Integrand size = 21, antiderivative size = 267 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}-\frac {b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{2 (d e-c f)^2 (e+f x)}-\frac {b d^2 F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^3}+\frac {b^2 d^2 f F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{2 (d e-c f)^4} \]
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Time = 1.18 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2255, 6874, 2240, 2241, 2254, 2260, 2209} \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\frac {b^2 d^2 f \log ^2(F) F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{2 (d e-c f)^4}-\frac {b d^2 \log (F) F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^3}+\frac {d^2 F^{a+\frac {b}{c+d x}}}{2 f (d e-c f)^2}-\frac {b d^2 \log (F) F^{a+\frac {b}{c+d x}}}{2 (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{2 f (e+f x)^2}+\frac {b d \log (F) F^{a+\frac {b}{c+d x}}}{2 (e+f x) (d e-c f)^2} \]
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Rule 2209
Rule 2240
Rule 2241
Rule 2254
Rule 2255
Rule 2260
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 ) \text {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*}
\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx \]
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Time = 0.67 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.90
method | result | size |
risch | \(-\frac {b \,d^{2} \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}}}{\left (c f -d e \right )^{3} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )}-\frac {b \,d^{2} \ln \left (F \right ) F^{\frac {a c f -a d e +b f}{c f -d e}} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}-a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c f +a e d \ln \left (F \right )-\ln \left (F \right ) b f}{c f -d e}\right )}{\left (c f -d e \right )^{3}}-\frac {b^{2} d^{2} \ln \left (F \right )^{2} f \,F^{a} F^{\frac {b}{d x +c}}}{2 \left (c f -d e \right )^{4} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )^{2}}-\frac {b^{2} d^{2} \ln \left (F \right )^{2} f \,F^{a} F^{\frac {b}{d x +c}}}{2 \left (c f -d e \right )^{4} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )}-\frac {b^{2} d^{2} \ln \left (F \right )^{2} f \,F^{\frac {a c f -a d e +b f}{c f -d e}} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}-a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c f +a e d \ln \left (F \right )-\ln \left (F \right ) b f}{c f -d e}\right )}{2 \left (c f -d e \right )^{4}}\) | \(506\) |
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Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (257) = 514\).
Time = 0.33 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.08 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\frac {{\left ({\left (b^{2} d^{2} f^{3} x^{2} + 2 \, b^{2} d^{2} e f^{2} x + b^{2} d^{2} e^{2} f\right )} \log \left (F\right )^{2} - 2 \, {\left (b d^{3} e^{3} - b c d^{2} e^{2} f + {\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 2 \, {\left (b d^{3} e^{2} f - b c d^{2} e f^{2}\right )} x\right )} \log \left (F\right )\right )} F^{\frac {a d e - {\left (a c + b\right )} f}{d e - c f}} {\rm Ei}\left (\frac {{\left (b d f x + b d e\right )} \log \left (F\right )}{c d e - c^{2} f + {\left (d^{2} e - c d f\right )} x}\right ) + {\left (2 \, c d^{3} e^{3} - 5 \, c^{2} d^{2} e^{2} f + 4 \, c^{3} d e f^{2} - c^{4} f^{3} + {\left (d^{4} e^{2} f - 2 \, c d^{3} e f^{2} + c^{2} d^{2} f^{3}\right )} x^{2} + 2 \, {\left (d^{4} e^{3} - 2 \, c d^{3} e^{2} f + c^{2} d^{2} e f^{2}\right )} x - {\left (b c d^{2} e^{2} f - b c^{2} d e f^{2} + {\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + {\left (b d^{3} e^{2} f - b c^{2} d f^{3}\right )} x\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{2 \, {\left (d^{4} e^{6} - 4 \, c d^{3} e^{5} f + 6 \, c^{2} d^{2} e^{4} f^{2} - 4 \, c^{3} d e^{3} f^{3} + c^{4} e^{2} f^{4} + {\left (d^{4} e^{4} f^{2} - 4 \, c d^{3} e^{3} f^{3} + 6 \, c^{2} d^{2} e^{2} f^{4} - 4 \, c^{3} d e f^{5} + c^{4} f^{6}\right )} x^{2} + 2 \, {\left (d^{4} e^{5} f - 4 \, c d^{3} e^{4} f^{2} + 6 \, c^{2} d^{2} e^{3} f^{3} - 4 \, c^{3} d e^{2} f^{4} + c^{4} e f^{5}\right )} x\right )}} \]
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\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\int \frac {F^{a + \frac {b}{c + d x}}}{\left (e + f x\right )^{3}}\, dx \]
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\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{3}} \,d x } \]
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\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx=\int \frac {F^{a+\frac {b}{c+d\,x}}}{{\left (e+f\,x\right )}^3} \,d x \]
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