Optimal. Leaf size=63 \[ \frac {2 \pi ^n (b x)^{7/2} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} F_1\left (\frac {7}{2};-n,-p;\frac {9}{2};-\frac {d x}{\pi },-\frac {f x}{e}\right )}{7 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {140, 138}
\begin {gather*} \frac {2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} F_1\left (\frac {7}{2};-n,-p;\frac {9}{2};-\frac {d x}{\pi },-\frac {f x}{e}\right )}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rubi steps
\begin {align*} \int (b x)^{5/2} (\pi +d x)^n (e+f x)^p \, dx &=\left ((e+f x)^p \left (1+\frac {f x}{e}\right )^{-p}\right ) \int (b x)^{5/2} (\pi +d x)^n \left (1+\frac {f x}{e}\right )^p \, dx\\ &=\frac {2 \pi ^n (b x)^{7/2} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} F_1\left (\frac {7}{2};-n,-p;\frac {9}{2};-\frac {d x}{\pi },-\frac {f x}{e}\right )}{7 b}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 62, normalized size = 0.98 \begin {gather*} \frac {2}{7} \pi ^n x (b x)^{5/2} (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} F_1\left (\frac {7}{2};-n,-p;\frac {9}{2};-\frac {d x}{\pi },-\frac {f x}{e}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{\frac {5}{2}} \left (d x +\pi \right )^{n} \left (f x +e \right )^{p}\, dx\]
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 [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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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 [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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (e+f\,x\right )}^p\,{\left (b\,x\right )}^{5/2}\,{\left (\Pi +d\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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