 (n-2) (n-1) (3 n-2) x^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 145, normalized size = 0.7 \begin{align*} -{\frac{{a}^{3}}{ \left ( 2\,a+2\,b{x}^{n} \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3} \left ({x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) \left ( -2+3\,n \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{3\,a{b}^{2} \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) \left ( -1+n \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( -2+n \right ){x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01281, size = 136, normalized size = 0.62 \begin{align*} \frac{2 \,{\left (n^{2} - 3 \, n + 2\right )} b^{3} x^{3 \, n} + 3 \,{\left (3 \, n^{2} - 8 \, n + 4\right )} a b^{2} x^{2 \, n} + 6 \,{\left (3 \, n^{2} - 5 \, n + 2\right )} a^{2} b x^{n} -{\left (3 \, n^{3} - 11 \, n^{2} + 12 \, n - 4\right )} a^{3}}{2 \,{\left (3 \, n^{3} - 11 \, n^{2} + 12 \, n - 4\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60612, size = 292, normalized size = 1.34 \begin{align*} -\frac{3 \, a^{3} n^{3} - 11 \, a^{3} n^{2} + 12 \, a^{3} n - 4 \, a^{3} - 2 \,{\left (b^{3} n^{2} - 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3 \, n} - 3 \,{\left (3 \, a b^{2} n^{2} - 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2 \, n} - 6 \,{\left (3 \, a^{2} b n^{2} - 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{n}}{2 \,{\left (3 \, n^{3} - 11 \, n^{2} + 12 \, n - 4\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{2}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]