Optimal. Leaf size=53 \[ \frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3}{3 d}-\frac {b F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log (F)}{3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 = {2245, 2241}
\begin {gather*} \frac {(c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{3 d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2245
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^2 \, dx &=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3}{3 d}+(b \log (F)) \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3}{3 d}-\frac {b F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log (F)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.89 \begin {gather*} \frac {F^a \left (F^{\frac {b}{(c+d x)^3}} (c+d x)^3-b \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right ) \log (F)\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}} \left (d x +c \right )^{2}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (49) = 98\).
time = 0.36, size = 141, normalized size = 2.66 \begin {gather*} -\frac {F^{a} b {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right ) - {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, d} \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*} \int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{2}\, dx \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 = 3.72, size = 51, normalized size = 0.96 \begin {gather*} \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,{\left (c+d\,x\right )}^3}{3\,d}+\frac {F^a\,b\,\ln \left (F\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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