Optimal. Leaf size=83 \[ \frac {x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac {a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^2 (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {368, 43} \[ \frac {x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac {a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int x (a+b x)^p \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^p}{b}+\frac {(a+b x)^{1+p}}{b}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=-\frac {a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^2 (1+p)}+\frac {x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^2 (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 0.76 \[ \frac {x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1} \left (b (p+1) \left (c x^n\right )^{\frac {1}{n}}-a\right )}{b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 79, normalized size = 0.95 \[ \frac {{\left (a b c^{\left (\frac {1}{n}\right )} p x + {\left (b^{2} p + b^{2}\right )} c^{\frac {2}{n}} x^{2} - a^{2}\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )} c^{\frac {2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 136, normalized size = 1.64 \[ \frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac {2}{n}} p x^{2} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b c^{\left (\frac {1}{n}\right )} p x + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac {2}{n}} x^{2} - {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2}}{b^{2} c^{\frac {2}{n}} p^{2} + 3 \, b^{2} c^{\frac {2}{n}} p + 2 \, b^{2} c^{\frac {2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 304, normalized size = 3.66 \[ \frac {x^{2} c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{p +1} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{\left (p +1\right ) b}-\frac {x^{2} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{p +2} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{n}}}{\left (p +1\right ) \left (p +2\right ) b^{2}} \]
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 \[ \int {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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