Optimal. Leaf size=380 \[ \frac {3 (b c-a d) \sqrt [3]{a+b x}}{d (d e-c f) \sqrt [3]{c+d x}}-\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{d^{4/3} f}+\frac {\sqrt {3} (b e-a f)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{f (d e-c f)^{4/3}}-\frac {b^{4/3} \log (a+b x)}{2 d^{4/3} f}-\frac {(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac {3 (b e-a f)^{4/3} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f (d e-c f)^{4/3}}-\frac {3 b^{4/3} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 d^{4/3} f} \]
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Rubi [A]
time = 0.24, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {100, 163, 61,
93} \begin {gather*} -\frac {\sqrt {3} b^{4/3} \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 )}{d^{4/3} f}+\frac {\sqrt {3} (b e-a f)^{4/3} \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 )}{f (d e-c f)^{4/3}}-\frac {3 b^{4/3} \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3} f}-\frac {b^{4/3} \log (a+b x)}{2 d^{4/3} f}+\frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (d e-c f)}-\frac {(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac {3 (b e-a f)^{4/3} \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 )}{2 f (d e-c f)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 61
Rule 93
Rule 100
Rule 163
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx &=\frac {3 (b c-a d) \sqrt [3]{a+b x}}{d (d e-c f) \sqrt [3]{c+d x}}-\frac {3 \int \frac {\frac {1}{3} \left (b^2 c e-2 a b d e+a^2 d f\right )-\frac {1}{3} b^2 (d e-c f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{d (d e-c f)}\\ &=\frac {3 (b c-a d) \sqrt [3]{a+b x}}{d (d e-c f) \sqrt [3]{c+d x}}+\frac {b^2 \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{d f}-\frac {(b e-a f)^2 \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{f (d e-c f)}\\ &=\frac {3 (b c-a d) \sqrt [3]{a+b x}}{d (d e-c f) \sqrt [3]{c+d x}}-\frac {\sqrt {3} b^{4/3} \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 )}{d^{4/3} f}+\frac {\sqrt {3} (b e-a f)^{4/3} \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 )}{f (d e-c f)^{4/3}}-\frac {b^{4/3} \log (a+b x)}{2 d^{4/3} f}-\frac {(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac {3 (b e-a f)^{4/3} \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 )}{2 f (d e-c f)^{4/3}}-\frac {3 b^{4/3} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 d^{4/3} f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.17, size = 148, normalized size = 0.39 \begin {gather*} \frac {3 \sqrt [3]{a+b x} \left (-\frac {4 (b e-a f) \left (-1+\, _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}+\frac {(a+b x) \left (\frac {b (c+d x)}{b c-a d}\right )^{4/3} \, _2F_1\left (\frac {4}{3},\frac {4}{3};\frac {7}{3};\frac {d (a+b x)}{-b c+a d}\right )}{c+d x}\right )}{4 f \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )}\, dx\]
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. 751 vs.
\(2 (316) = 632\).
time = 2.20, size = 751, normalized size = 1.98 \begin {gather*} -\frac {6 \, {\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} f + 2 \, \sqrt {3} {\left (a d^{2} f x + a c d f - {\left (b d^{2} x + b c d\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 \, \sqrt {3} {\left (b c d f x + b c^{2} f - {\left (b d^{2} x + b c d\right )} e\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 ) + {\left (a d^{2} f x + a c d f - {\left (b d^{2} x + b c d\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 ) + {\left (b c d f x + b c^{2} f - {\left (b d^{2} x + b c d\right )} e\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 ) - 2 \, {\left (a d^{2} f x + a c d f - {\left (b d^{2} x + b c d\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 ) - 2 \, {\left (b c d f x + b c^{2} f - {\left (b d^{2} x + b c d\right )} e\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 )}{2 \, {\left (c d^{2} f^{2} x + c^{2} d f^{2} - {\left (d^{3} f x + c d^{2} f\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 )}\, 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 )\,{\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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