Optimal. Leaf size=127 \[ \frac{3 (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac{3 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}+\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d} \]
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Rubi [A] time = 0.080577, antiderivative size = 127, 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 = {50, 63, 331, 298, 205, 208} \[ \frac{3 (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac{3 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}+\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 331
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx &=\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac{(3 (b c-a d)) \int \frac{1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{4 d}\\ &=\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac{(3 (b c-a d)) \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 d}\\ &=\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac{(3 (b c-a d)) \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 d}\\ &=\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac{(3 (b c-a d)) \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 )}{2 d^{3/2}}+\frac{(3 (b c-a d)) \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 )}{2 d^{3/2}}\\ &=\frac{(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}+\frac{3 (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac{3 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}\\ \end{align*}
Mathematica [C] time = 0.027152, size = 73, normalized size = 0.57 \[ \frac{4 (a+b x)^{7/4} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{7}{4};\frac{11}{4};\frac{d (a+b x)}{a d-b c}\right )}{7 b (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{3}{4}}} \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{{\left (b x + a\right )}^{\frac{3}{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 = 3.11552, size = 1721, normalized size = 13.55 \begin{align*} \text{result too large to display} \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{\left (a + b x\right )^{\frac{3}{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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