Optimal. Leaf size=301 \[ -\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {4 (b e-a f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f)^2 (e+f x)}+\frac {4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} (d e-c f)^{7/3}}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{(d e-c f)^{7/3}} \]
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Rubi [A]
time = 0.11, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {96, 93}
\begin {gather*} \frac {4 (b c-a d) \sqrt [3]{b e-a f} \text {ArcTan}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} (d e-c f)^{7/3}}-\frac {3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}+\frac {4 \sqrt [3]{a+b x} (c+d x)^{2/3} (b e-a f)}{(e+f x) (d e-c f)^2}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{(d e-c f)^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx &=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {(4 (b e-a f)) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx}{d e-c f}\\ &=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {4 (b e-a f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f)^2 (e+f x)}-\frac {(4 (b c-a d) (b e-a f)) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{3 (d e-c f)^2}\\ &=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {4 (b e-a f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f)^2 (e+f x)}+\frac {4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} (d e-c f)^{7/3}}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{(d e-c f)^{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 = 123, normalized size = 0.41 \begin {gather*} \frac {\sqrt [3]{a+b x} \left (b (4 c e+d e x+3 c f x)-a (3 d e+c f+4 d f x)-4 (b c-a d) (e+f x) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(d e-c f)^2 \sqrt [3]{c+d x} (e+f x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{2}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 662 vs.
\(2 (275) = 550\).
time = 1.79, size = 662, normalized size = 2.20 \begin {gather*} \frac {4 \, \sqrt {3} {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} f x + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (c f - d e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} + \sqrt {3} {\left (a d f x + a c f - {\left (b d x + b c\right )} e\right )}}{3 \, {\left (a d f x + a c f - {\left (b d x + b c\right )} e\right )}}\right ) + 2 \, {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} f x + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {a f - b e}{c f - d e}\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 ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} f x + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {a f - b e}{c f - d e}\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 (a c f - {\left (3 \, b c - 4 \, a d\right )} f x - {\left (b d x + 4 \, b c - 3 \, a d\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{3 \, {\left (c^{2} d f^{3} x^{2} + c^{3} f^{3} x + {\left (d^{3} x + c d^{2}\right )} e^{3} + {\left (d^{3} f x^{2} - c d^{2} f x - 2 \, c^{2} d f\right )} e^{2} - {\left (2 \, c d^{2} f^{2} x^{2} + c^{2} d f^{2} x - c^{3} f^{2}\right )} e\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}} \left (e + f x\right )^{2}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (e+f\,x\right )}^2\,{\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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