Optimal. Leaf size=328 \[ \frac {(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac {2 (b c-a d) (a d f-7 b c f+6 b d e) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3} d^{10/3}}+\frac {2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}-\frac {(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac {3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \]
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Rubi [A] time = 0.24, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 50, 59} \[ \frac {(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac {2 (b c-a d) (a d f-7 b c f+6 b d e) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3} d^{10/3}}-\frac {(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac {2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}+\frac {3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 50
Rule 59
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3} (e+f x)}{(c+d x)^{4/3}} \, dx &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}-\frac {(6 b d e-7 b c f+a d f) \int \frac {(a+b x)^{4/3}}{\sqrt [3]{c+d x}} \, dx}{d (b c-a d)}\\ &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}+\frac {(2 (6 b d e-7 b c f+a d f)) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{3 d^2}\\ &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}+\frac {2 (6 b d e-7 b c f+a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}-\frac {(2 (b c-a d) (6 b d e-7 b c f+a d f)) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 d^3}\\ &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}+\frac {2 (6 b d e-7 b c f+a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}+\frac {2 (b c-a d) (6 b d e-7 b c f+a d f) \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 )}{3 \sqrt {3} b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log (a+b x)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 b^{2/3} d^{10/3}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 114, normalized size = 0.35 \[ \frac {3 (a+b x)^{7/3} \left (\sqrt [3]{\frac {b (c+d x)}{b c-a d}} (a d f-7 b c f+6 b d e) \, _2F_1\left (\frac {1}{3},\frac {7}{3};\frac {10}{3};\frac {d (a+b x)}{a d-b c}\right )+b (7 c f-7 d e)\right )}{7 b d \sqrt [3]{c+d x} (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.30, size = 1386, normalized size = 4.23 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {4}{3}} {\left (f x + e\right )}}{{\left (d x + c\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {4}{3}} \left (f x +e \right )}{\left (d x +c \right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {4}{3}} {\left (f x + e\right )}}{{\left (d x + c\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^{4/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {4}{3}} \left (e + f x\right )}{\left (c + d x\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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