Optimal. Leaf size=72 \[ \frac {2 \sqrt {a+b x} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {3}{2};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [5]{c+d x}} \]
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Rubi [A] time = 0.02, antiderivative size = 72, 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 = {70, 69} \[ \frac {2 \sqrt {a+b x} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {3}{2};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt [5]{c+d x}} \, dx &=\frac {\sqrt [5]{\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \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 {a+b x} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {3}{2};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [5]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 0.99 \[ \frac {2 \sqrt {a+b x} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {3}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 5.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {4}{5}}}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {1}{5}}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{5}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{1/5}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b x} \sqrt [5]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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