Optimal. Leaf size=100 \[ \frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2244, 2240}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2240
Rule 2244
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx &=\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*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.01, size = 31, normalized size = 0.31 \begin {gather*} \frac {F^a \Gamma \left (3,-b (c+d x)^n \log (F)\right )}{b^3 d n \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 59, normalized size = 0.59
method | result | size |
risch | \(\frac {\left (b^{2} \left (d x +c \right )^{2 n} \ln \left (F \right )^{2}-2 b \left (d x +c \right )^{n} \ln \left (F \right )+2\right ) F^{a +b \left (d x +c \right )^{n}}}{b^{3} \ln \left (F \right )^{3} n d}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 66, normalized size = 0.66 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 62, normalized size = 0.62 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{3\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________