Optimal. Leaf size=87 \[ -\frac {b^2 F^a \log ^2(F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{4 d}+\frac {(c+d x)^4 F^{a+\frac {b}{(c+d x)^2}}}{4 d}+\frac {b \log (F) (c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{4 d} \]
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Rubi [A] time = 0.12, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ -\frac {b^2 F^a \log ^2(F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{4 d}+\frac {(c+d x)^4 F^{a+\frac {b}{(c+d x)^2}}}{4 d}+\frac {b \log (F) (c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{4 d} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \, dx &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4}{4 d}+\frac {1}{2} (b \log (F)) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x) \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4}{4 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log (F)}{4 d}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4}{4 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log (F)}{4 d}-\frac {b^2 F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right ) \log ^2(F)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.82 \[ \frac {F^a \left (b \log (F) \left ((c+d x)^2 F^{\frac {b}{(c+d x)^2}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )\right )+(c+d x)^4 F^{\frac {b}{(c+d x)^2}}\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 145, normalized size = 1.67 \[ -\frac {F^{a} b^{2} {\rm Ei}\left (\frac {b \log \relax (F)}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \relax (F)^{2} - {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4} + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{4 \, d} \]
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 \[ \int {\left (d x + c\right )}^{3} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 208, normalized size = 2.39 \[ \frac {d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4}+c \,d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{4}+\frac {3 c^{2} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+\frac {b c x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{2}+c^{3} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b^{2} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )^{2}}{4 d}+\frac {b \,c^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{4 d}+\frac {c^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4 d} \]
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 \[ \frac {1}{4} \, {\left (F^{a} d^{3} x^{4} + 4 \, F^{a} c d^{2} x^{3} + {\left (6 \, F^{a} c^{2} d + F^{a} b d \log \relax (F)\right )} x^{2} + 2 \, {\left (2 \, F^{a} c^{3} + F^{a} b c \log \relax (F)\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {{\left (F^{a} b^{2} d^{2} x^{2} \log \relax (F)^{2} + 2 \, F^{a} b^{2} c d x \log \relax (F)^{2} - F^{a} b c^{4} \log \relax (F)\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 76, normalized size = 0.87 \[ \frac {F^a\,b^2\,{\ln \relax (F)}^2\,\left (\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}{2}+F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\left (\frac {{\left (c+d\,x\right )}^2}{2\,b\,\ln \relax (F)}+\frac {{\left (c+d\,x\right )}^4}{2\,b^2\,{\ln \relax (F)}^2}\right )\right )}{2\,d} \]
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 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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