Optimal. Leaf size=74 \[ -\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {3}{2},\frac {1}{5};-\frac {1}{2};-\frac {d (a+b x)}{b c-a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {72, 71}
\begin {gather*} -\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {3}{2},\frac {1}{5};-\frac {1}{2};-\frac {d (a+b x)}{b c-a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} \sqrt [5]{c+d x}} \, dx &=\frac {\sqrt [5]{\frac {b (c+d x)}{b c-a d}} \int \frac {1}{(a+b x)^{5/2} \sqrt [5]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{\sqrt [5]{c+d x}}\\ &=-\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {3}{2},\frac {1}{5};-\frac {1}{2};-\frac {d (a+b x)}{b c-a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}}\\ \end {align*}
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Mathematica [A]
time = 10.03, size = 73, normalized size = 0.99 \begin {gather*} -\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {3}{2},\frac {1}{5};-\frac {1}{2};\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {1}{5}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \sqrt [5]{c + d x}}\, dx \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{1/5}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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