Optimal. Leaf size=369 \[ -\frac {\log (c+d x) \left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\right )\right )}{18 b^{7/3} d^{8/3}}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\right )\right ) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 b^{7/3} d^{8/3}}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{7/3} d^{8/3}}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-4 a d f-5 b c f+9 b d e)}{6 b^2 d^2}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}{2 b d} \]
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Rubi [A] time = 0.30, antiderivative size = 369, 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 = {90, 80, 59} \begin {gather*} -\frac {\log (c+d x) \left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\right )\right )}{18 b^{7/3} d^{8/3}}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\right )\right ) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 b^{7/3} d^{8/3}}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\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 )}{3 \sqrt {3} b^{7/3} d^{8/3}}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-4 a d f-5 b c f+9 b d e)}{6 b^2 d^2}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 80
Rule 90
Rubi steps
\begin {align*} \int \frac {(e+f x)^2}{\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 b d}+\frac {\int \frac {\frac {1}{3} \left (6 b d e^2-f (2 b c e+a d e+3 a c f)\right )+\frac {1}{3} f (9 b d e-5 b c f-4 a d f) x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{2 b d}\\ &=\frac {f (9 b d e-5 b c f-4 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b^2 d^2}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}{2 b d}+\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (9 d^2 e^2-12 c d e f+5 c^2 f^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{9 b^2 d^2}\\ &=\frac {f (9 b d e-5 b c f-4 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b^2 d^2}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}{2 b d}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (9 d^2 e^2-12 c d e f+5 c^2 f^2\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 )}{3 \sqrt {3} b^{7/3} d^{8/3}}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (9 d^2 e^2-12 c d e f+5 c^2 f^2\right )\right ) \log (c+d x)}{18 b^{7/3} d^{8/3}}-\frac {\left (2 a^2 d^2 f^2-2 a b d f (3 d e-c f)+b^2 \left (9 d^2 e^2-12 c d e f+5 c^2 f^2\right )\right ) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{6 b^{7/3} d^{8/3}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 174, normalized size = 0.47 \begin {gather*} \frac {(a+b x)^{2/3} \left (2 \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3} \left (2 a^2 d^2 f^2+2 a b d f (c f-3 d e)+b^2 \left (5 c^2 f^2-12 c d e f+9 d^2 e^2\right )\right ) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {d (a+b x)}{a d-b c}\right )-2 b f (c+d x) (4 a d f+5 b c f-9 b d e)+6 b^2 d f (c+d x) (e+f x)\right )}{12 b^3 d^2 (c+d x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.87, size = 499, normalized size = 1.35 \begin {gather*} \frac {\left (-2 a^2 d^2 f^2-2 a b c d f^2+6 a b d^2 e f-5 b^2 c^2 f^2+12 b^2 c d e f-9 b^2 d^2 e^2\right ) \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )}{9 b^{7/3} d^{8/3}}+\frac {\left (2 a^2 d^2 f^2+2 a b c d f^2-6 a b d^2 e f+5 b^2 c^2 f^2-12 b^2 c d e f+9 b^2 d^2 e^2\right ) \log \left (\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+d^{2/3}\right )}{18 b^{7/3} d^{8/3}}+\frac {\left (2 a^2 d^2 f^2+2 a b c d f^2-6 a b d^2 e f+5 b^2 c^2 f^2-12 b^2 c d e f+9 b^2 d^2 e^2\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 )}{3 \sqrt {3} b^{7/3} d^{8/3}}+\frac {f \sqrt [3]{c+d x} (b c-a d) \left (\frac {12 b^2 d e (c+d x)}{a+b x}-\frac {5 b^2 c f (c+d x)}{a+b x}-\frac {7 a b d f (c+d x)}{a+b x}+4 a d^2 f+8 b c d f-12 b d^2 e\right )}{6 b^2 d^2 \sqrt [3]{a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.02, size = 1001, normalized size = 2.71
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 )}^{2}}{{\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 = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f x +e \right )^{2}}{\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 )}^{2}}{{\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 )}^2}{{\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 )^{2}}{\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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