Optimal. Leaf size=134 \[ -\frac{2 d^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac{2 d^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}} \]
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Rubi [A] time = 0.0876632, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {47, 63, 331, 298, 205, 208} \[ -\frac{2 d^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac{2 d^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 331
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/4}}{(a+b x)^{9/4}} \, dx &=-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac{d \int \frac{\sqrt [4]{c+d x}}{(a+b x)^{5/4}} \, dx}{b}\\ &=-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac{d^2 \int \frac{1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{b^2}\\ &=-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (c-\frac{a d}{b}+\frac{d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{b^3}\\ &=-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{d x^4}{b}} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{b^3}\\ &=-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}+\frac{\left (2 d^{3/2}\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 )}{b^2}-\frac{\left (2 d^{3/2}\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 )}{b^2}\\ &=-\frac{4 d \sqrt [4]{c+d x}}{b^2 \sqrt [4]{a+b x}}-\frac{4 (c+d x)^{5/4}}{5 b (a+b x)^{5/4}}-\frac{2 d^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}+\frac{2 d^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0580159, size = 73, normalized size = 0.54 \[ -\frac{4 (c+d x)^{5/4} \, _2F_1\left (-\frac{5}{4},-\frac{5}{4};-\frac{1}{4};\frac{d (a+b x)}{a d-b c}\right )}{5 b (a+b x)^{5/4} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \right ) ^{-{\frac{9}{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 (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.78767, size = 842, normalized size = 6.28 \begin{align*} -\frac{20 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} b^{7} d \left (\frac{d^{5}}{b^{9}}\right )^{\frac{3}{4}} -{\left (b^{8} x + a b^{7}\right )} \sqrt{\frac{\sqrt{b x + a} \sqrt{d x + c} d^{2} +{\left (b^{5} x + a b^{4}\right )} \sqrt{\frac{d^{5}}{b^{9}}}}{b x + a}} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{3}{4}}}{b d^{5} x + a d^{5}}\right ) - 5 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} d +{\left (b^{3} x + a b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}}}{b x + a}\right ) + 5 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} d -{\left (b^{3} x + a b^{2}\right )} \left (\frac{d^{5}}{b^{9}}\right )^{\frac{1}{4}}}{b x + a}\right ) + 4 \,{\left (6 \, 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 (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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