Optimal. Leaf size=108 \[ \frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}} \]
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Rubi [A] time = 0.0652558, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {47, 63, 240, 212, 208, 205} \[ \frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 240
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx &=-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}+\frac{b \int \frac{1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{d}\\ &=-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{c-\frac{a d}{b}+\frac{d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{d}\\ &=-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^4}{b}} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{d}\\ &=-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}+\frac{\left (2 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}-\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{d}+\frac{\left (2 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}+\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{d}\\ &=-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}+\frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}\\ \end{align*}
Mathematica [C] time = 0.0434086, size = 73, normalized size = 0.68 \[ \frac{4 (a+b x)^{5/4} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};\frac{d (a+b x)}{a d-b c}\right )}{5 b (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{bx+a} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, 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{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47309, size = 656, normalized size = 6.07 \begin{align*} -\frac{4 \,{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} d^{4} \left (\frac{b}{d^{5}}\right )^{\frac{3}{4}} -{\left (d^{5} x + c d^{4}\right )} \sqrt{\frac{{\left (d^{3} x + c d^{2}\right )} \sqrt{\frac{b}{d^{5}}} + \sqrt{b x + a} \sqrt{d x + c}}{d x + c}} \left (\frac{b}{d^{5}}\right )^{\frac{3}{4}}}{b d x + b c}\right ) -{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) +{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) + 4 \,{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d^{2} x + c d} \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{\sqrt [4]{a + b x}}{\left (c + d x\right )^{\frac{5}{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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