Optimal. Leaf size=562 \[ \frac {3 (d e-c f)^2 (a+b x)^{7/3}}{d^2 (b c-a d) \sqrt [3]{c+d x}}-\frac {4 \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^4}+\frac {\left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{9 b d^3 (b c-a d)}+\frac {f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2}-\frac {4 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} b^{5/3} d^{13/3}}-\frac {2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \log (a+b x)}{81 b^{5/3} d^{13/3}}-\frac {2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{27 b^{5/3} d^{13/3}} \]
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Rubi [A]
time = 0.35, antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {91, 81, 52, 61}
\begin {gather*} -\frac {4 (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 ) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-\left (b^2 \left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{27 \sqrt {3} b^{5/3} d^{13/3}}-\frac {4 \sqrt [3]{a+b x} (c+d x)^{2/3} \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-\left (b^2 \left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{27 b d^4}+\frac {(a+b x)^{4/3} (c+d x)^{2/3} \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-\left (b^2 \left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{9 b d^3 (b c-a d)}-\frac {2 (b c-a d) \log (a+b x) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-\left (b^2 \left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{81 b^{5/3} d^{13/3}}-\frac {2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-\left (b^2 \left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right ) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{27 b^{5/3} d^{13/3}}+\frac {3 (a+b x)^{7/3} (d e-c f)^2}{d^2 \sqrt [3]{c+d x} (b c-a d)}+\frac {f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rule 81
Rule 91
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3} (e+f x)^2}{(c+d x)^{4/3}} \, dx &=\frac {3 (d e-c f)^2 (a+b x)^{7/3}}{d^2 (b c-a d) \sqrt [3]{c+d x}}-\frac {3 \int \frac {(a+b x)^{4/3} \left (\frac {1}{3} \left (a d f (2 d e-c f)+b \left (6 d^2 e^2-14 c d e f+7 c^2 f^2\right )\right )-\frac {1}{3} d (b c-a d) f^2 x\right )}{\sqrt [3]{c+d x}} \, dx}{d^2 (b c-a d)}\\ &=\frac {3 (d e-c f)^2 (a+b x)^{7/3}}{d^2 (b c-a d) \sqrt [3]{c+d x}}+\frac {f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2}+\frac {\left (2 \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right )\right ) \int \frac {(a+b x)^{4/3}}{\sqrt [3]{c+d x}} \, dx}{9 b d^2 (b c-a d)}\\ &=\frac {3 (d e-c f)^2 (a+b x)^{7/3}}{d^2 (b c-a d) \sqrt [3]{c+d x}}+\frac {\left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{9 b d^3 (b c-a d)}+\frac {f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2}-\frac {\left (4 \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right )\right ) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{27 b d^3}\\ &=\frac {3 (d e-c f)^2 (a+b x)^{7/3}}{d^2 (b c-a d) \sqrt [3]{c+d x}}-\frac {4 \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^4}+\frac {\left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{9 b d^3 (b c-a d)}+\frac {f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2}+\frac {\left (4 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right )\right ) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{81 b d^4}\\ &=\frac {3 (d e-c f)^2 (a+b x)^{7/3}}{d^2 (b c-a d) \sqrt [3]{c+d x}}-\frac {4 \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^4}+\frac {\left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{9 b d^3 (b c-a d)}+\frac {f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2}-\frac {4 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \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 )}{27 \sqrt {3} b^{5/3} d^{13/3}}-\frac {2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \log (a+b x)}{81 b^{5/3} d^{13/3}}-\frac {2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)-b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{27 b^{5/3} d^{13/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.19, size = 175, normalized size = 0.31 \begin {gather*} \frac {(a+b x)^{7/3} \left (-63 (d e-c f)^2-\frac {7 (b c-a d) f^2 (c+d x)}{b}+\frac {2 \left (-a^2 d^2 f^2+a b d f (9 d e-7 c f)+b^2 \left (27 d^2 e^2-63 c d e f+35 c^2 f^2\right )\right ) \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 )}{b^2}\right )}{21 d^2 (-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 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}} \left (f x +e \right )^{2}}{\left (d x +c \right )^{\frac {4}{3}}}\, dx\]
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. 1106 vs.
\(2 (503) = 1006\).
time = 2.52, size = 2258, normalized size = 4.02 \begin {gather*} \text {Too large to display} \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 )^{2}}{\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 )}^2\,{\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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