Optimal. Leaf size=44 \[ \frac {3 x^{4 (1-n)} \left (a x^{-3 (1-n)}+b x^{-3+4 n}\right )^{4/3}}{4 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2004, 2025}
\begin {gather*} \frac {3 x^{4 (1-n)} \left (a x^{-3 (1-n)}+b x^{4 n-3}\right )^{4/3}}{4 b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2004
Rule 2025
Rubi steps
\begin {align*} \int \sqrt [3]{x^{3 (-1+n)} \left (a+b x^n\right )} \, dx &=\int \sqrt [3]{a x^{3 (-1+n)}+b x^{3 (-1+n)+n}} \, dx\\ &=\frac {3 x^{4 (1-n)} \left (a x^{-3 (1-n)}+b x^{-3+4 n}\right )^{4/3}}{4 b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 36, normalized size = 0.82 \begin {gather*} \frac {3 x^{4-4 n} \left (x^{-3+3 n} \left (a+b x^n\right )\right )^{4/3}}{4 b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.43, size = 40, normalized size = 0.91
method | result | size |
risch | \(\frac {3 \left (\frac {x^{3 n} \left (a +b \,x^{n}\right )}{x^{3}}\right )^{\frac {1}{3}} x \,x^{-n} \left (a +b \,x^{n}\right )}{4 b n}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 17, normalized size = 0.39 \begin {gather*} \frac {3 \, {\left (b x^{n} + a\right )}^{\frac {4}{3}}}{4 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.98, size = 44, normalized size = 1.00 \begin {gather*} \frac {3 \, {\left (b x x^{n} + a x\right )} \left (\frac {b x^{4 \, n} + a x^{3 \, n}}{x^{3}}\right )^{\frac {1}{3}}}{4 \, b n x^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (x^{3\,n-3}\,\left (a+b\,x^n\right )\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________