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 = 0.0205612, antiderivative size = 74, normalized size of antiderivative = 1., 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 = {70, 69} \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{3}{2},\frac{1}{5};-\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
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*}
Mathematica [A] time = 0.0246489, size = 73, normalized size = 0.99 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{3}{2},\frac{1}{5};-\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [5]{dx+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{4}{5}}}{b^{3} d x^{4} + a^{3} c +{\left (b^{3} c + 3 \, a b^{2} d\right )} x^{3} + 3 \,{\left (a b^{2} c + a^{2} b d\right )} x^{2} +{\left (3 \, a^{2} b c + a^{3} d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \sqrt [5]{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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