Optimal. Leaf size=30 \[ -\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ -\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx &=-\frac {4 \sqrt [4]{c+d x}}{(b c-a d) \sqrt [4]{a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x} (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 42, normalized size = 1.40 \[ -\frac {4 \, {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.90 \[ \frac {4 \left (d x +c \right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {1}{4}} \left (a d -b c \right )} \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 26, normalized size = 0.87 \[ \frac {4\,{\left (c+d\,x\right )}^{1/4}}{\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^{1/4}} \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{4}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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