Optimal. Leaf size=119 \[ -\frac {b^3 F^a \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{6 d}+\frac {b^2 \log ^2(F) (c+d x) F^{a+\frac {b}{c+d x}}}{6 d}+\frac {(c+d x)^3 F^{a+\frac {b}{c+d x}}}{3 d}+\frac {b \log (F) (c+d x)^2 F^{a+\frac {b}{c+d x}}}{6 d} \]
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Rubi [A] time = 0.13, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2214, 2206, 2210} \[ -\frac {b^3 F^a \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{6 d}+\frac {b^2 \log ^2(F) (c+d x) F^{a+\frac {b}{c+d x}}}{6 d}+\frac {(c+d x)^3 F^{a+\frac {b}{c+d x}}}{3 d}+\frac {b \log (F) (c+d x)^2 F^{a+\frac {b}{c+d x}}}{6 d} \]
Antiderivative was successfully verified.
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Rule 2206
Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{c+d x}} (c+d x)^2 \, dx &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^3}{3 d}+\frac {1}{3} (b \log (F)) \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx\\ &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^3}{3 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x)^2 \log (F)}{6 d}+\frac {1}{6} \left (b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{c+d x}} \, dx\\ &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^3}{3 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x)^2 \log (F)}{6 d}+\frac {b^2 F^{a+\frac {b}{c+d x}} (c+d x) \log ^2(F)}{6 d}+\frac {1}{6} \left (b^3 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^3}{3 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x)^2 \log (F)}{6 d}+\frac {b^2 F^{a+\frac {b}{c+d x}} (c+d x) \log ^2(F)}{6 d}-\frac {b^3 F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log ^3(F)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 76, normalized size = 0.64 \[ \frac {F^a \left ((c+d x) F^{\frac {b}{c+d x}} \left (b^2 \log ^2(F)+b \log (F) (c+d x)+2 (c+d x)^2\right )-b^3 \log ^3(F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 120, normalized size = 1.01 \[ -\frac {F^{a} b^{3} {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right ) \log \relax (F)^{3} - {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 6 \, c^{2} d x + 2 \, c^{3} + {\left (b^{2} d x + b^{2} c\right )} \log \relax (F)^{2} + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{6 \, d} \]
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 \[ \int {\left (d x + c\right )}^{2} F^{a + \frac {b}{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 234, normalized size = 1.97 \[ \frac {b^{3} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right ) \ln \relax (F )^{3}}{6 d}+\frac {b^{2} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{6}+\frac {b d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{6}+\frac {d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}}}{3}+\frac {b^{2} c \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{6 d}+\frac {b c x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{3}+c d \,x^{2} F^{a} F^{\frac {b}{d x +c}}+\frac {b \,c^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{6 d}+c^{2} x \,F^{a} F^{\frac {b}{d x +c}}+\frac {c^{3} F^{a} F^{\frac {b}{d x +c}}}{3 d} \]
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 \[ \frac {1}{6} \, {\left (2 \, F^{a} d^{2} x^{3} + {\left (F^{a} b d \log \relax (F) + 6 \, F^{a} c d\right )} x^{2} + {\left (F^{a} b^{2} \log \relax (F)^{2} + 2 \, F^{a} b c \log \relax (F) + 6 \, F^{a} c^{2}\right )} x\right )} F^{\frac {b}{d x + c}} + \int \frac {{\left (F^{a} b^{3} d x \log \relax (F)^{3} - F^{a} b^{2} c^{2} \log \relax (F)^{2} - 2 \, F^{a} b c^{3} \log \relax (F)\right )} F^{\frac {b}{d x + c}}}{6 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.88, size = 89, normalized size = 0.75 \[ \frac {F^a\,b^3\,{\ln \relax (F)}^3\,\left (\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{c+d\,x}\right )}{6}+F^{\frac {b}{c+d\,x}}\,\left (\frac {c+d\,x}{6\,b\,\ln \relax (F)}+\frac {{\left (c+d\,x\right )}^2}{6\,b^2\,{\ln \relax (F)}^2}+\frac {{\left (c+d\,x\right )}^3}{3\,b^3\,{\ln \relax (F)}^3}\right )\right )}{d} \]
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 \[ \int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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