Optimal. Leaf size=328 \[ \frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}+\frac {2 (6 b d e-7 b c f+a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}+\frac {2 (b c-a d) (6 b d e-7 b c f+a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log (a+b x)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 b^{2/3} d^{10/3}} \]
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Rubi [A]
time = 0.16, 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 = {79, 52, 61}
\begin {gather*} \frac {2 (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 ) (a d f-7 b c f+6 b d e)}{3 \sqrt {3} b^{2/3} d^{10/3}}+\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 \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rule 79
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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.12, size = 114, normalized size = 0.35 \begin {gather*} \frac {3 (a+b x)^{7/3} \left (b (-7 d e+7 c f)+(6 b d e-7 b c f+a d f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{3},\frac {7}{3};\frac {10}{3};\frac {d (a+b x)}{-b c+a d}\right )\right )}{7 b d (-b c+a d) \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \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. 656 vs.
\(2 (285) = 570\).
time = 1.72, size = 1358, normalized size = 4.14 \begin {gather*} \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, b^{3} c^{2} d^{2} - 8 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} f x + {\left (7 \, b^{3} c^{3} d - 8 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} f - 6 \, {\left (b^{3} c^{2} d^{2} - a b^{2} c d^{3} + {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x\right )} e\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (3 \, b^{2} d x + b^{2} c + 2 \, a b d + 3 \, \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}\right ) + 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 4 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{3} f x^{2} - 7 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f x - {\left (28 \, b^{3} c^{2} d - 25 \, a b^{2} c d^{2}\right )} f + 6 \, {\left (b^{3} d^{3} x + 4 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}, \frac {12 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, b^{3} c^{2} d^{2} - 8 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} f x + {\left (7 \, b^{3} c^{3} d - 8 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} f - 6 \, {\left (b^{3} c^{2} d^{2} - a b^{2} c d^{3} + {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x\right )} e\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}}{b^{2} d x + b^{2} c}\right ) + 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 4 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{3} f x^{2} - 7 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f x - {\left (28 \, b^{3} c^{2} d - 25 \, a b^{2} c d^{2}\right )} f + 6 \, {\left (b^{3} d^{3} x + 4 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}\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 (e + f x\right )}{\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 = -0.00 \begin {gather*} \int \frac {\left (e+f\,x\right )\,{\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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