Optimal. Leaf size=136 \[ -\frac {4 \sqrt {\pi } b^{5/2} F^a \log ^{\frac {5}{2}}(F) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{15 d}+\frac {4 b^2 \log ^2(F) (c+d x) F^{a+\frac {b}{(c+d x)^2}}}{15 d}+\frac {(c+d x)^5 F^{a+\frac {b}{(c+d x)^2}}}{5 d}+\frac {2 b \log (F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^2}}}{15 d} \]
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Rubi [A] time = 0.17, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2214, 2206, 2211, 2204} \[ -\frac {4 \sqrt {\pi } b^{5/2} F^a \log ^{\frac {5}{2}}(F) \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{15 d}+\frac {4 b^2 \log ^2(F) (c+d x) F^{a+\frac {b}{(c+d x)^2}}}{15 d}+\frac {(c+d x)^5 F^{a+\frac {b}{(c+d x)^2}}}{5 d}+\frac {2 b \log (F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^2}}}{15 d} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2206
Rule 2211
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \, dx &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5}{5 d}+\frac {1}{5} (2 b \log (F)) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5}{5 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log (F)}{15 d}+\frac {1}{15} \left (4 b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5}{5 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log (F)}{15 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^2(F)}{15 d}+\frac {1}{15} \left (8 b^3 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5}{5 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log (F)}{15 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^2(F)}{15 d}-\frac {\left (8 b^3 \log ^3(F)\right ) \operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{15 d}\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5}{5 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log (F)}{15 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^2(F)}{15 d}-\frac {4 b^{5/2} F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right ) \log ^{\frac {5}{2}}(F)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 97, normalized size = 0.71 \[ \frac {F^a \left ((c+d x) F^{\frac {b}{(c+d x)^2}} \left (4 b^2 \log ^2(F)+2 b \log (F) (c+d x)^2+3 (c+d x)^4\right )-4 \sqrt {\pi } b^{5/2} \log ^{\frac {5}{2}}(F) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 201, normalized size = 1.48 \[ \frac {4 \, \sqrt {\pi } F^{a} b^{2} d \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) \log \relax (F)^{2} + {\left (3 \, d^{5} x^{5} + 15 \, c d^{4} x^{4} + 30 \, c^{2} d^{3} x^{3} + 30 \, c^{3} d^{2} x^{2} + 15 \, c^{4} d x + 3 \, c^{5} + 4 \, {\left (b^{2} d x + b^{2} c\right )} \log \relax (F)^{2} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{15 \, d} \]
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 \[ \int {\left (d x + c\right )}^{4} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 324, normalized size = 2.38 \[ \frac {d^{4} x^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{5}+c \,d^{3} x^{4} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {2 b \,d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{15}+2 c^{2} d^{2} x^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {2 b c d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{5}+2 c^{3} d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}-\frac {4 \sqrt {\pi }\, b^{3} F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right ) \ln \relax (F )^{3}}{15 \sqrt {-b \ln \relax (F )}\, d}+\frac {4 b^{2} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{15}+\frac {2 b \,c^{2} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{5}+c^{4} x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {4 b^{2} c \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )^{2}}{15 d}+\frac {2 b \,c^{3} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} \ln \relax (F )}{15 d}+\frac {c^{5} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{5 d} \]
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 \[ \frac {1}{15} \, {\left (3 \, F^{a} d^{4} x^{5} + 15 \, F^{a} c d^{3} x^{4} + 2 \, {\left (15 \, F^{a} c^{2} d^{2} + F^{a} b d^{2} \log \relax (F)\right )} x^{3} + 6 \, {\left (5 \, F^{a} c^{3} d + F^{a} b c d \log \relax (F)\right )} x^{2} + {\left (15 \, F^{a} c^{4} + 6 \, F^{a} b c^{2} \log \relax (F) + 4 \, F^{a} b^{2} \log \relax (F)^{2}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {2 \, {\left (4 \, F^{a} b^{3} d x \log \relax (F)^{3} - 3 \, F^{a} b c^{5} \log \relax (F) - 2 \, F^{a} b^{2} c^{3} \log \relax (F)^{2}\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{15 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 166, normalized size = 1.22 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^5}{5\,d}+\frac {4\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^5\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{5/2}}{15\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3}{15\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \relax (F)}^2\,\left (c+d\,x\right )}{15\,d}-\frac {4\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^5\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}^{5/2}}{15\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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