Optimal. Leaf size=195 \[ \frac{4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}+\frac{2 \sqrt [3]{b} (b c-a d) \log (a+b x)}{3 d^{7/3}}+\frac{2 \sqrt [3]{b} (b c-a d) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{d^{7/3}}+\frac{4 \sqrt [3]{b} (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} d^{7/3}}-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0706522, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {47, 50, 59} \[ \frac{4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}+\frac{2 \sqrt [3]{b} (b c-a d) \log (a+b x)}{3 d^{7/3}}+\frac{2 \sqrt [3]{b} (b c-a d) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{d^{7/3}}+\frac{4 \sqrt [3]{b} (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} d^{7/3}}-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3}} \, dx &=-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}}+\frac{(4 b) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{d}\\ &=-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}}+\frac{4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}-\frac{(4 b (b c-a d)) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{3 d^2}\\ &=-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}}+\frac{4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}+\frac{4 \sqrt [3]{b} (b c-a d) \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{\sqrt{3} d^{7/3}}+\frac{2 \sqrt [3]{b} (b c-a d) \log (a+b x)}{3 d^{7/3}}+\frac{2 \sqrt [3]{b} (b c-a d) \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{d^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.0482111, size = 73, normalized size = 0.37 \[ \frac{3 (a+b x)^{7/3} \left (\frac{b (c+d x)}{b c-a d}\right )^{4/3} \, _2F_1\left (\frac{4}{3},\frac{7}{3};\frac{10}{3};\frac{d (a+b x)}{a d-b c}\right )}{7 b (c+d x)^{4/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.39372, size = 745, normalized size = 3.82 \begin{align*} \frac{4 \, \sqrt{3}{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} d \left (-\frac{b}{d}\right )^{\frac{2}{3}} + \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right ) + 2 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 4 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) + 3 \,{\left (b d x + 4 \, b c - 3 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{3 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{4}{3}}}{\left (c + d x\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]