Optimal. Leaf size=195 \[ -\frac {3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}}+\frac {4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}+\frac {4 \sqrt [3]{b} (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{\sqrt {3} d^{7/3}}+\frac {2 \sqrt [3]{b} (b c-a d) \log (a+b x)}{3 d^{7/3}}+\frac {2 \sqrt [3]{b} (b c-a d) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{d^{7/3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 195, normalized size of antiderivative = 1.00, 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 = {49, 52, 61}
\begin {gather*} \frac {4 \sqrt [3]{b} (b c-a d) \text {ArcTan}\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 {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 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}-\frac {3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 61
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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.05, size = 73, normalized size = 0.37 \begin {gather*} \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)}{-b c+a d}\right )}{7 b (c+d x)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.68, size = 306, normalized size = 1.57 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {4}{3}}}{\left (c + d x\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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