Optimal. Leaf size=70 \[ \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5}{5 b^2}-\frac {a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}{4 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {15, 368, 43} \begin {gather*} \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5}{5 b^2}-\frac {a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rule 368
Rubi steps
\begin {align*} \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3 \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3 \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x (a+b x)^3 \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}{4 b^2}+\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^5}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.69 \begin {gather*} -\frac {x \left (c x^n\right )^{-1/n} \left (a-4 b \left (c x^n\right )^{\frac {1}{n}}\right ) \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}{20 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.72, size = 60, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, b^{3} c^{\frac {4}{n}} x^{5} + \frac {3}{4} \, a b^{2} c^{\frac {3}{n}} x^{4} + a^{2} b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a^{3} c^{\left (\frac {1}{n}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 60, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, b^{3} c^{\frac {4}{n}} x^{5} + \frac {3}{4} \, a b^{2} c^{\frac {3}{n}} x^{4} + a^{2} b c^{\frac {2}{n}} x^{3} + \frac {1}{2} \, a^{3} c^{\left (\frac {1}{n}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{3} \left (c \,x^{n}\right )^{\frac {1}{n}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{3} \left (c x^{n}\right )^{\left (\frac {1}{n}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 68, normalized size = 0.97 \begin {gather*} \frac {b^3\,x\,{\left (c\,x^n\right )}^{4/n}}{5}+\frac {a^3\,x\,{\left (c\,x^n\right )}^{1/n}}{2}+a^2\,b\,x\,{\left (c\,x^n\right )}^{2/n}+\frac {3\,a\,b^2\,x\,{\left (c\,x^n\right )}^{3/n}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.84, size = 76, normalized size = 1.09 \begin {gather*} \frac {a^{3} c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2} + a^{2} b c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}} + \frac {3 a b^{2} c^{\frac {3}{n}} x \left (x^{n}\right )^{\frac {3}{n}}}{4} + \frac {b^{3} c^{\frac {4}{n}} x \left (x^{n}\right )^{\frac {4}{n}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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