Optimal. Leaf size=208 \[ \frac {105 e^{7/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 e^3 F^{c (a+b x)} \sqrt {d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac {35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)} \]
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Rubi [A]
time = 0.20, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2207, 2211,
2235} \begin {gather*} \frac {105 \sqrt {\pi } e^{7/2} F^{c \left (a-\frac {b d}{e}\right )} \text {Erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 e^3 \sqrt {d+e x} F^{c (a+b x)}}{8 b^4 c^4 \log ^4(F)}+\frac {35 e^2 (d+e x)^{3/2} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e (d+e x)^{5/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2211
Rule 2235
Rubi steps
\begin {align*} \int F^{c (a+b x)} (d+e x)^{7/2} \, dx &=\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}-\frac {(7 e) \int F^{c (a+b x)} (d+e x)^{5/2} \, dx}{2 b c \log (F)}\\ &=-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}+\frac {\left (35 e^2\right ) \int F^{c (a+b x)} (d+e x)^{3/2} \, dx}{4 b^2 c^2 \log ^2(F)}\\ &=\frac {35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}-\frac {\left (105 e^3\right ) \int F^{c (a+b x)} \sqrt {d+e x} \, dx}{8 b^3 c^3 \log ^3(F)}\\ &=-\frac {105 e^3 F^{c (a+b x)} \sqrt {d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac {35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}+\frac {\left (105 e^4\right ) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{16 b^4 c^4 \log ^4(F)}\\ &=-\frac {105 e^3 F^{c (a+b x)} \sqrt {d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac {35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}+\frac {\left (105 e^3\right ) \text {Subst}\left (\int F^{c \left (a-\frac {b d}{e}\right )+\frac {b c x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^4 c^4 \log ^4(F)}\\ &=\frac {105 e^{7/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 e^3 F^{c (a+b x)} \sqrt {d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac {35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 72, normalized size = 0.35 \begin {gather*} \frac {e^4 F^{c \left (a-\frac {b d}{e}\right )} \Gamma \left (\frac {9}{2},-\frac {b c (d+e x) \log (F)}{e}\right ) \sqrt {-\frac {b c (d+e x) \log (F)}{e}}}{b^5 c^5 \sqrt {d+e x} \log ^5(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (e x +d \right )^{\frac {7}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 224, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (8 \, {\left (b^{4} c^{4} x^{3} e^{3} + 3 \, b^{4} c^{4} d x^{2} e^{2} + 3 \, b^{4} c^{4} d^{2} x e + b^{4} c^{4} d^{3}\right )} \log \left (F\right )^{4} - 105 \, b c e^{3} \log \left (F\right ) - 28 \, {\left (b^{3} c^{3} x^{2} e^{3} + 2 \, b^{3} c^{3} d x e^{2} + b^{3} c^{3} d^{2} e\right )} \log \left (F\right )^{3} + 70 \, {\left (b^{2} c^{2} x e^{3} + b^{2} c^{2} d e^{2}\right )} \log \left (F\right )^{2}\right )} \sqrt {x e + d} F^{b c x + a c} - \frac {105 \, \sqrt {\pi } \sqrt {-b c e^{\left (-1\right )} \log \left (F\right )} \operatorname {erf}\left (\sqrt {-b c e^{\left (-1\right )} \log \left (F\right )} \sqrt {x e + d}\right ) e^{4}}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}}}{16 \, b^{5} c^{5} \log \left (F\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1091 vs.
\(2 (176) = 352\).
time = 2.59, size = 1091, normalized size = 5.25 \begin {gather*} -\frac {1}{16} \, {\left (\frac {16 \, \sqrt {\pi } d^{4} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )}} - 32 \, d^{3} {\left (\frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} + \frac {2 \, \sqrt {x e + d} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c \log \left (F\right )}\right )} + 24 \, d^{2} {\left (\frac {\sqrt {\pi } {\left (4 \, b^{2} c^{2} d^{2} \log \left (F\right )^{2} + 4 \, b c d e \log \left (F\right ) + 3 \, e^{2}\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right ) + 2 \, e\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b^{2} c^{2} \log \left (F\right )^{2}} - \frac {2 \, {\left (2 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e \log \left (F\right ) - 4 \, \sqrt {x e + d} b c d e \log \left (F\right ) - 3 \, \sqrt {x e + d} e^{2}\right )} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right ) - 2 \, e\right )} e^{\left (-1\right )}\right )}}{b^{2} c^{2} \log \left (F\right )^{2}}\right )} e^{2} - 8 \, d {\left (\frac {\sqrt {\pi } {\left (8 \, b^{3} c^{3} d^{3} \log \left (F\right )^{3} + 12 \, b^{2} c^{2} d^{2} e \log \left (F\right )^{2} + 18 \, b c d e^{2} \log \left (F\right ) + 15 \, e^{3}\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right ) + 3 \, e\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b^{3} c^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (4 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{2} e \log \left (F\right )^{2} - 12 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e \log \left (F\right )^{2} + 12 \, \sqrt {x e + d} b^{2} c^{2} d^{2} e \log \left (F\right )^{2} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e^{2} \log \left (F\right ) + 18 \, \sqrt {x e + d} b c d e^{2} \log \left (F\right ) + 15 \, \sqrt {x e + d} e^{3}\right )} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right ) - 3 \, e\right )} e^{\left (-1\right )}\right )}}{b^{3} c^{3} \log \left (F\right )^{3}}\right )} e^{3} + {\left (\frac {\sqrt {\pi } {\left (16 \, b^{4} c^{4} d^{4} \log \left (F\right )^{4} + 32 \, b^{3} c^{3} d^{3} e \log \left (F\right )^{3} + 72 \, b^{2} c^{2} d^{2} e^{2} \log \left (F\right )^{2} + 120 \, b c d e^{3} \log \left (F\right ) + 105 \, e^{4}\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right ) + 4 \, e\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b^{4} c^{4} \log \left (F\right )^{4}} - \frac {2 \, {\left (8 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{3} e \log \left (F\right )^{3} - 32 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{3} d e \log \left (F\right )^{3} + 48 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{3} d^{2} e \log \left (F\right )^{3} - 32 \, \sqrt {x e + d} b^{3} c^{3} d^{3} e \log \left (F\right )^{3} - 28 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 80 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e^{2} \log \left (F\right )^{2} - 72 \, \sqrt {x e + d} b^{2} c^{2} d^{2} e^{2} \log \left (F\right )^{2} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e^{3} \log \left (F\right ) - 120 \, \sqrt {x e + d} b c d e^{3} \log \left (F\right ) - 105 \, \sqrt {x e + d} e^{4}\right )} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right ) - 4 \, e\right )} e^{\left (-1\right )}\right )}}{b^{4} c^{4} \log \left (F\right )^{4}}\right )} e^{4}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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