Integrand size = 25, antiderivative size = 100 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)} \]
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Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2244, 2240} \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {2 (c+d x)^n F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {(c+d x)^{2 n} F^{a+b (c+d x)^n}}{b d n \log (F)} \]
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Rule 2240
Rule 2244
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}-\frac {2 \int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx}{b \log (F)} \\ & = -\frac {2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}+\frac {2 \int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b^2 \log ^2(F)} \\ & = \frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.31 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\frac {F^a \Gamma \left (3,-b (c+d x)^n \log (F)\right )}{b^3 d n \log ^3(F)} \]
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Time = 0.44 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {\left (b^{2} \left (d x +c \right )^{2 n} \ln \left (F \right )^{2}-2 b \left (d x +c \right )^{n} \ln \left (F \right )+2\right ) F^{a +b \left (d x +c \right )^{n}}}{b^{3} \ln \left (F \right )^{3} n d}\) | \(59\) |
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Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.62 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\frac {{\left ({\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} - 2 \, {\left (d x + c\right )}^{n} b \log \left (F\right ) + 2\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{3} d n \log \left (F\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (85) = 170\).
Time = 10.94 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.85 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\begin {cases} \frac {x}{c} & \text {for}\: F = 1 \wedge b = 0 \wedge d = 0 \wedge n = 0 \\F^{a} \left (\frac {c \left (c + d x\right )^{3 n - 1}}{3 d n} + \frac {x \left (c + d x\right )^{3 n - 1}}{3 n}\right ) & \text {for}\: b = 0 \\F^{a + b c^{n}} c^{3 n - 1} x & \text {for}\: d = 0 \\\frac {F^{a + b} \log {\left (\frac {c}{d} + x \right )}}{d} & \text {for}\: n = 0 \\\frac {c \left (c + d x\right )^{3 n - 1}}{3 d n} + \frac {x \left (c + d x\right )^{3 n - 1}}{3 n} & \text {for}\: F = 1 \\\frac {F^{a + b \left (c + d x\right )^{n}} \left (c + d x\right )^{2 n}}{b d n \log {\left (F \right )}} - \frac {2 F^{a + b \left (c + d x\right )^{n}} \left (c + d x\right )^{n}}{b^{2} d n \log {\left (F \right )}^{2}} + \frac {2 F^{a + b \left (c + d x\right )^{n}}}{b^{3} d n \log {\left (F \right )}^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.66 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\frac {{\left ({\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \left (F\right )^{2} - 2 \, {\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) + 2 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{3} d n \log \left (F\right )^{3}} \]
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\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\int { {\left (d x + c\right )}^{3 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]
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Timed out. \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx=\int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{3\,n-1} \,d x \]
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