Optimal. Leaf size=62 \[ \frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^3 \log (F)} \]
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Rubi [A]
time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2240}
\begin {gather*} \frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^7} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^3 \log (F)}-\frac {\int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^4} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^3}}}{3 b d (c+d x)^3 \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.76 \begin {gather*} \frac {F^{a+\frac {b}{(c+d x)^3}} \left ((c+d x)^3-b \log (F)\right )}{3 b^2 d (c+d x)^3 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 97, normalized size = 1.56
method | result | size |
risch | \(-\frac {\left (-d^{3} x^{3}-3 c \,d^{2} x^{2}-3 c^{2} d x -c^{3}+b \ln \left (F \right )\right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 d \ln \left (F \right )^{2} b^{2} \left (d x +c \right )^{3}}\) | \(97\) |
norman | \(\frac {\frac {d^{5} x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{2} b^{2}}-\frac {c^{2} \left (-2 c^{3}+b \ln \left (F \right )\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}-\frac {c^{3} \left (-c^{3}+b \ln \left (F \right )\right ) {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 d \ln \left (F \right )^{2} b^{2}}-\frac {d^{2} \left (-20 c^{3}+b \ln \left (F \right )\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{2} b^{2}}+\frac {5 d^{3} c^{2} x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {2 d^{4} c \,x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}-\frac {c d \left (-5 c^{3}+b \ln \left (F \right )\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{3}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}}{\left (d x +c \right )^{6}}\) | \(261\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (58) = 116\).
time = 0.28, size = 144, normalized size = 2.32 \begin {gather*} \frac {{\left (F^{a} d^{3} x^{3} + 3 \, F^{a} c d^{2} x^{2} + 3 \, F^{a} c^{2} d x + F^{a} c^{3} - F^{a} b \log \left (F\right )\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{2} d^{4} x^{3} \log \left (F\right )^{2} + 3 \, b^{2} c d^{3} x^{2} \log \left (F\right )^{2} + 3 \, b^{2} c^{2} d^{2} x \log \left (F\right )^{2} + b^{2} c^{3} d \log \left (F\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs.
\(2 (58) = 116\).
time = 0.36, size = 148, normalized size = 2.39 \begin {gather*} \frac {{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3} - b \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (49) = 98\).
time = 0.15, size = 114, normalized size = 1.84 \begin {gather*} \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b \log {\left (F \right )} + c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}\right )}{3 b^{2} c^{3} d \log {\left (F \right )}^{2} + 9 b^{2} c^{2} d^{2} x \log {\left (F \right )}^{2} + 9 b^{2} c d^{3} x^{2} \log {\left (F \right )}^{2} + 3 b^{2} d^{4} x^{3} \log {\left (F \right )}^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.89, size = 136, normalized size = 2.19 \begin {gather*} \frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {x^3}{3\,b^2\,d\,{\ln \left (F\right )}^2}-\frac {b\,\ln \left (F\right )-c^3}{3\,b^2\,d^4\,{\ln \left (F\right )}^2}+\frac {c\,x^2}{b^2\,d^2\,{\ln \left (F\right )}^2}+\frac {c^2\,x}{b^2\,d^3\,{\ln \left (F\right )}^2}\right )}{x^3+\frac {c^3}{d^3}+\frac {3\,c\,x^2}{d}+\frac {3\,c^2\,x}{d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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