Optimal. Leaf size=54 \[ -\frac {F^a (c+d x)^3 \Gamma \left (\frac {3}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-3/n}}{d n} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250}
\begin {gather*} -\frac {F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-b \log (F) (c+d x)^n\right )}{d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^2 \, dx &=-\frac {F^a (c+d x)^3 \Gamma \left (\frac {3}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-3/n}}{d n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 54, normalized size = 1.00 \begin {gather*} -\frac {F^a (c+d x)^3 \Gamma \left (\frac {3}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-3/n}}{d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int F^{a +b \left (d x +c \right )^{n}} \left (d x +c \right )^{2}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} F^{a + \frac {b}{c^{3}}} c^{2} x & \text {for}\: d = 0 \wedge n = -3 \\F^{a + b c^{n}} c^{2} x & \text {for}\: d = 0 \\\int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{2}\, dx & \text {for}\: n = -3 \\\frac {3 F^{a} F^{b \left (c + d x\right )^{n}} b c^{3} n \left (c + d x\right )^{n} \log {\left (F \right )}}{3 d n + 9 d} - \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} b c^{2} d n x \left (c + d x\right )^{n} \log {\left (F \right )}}{3 d n + 9 d} - \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} b c d^{2} n x^{2} \left (c + d x\right )^{n} \log {\left (F \right )}}{3 d n + 9 d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} b d^{3} n x^{3} \left (c + d x\right )^{n} \log {\left (F \right )}}{3 d n + 9 d} - \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} c^{3} n}{3 d n + 9 d} + \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} c^{3}}{3 d n + 9 d} + \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} c^{2} d n x}{3 d n + 9 d} + \frac {9 F^{a} F^{b \left (c + d x\right )^{n}} c^{2} d x}{3 d n + 9 d} + \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} c d^{2} n x^{2}}{3 d n + 9 d} + \frac {9 F^{a} F^{b \left (c + d x\right )^{n}} c d^{2} x^{2}}{3 d n + 9 d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d^{3} n x^{3}}{3 d n + 9 d} + \frac {3 F^{a} F^{b \left (c + d x\right )^{n}} d^{3} x^{3}}{3 d n + 9 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 3.90, size = 73, normalized size = 1.35 \begin {gather*} \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^n}{2}}\,{\left (c+d\,x\right )}^3\,{\mathrm {M}}_{\frac {1}{2}-\frac {3}{2\,n},\frac {3}{2\,n}}\left (b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^n\right )}{3\,d\,{\left (b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^n\right )}^{\frac {3}{2\,n}+\frac {1}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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