Optimal. Leaf size=85 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \]
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Rubi [A] time = 0.0665139, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {63, 331, 298, 205, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 331
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx &=\frac{4 \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}\\ &=\frac{4 \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}\\ &=\frac{2 \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 )}{\sqrt{d}}-\frac{2 \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 )}{\sqrt{d}}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.029191, size = 73, normalized size = 0.86 \[ \frac{4 (a+b x)^{3/4} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{3}{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{1}{{\left (b x + a\right )}^{\frac{1}{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 = 2.43529, size = 581, normalized size = 6.84 \begin{align*} -4 \, \left (\frac{1}{b d^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}} b d^{2} \left (\frac{1}{b d^{3}}\right )^{\frac{3}{4}} -{\left (b^{2} d^{2} x + a b d^{2}\right )} \sqrt{\frac{{\left (b d^{2} x + a d^{2}\right )} \sqrt{\frac{1}{b d^{3}}} + \sqrt{b x + a} \sqrt{d x + c}}{b x + a}} \left (\frac{1}{b d^{3}}\right )^{\frac{3}{4}}}{b x + a}\right ) + \left (\frac{1}{b d^{3}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (b d x + a d\right )} \left (\frac{1}{b d^{3}}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{b x + a}\right ) - \left (\frac{1}{b d^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (b d x + a d\right )} \left (\frac{1}{b d^{3}}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{b x + a}\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}{\sqrt [4]{a + b x} \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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