Optimal. Leaf size=267 \[ \frac {b^2 d^2 f \log ^2(F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{2 (d e-c f)^4}-\frac {b d^2 \log (F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^3}+\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {b d^2 \log (F) F^{a+\frac {b}{c+d x}}}{2 (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {b d \log (F) F^{a+\frac {b}{c+d x}}}{2 (e+f x) (d e-c f)^2} \]
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Rubi [A] time = 1.91, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2223, 6742, 2209, 2210, 2222, 2228, 2178} \[ \frac {b^2 d^2 f \log ^2(F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{2 (d e-c f)^4}-\frac {b d^2 \log (F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^3}+\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {b d^2 \log (F) F^{a+\frac {b}{c+d x}}}{2 (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {b d \log (F) F^{a+\frac {b}{c+d x}}}{2 (e+f x) (d e-c f)^2} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2209
Rule 2210
Rule 2222
Rule 2223
Rule 2228
Rule 6742
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx &=-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}-\frac {(b d \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)^2} \, dx}{2 f}\\ &=-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}-\frac {(b d \log (F)) \int \left (\frac {d^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (c+d x)^2}-\frac {2 d^2 f F^{a+\frac {b}{c+d x}}}{(d e-c f)^3 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)^2}+\frac {2 d f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^3 (e+f x)}\right ) \, dx}{2 f}\\ &=-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {\left (b d^3 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^3}-\frac {\left (b d^2 f \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{(d e-c f)^3}-\frac {\left (b d^3 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{2 f (d e-c f)^2}-\frac {(b d f \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^2} \, dx}{2 (d e-c f)^2}\\ &=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {b d^2 F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)}{(d e-c f)^3}-\frac {\left (b d^3 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^3}+\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{(d e-c f)^2}+\frac {\left (b^2 d^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)} \, dx}{2 (d e-c f)^2}\\ &=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {\left (b d^2 \log (F)\right ) \operatorname {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{(d e-c f)^3}+\frac {\left (b^2 d^2 \log ^2(F)\right ) \int \left (\frac {d F^{a+\frac {b}{c+d x}}}{(d e-c f) (c+d x)^2}-\frac {d f F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)}\right ) \, dx}{2 (d e-c f)^2}\\ &=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {b d^2 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^3}-\frac {\left (b^2 d^3 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{2 (d e-c f)^4}+\frac {\left (b^2 d^2 f^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{2 (d e-c f)^4}+\frac {\left (b^2 d^3 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{2 (d e-c f)^3}\\ &=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}-\frac {b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {b d^2 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^3}+\frac {b^2 d^2 f F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log ^2(F)}{2 (d e-c f)^4}+\frac {\left (b^2 d^3 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{2 (d e-c f)^4}-\frac {\left (b^2 d^2 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{2 (d e-c f)^3}\\ &=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}-\frac {b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {b d^2 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^3}+\frac {\left (b^2 d^2 f \log ^2(F)\right ) \operatorname {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{2 (d e-c f)^4}\\ &=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}-\frac {b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {b d^2 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^3}+\frac {b^2 d^2 f F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{2 (d e-c f)^4}\\ \end {align*}
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Mathematica [F] time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.47, size = 555, normalized size = 2.08 \[ \frac {{\left ({\left (b^{2} d^{2} f^{3} x^{2} + 2 \, b^{2} d^{2} e f^{2} x + b^{2} d^{2} e^{2} f\right )} \log \relax (F)^{2} - 2 \, {\left (b d^{3} e^{3} - b c d^{2} e^{2} f + {\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 2 \, {\left (b d^{3} e^{2} f - b c d^{2} e f^{2}\right )} x\right )} \log \relax (F)\right )} F^{\frac {a d e - {\left (a c + b\right )} f}{d e - c f}} {\rm Ei}\left (\frac {{\left (b d f x + b d e\right )} \log \relax (F)}{c d e - c^{2} f + {\left (d^{2} e - c d f\right )} x}\right ) + {\left (2 \, c d^{3} e^{3} - 5 \, c^{2} d^{2} e^{2} f + 4 \, c^{3} d e f^{2} - c^{4} f^{3} + {\left (d^{4} e^{2} f - 2 \, c d^{3} e f^{2} + c^{2} d^{2} f^{3}\right )} x^{2} + 2 \, {\left (d^{4} e^{3} - 2 \, c d^{3} e^{2} f + c^{2} d^{2} e f^{2}\right )} x - {\left (b c d^{2} e^{2} f - b c^{2} d e f^{2} + {\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + {\left (b d^{3} e^{2} f - b c^{2} d f^{3}\right )} x\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{2 \, {\left (d^{4} e^{6} - 4 \, c d^{3} e^{5} f + 6 \, c^{2} d^{2} e^{4} f^{2} - 4 \, c^{3} d e^{3} f^{3} + c^{4} e^{2} f^{4} + {\left (d^{4} e^{4} f^{2} - 4 \, c d^{3} e^{3} f^{3} + 6 \, c^{2} d^{2} e^{2} f^{4} - 4 \, c^{3} d e f^{5} + c^{4} f^{6}\right )} x^{2} + 2 \, {\left (d^{4} e^{5} f - 4 \, c d^{3} e^{4} f^{2} + 6 \, c^{2} d^{2} e^{3} f^{3} - 4 \, c^{3} d e^{2} f^{4} + c^{4} e f^{5}\right )} x\right )}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 506, normalized size = 1.90 \[ -\frac {b^{2} d^{2} f \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{2 \left (c f -d e \right )^{4} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )^{2}}-\frac {b^{2} d^{2} f \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{2 \left (c f -d e \right )^{4} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )}-\frac {b^{2} d^{2} f \,F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right ) \ln \relax (F )^{2}}{2 \left (c f -d e \right )^{4}}-\frac {b \,d^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{\left (c f -d e \right )^{3} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )}-\frac {b \,d^{2} F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right ) \ln \relax (F )}{\left (c f -d e \right )^{3}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {F^{a+\frac {b}{c+d\,x}}}{{\left (e+f\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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