Optimal. Leaf size=587 \[ \frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2+3 b d f x (-7 a d f-8 b c f+15 b d e)-a b d f (108 d e-31 c f)+b^2 \left (40 c^2 f^2-135 c d e f+144 d^2 e^2\right )\right )}{54 b^3 d^3}+\frac {\log (c+d x) \left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )-\left (b^3 \left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right )}{162 b^{10/3} d^{11/3}}+\frac {\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )-\left (b^3 \left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{54 b^{10/3} d^{11/3}}+\frac {\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )-\left (b^3 \left (-40 c^3 f^3+135 c^2 d e f^2-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{27 \sqrt {3} b^{10/3} d^{11/3}}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d} \]
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Rubi [A] time = 0.48, antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {100, 147, 59} \begin {gather*} \frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2+3 b d f x (-7 a d f-8 b c f+15 b d e)-a b d f (108 d e-31 c f)+b^2 \left (40 c^2 f^2-135 c d e f+144 d^2 e^2\right )\right )}{54 b^3 d^3}+\frac {\log (c+d x) \left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right )}{162 b^{10/3} d^{11/3}}+\frac {\left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{54 b^{10/3} d^{11/3}}+\frac {\left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{27 \sqrt {3} b^{10/3} d^{11/3}}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 100
Rule 147
Rubi steps
\begin {align*} \int \frac {(e+f x)^3}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx &=\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d}+\frac {\int \frac {(e+f x) \left (\frac {1}{3} \left (9 b d e^2-f (2 b c e+a d e+6 a c f)\right )+\frac {1}{3} f (15 b d e-8 b c f-7 a d f) x\right )}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{3 b d}\\ &=\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2-a b d f (108 d e-31 c f)+b^2 \left (144 d^2 e^2-135 c d e f+40 c^2 f^2\right )+3 b d f (15 b d e-8 b c f-7 a d f) x\right )}{54 b^3 d^3}-\frac {\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{81 b^3 d^3}\\ &=\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2-a b d f (108 d e-31 c f)+b^2 \left (144 d^2 e^2-135 c d e f+40 c^2 f^2\right )+3 b d f (15 b d e-8 b c f-7 a d f) x\right )}{54 b^3 d^3}+\frac {\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{27 \sqrt {3} b^{10/3} d^{11/3}}+\frac {\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \log (c+d x)}{162 b^{10/3} d^{11/3}}+\frac {\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{54 b^{10/3} d^{11/3}}\\ \end {align*}
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Mathematica [C] time = 0.79, size = 291, normalized size = 0.50 \begin {gather*} \frac {(a+b x)^{2/3} \left (\frac {f^2 (c+d x)^2 (-7 a d f-8 b c f+15 b d e) \, _2F_1\left (-\frac {4}{3},\frac {2}{3};\frac {5}{3};\frac {d (a+b x)}{a d-b c}\right )}{b d^2 \left (\frac {b (c+d x)}{b c-a d}\right )^{4/3}}+\frac {(d e-c f)^2 \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3} (-a d f-8 b c f+9 b d e) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {d (a+b x)}{a d-b c}\right )}{b d^2}+\frac {8 f (c+d x) (c f-d e) (a d f+2 b c f-3 b d e) \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {d (a+b x)}{a d-b c}\right )}{b d^2 \sqrt [3]{\frac {b (c+d x)}{b c-a d}}}+2 f (c+d x) (e+f x)^2\right )}{6 b d (c+d x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.42, size = 978, normalized size = 1.67 \begin {gather*} \frac {(b c-a d) f \sqrt [3]{c+d x} \left (\frac {162 d^2 e^2 (c+d x)^2 b^4}{(a+b x)^2}+\frac {40 c^2 f^2 (c+d x)^2 b^4}{(a+b x)^2}-\frac {135 c d e f (c+d x)^2 b^4}{(a+b x)^2}+\frac {55 a c d f^2 (c+d x)^2 b^3}{(a+b x)^2}-\frac {189 a d^2 e f (c+d x)^2 b^3}{(a+b x)^2}-\frac {324 d^3 e^2 (c+d x) b^3}{a+b x}-\frac {104 c^2 d f^2 (c+d x) b^3}{a+b x}+\frac {351 c d^2 e f (c+d x) b^3}{a+b x}+162 d^4 e^2 b^2+82 c^2 d^2 f^2 b^2+\frac {67 a^2 d^2 f^2 (c+d x)^2 b^2}{(a+b x)^2}-216 c d^3 e f b^2-\frac {143 a c d^2 f^2 (c+d x) b^2}{a+b x}+\frac {297 a d^3 e f (c+d x) b^2}{a+b x}+52 a c d^3 f^2 b-108 a d^4 e f b-\frac {77 a^2 d^3 f^2 (c+d x) b}{a+b x}+28 a^2 d^4 f^2\right )}{54 b^3 d^3 \sqrt [3]{a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^3}+\frac {\left (81 d^3 e^3 b^3-40 c^3 f^3 b^3+135 c^2 d e f^2 b^3-162 c d^2 e^2 f b^3-15 a c^2 d f^3 b^2+54 a c d^2 e f^2 b^2-81 a d^3 e^2 f b^2-12 a^2 c d^2 f^3 b+54 a^2 d^3 e f^2 b-14 a^3 d^3 f^3\right ) \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 )}{27 \sqrt {3} b^{10/3} d^{11/3}}+\frac {\left (-81 d^3 e^3 b^3+40 c^3 f^3 b^3-135 c^2 d e f^2 b^3+162 c d^2 e^2 f b^3+15 a c^2 d f^3 b^2-54 a c d^2 e f^2 b^2+81 a d^3 e^2 f b^2+12 a^2 c d^2 f^3 b-54 a^2 d^3 e f^2 b+14 a^3 d^3 f^3\right ) \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )}{81 b^{10/3} d^{11/3}}+\frac {\left (81 d^3 e^3 b^3-40 c^3 f^3 b^3+135 c^2 d e f^2 b^3-162 c d^2 e^2 f b^3-15 a c^2 d f^3 b^2+54 a c d^2 e f^2 b^2-81 a d^3 e^2 f b^2-12 a^2 c d^2 f^3 b+54 a^2 d^3 e f^2 b-14 a^3 d^3 f^3\right ) \log \left (d^{2/3}+\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{d}}{\sqrt [3]{a+b x}}+\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{162 b^{10/3} d^{11/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.40, size = 1481, normalized size = 2.52
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 \begin {gather*} \int \frac {{\left (f x + e\right )}^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f x +e \right )^{3}}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}\, dx \end {gather*}
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*} \int \frac {{\left (f x + e\right )}^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}\,{d x} \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 )}^3}{{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \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 (e + f x\right )^{3}}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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