Integrand size = 19, antiderivative size = 147 \[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b}+\frac {c^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {c d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}-\frac {c^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}} \]
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Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1900, 267, 1907, 245, 372, 371} \[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {c^2 \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {c^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}+\frac {c d x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}+\frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b} \]
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Rule 245
Rule 267
Rule 371
Rule 372
Rule 1900
Rule 1907
Rubi steps \begin{align*} \text {integral}& = d^2 \int \frac {x^2}{\sqrt [3]{a+b x^3}} \, dx+\int \frac {c^2+2 c d x}{\sqrt [3]{a+b x^3}} \, dx \\ & = \frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b}+\int \left (\frac {c^2}{\sqrt [3]{a+b x^3}}+\frac {2 c d x}{\sqrt [3]{a+b x^3}}\right ) \, dx \\ & = \frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b}+c^2 \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx+(2 c d) \int \frac {x}{\sqrt [3]{a+b x^3}} \, dx \\ & = \frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b}+\frac {c^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {c^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}}+\frac {\left (2 c d \sqrt [3]{1+\frac {b x^3}{a}}\right ) \int \frac {x}{\sqrt [3]{1+\frac {b x^3}{a}}} \, dx}{\sqrt [3]{a+b x^3}} \\ & = \frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b}+\frac {c^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {c d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}-\frac {c^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}} \\ \end{align*}
Time = 9.64 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.37 \[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {d^2 \left (a+b x^3\right )^{2/3}}{2 b}+\frac {c^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {c d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}-\frac {c^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {c^2 \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{b}} \]
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\[\int \frac {\left (d x +c \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 1.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.75 \[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=d^{2} \left (\begin {cases} \frac {x^{3}}{3 \sqrt [3]{a}} & \text {for}\: b = 0 \\\frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{2 b} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {2 c d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} \]
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\[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^2}{\sqrt [3]{a+b x^3}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (b\,x^3+a\right )}^{1/3}} \,d x \]
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