Optimal. Leaf size=152 \[ \frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{5 \sqrt [4]{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac{5 \sqrt [4]{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]
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Rubi [A] time = 0.0871252, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {47, 50, 63, 240, 212, 208, 205} \[ \frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{5 \sqrt [4]{b} (b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac{5 \sqrt [4]{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 240
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx &=-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac{(5 b) \int \frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx}{d}\\ &=-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{(5 b (b c-a d)) \int \frac{1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 d^2}\\ &=-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{(5 (b c-a d)) \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^2}\\ &=-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{(5 (b c-a d)) \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^2}\\ &=-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{\left (5 \sqrt{b} (b c-a d)\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 )}{2 d^2}-\frac{\left (5 \sqrt{b} (b c-a d)\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 )}{2 d^2}\\ &=-\frac{4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac{5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac{5 \sqrt [4]{b} (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 d^{9/4}}-\frac{5 \sqrt [4]{b} (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 d^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0528923, size = 73, normalized size = 0.48 \[ \frac{4 (a+b x)^{9/4} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac{5}{4},\frac{9}{4};\frac{13}{4};\frac{d (a+b x)}{a d-b c}\right )}{9 b (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{5}{4}}} \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{5}{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 = 3.15997, size = 1818, normalized size = 11.96 \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{5}{4}}}{\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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