Optimal. Leaf size=70 \[ \frac{b x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a^2 (n+1) (n+2)}-\frac{x^{-3 n-4} \left (a x^2+b x^3\right )^{n+1}}{a (n+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0543414, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac{b x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a^2 (n+1) (n+2)}-\frac{x^{-3 n-4} \left (a x^2+b x^3\right )^{n+1}}{a (n+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int x^{-3-3 n} \left (a x^2+b x^3\right )^n \, dx &=-\frac{x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (2+n)}-\frac{b \int x^{-2-3 n} \left (a x^2+b x^3\right )^n \, dx}{a (2+n)}\\ &=-\frac{x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (2+n)}+\frac{b x^{-3 (1+n)} \left (a x^2+b x^3\right )^{1+n}}{a^2 (1+n) (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0208736, size = 44, normalized size = 0.63 \[ -\frac{x^{-3 n-4} (a n+a-b x) \left (x^2 (a+b x)\right )^{n+1}}{a^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 50, normalized size = 0.7 \begin{align*} -{\frac{{x}^{-2-3\,n} \left ( b{x}^{3}+a{x}^{2} \right ) ^{n} \left ( an-bx+a \right ) \left ( bx+a \right ) }{ \left ( 2+n \right ) \left ( 1+n \right ){a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.811785, size = 136, normalized size = 1.94 \begin{align*} -\frac{{\left (a b n x^{2} - b^{2} x^{3} +{\left (a^{2} n + a^{2}\right )} x\right )}{\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 3}}{a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]