Optimal. Leaf size=66 \[ \frac{16 d \sqrt [4]{c+d x}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac{4 \sqrt [4]{c+d x}}{5 (a+b x)^{5/4} (b c-a d)} \]
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Rubi [A] time = 0.0093289, antiderivative size = 66, 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 = {45, 37} \[ \frac{16 d \sqrt [4]{c+d x}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac{4 \sqrt [4]{c+d x}}{5 (a+b x)^{5/4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{9/4} (c+d x)^{3/4}} \, dx &=-\frac{4 \sqrt [4]{c+d x}}{5 (b c-a d) (a+b x)^{5/4}}-\frac{(4 d) \int \frac{1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx}{5 (b c-a d)}\\ &=-\frac{4 \sqrt [4]{c+d x}}{5 (b c-a d) (a+b x)^{5/4}}+\frac{16 d \sqrt [4]{c+d x}}{5 (b c-a d)^2 \sqrt [4]{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0145438, size = 46, normalized size = 0.7 \[ \frac{4 \sqrt [4]{c+d x} (5 a d-b c+4 b d x)}{5 (a+b x)^{5/4} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 54, normalized size = 0.8 \begin{align*}{\frac{16\,bdx+20\,ad-4\,bc}{5\,{a}^{2}{d}^{2}-10\,abcd+5\,{b}^{2}{c}^{2}}\sqrt [4]{dx+c} \left ( bx+a \right ) ^{-{\frac{5}{4}}}} \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{9}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23628, size = 255, normalized size = 3.86 \begin{align*} \frac{4 \,{\left (4 \, b d x - b c + 5 \, a d\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{5 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} 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{9}{4}} \left (c + d x\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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