Optimal. Leaf size=141 \[ -\frac {c d \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{b^{2/3}}-\frac {2 c d \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {c^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac {d^2 \sqrt [3]{a+b x^3}}{b} \]
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Rubi [A] time = 0.15, antiderivative size = 195, normalized size of antiderivative = 1.38, number of steps used = 13, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {1886, 261, 1893, 246, 245, 331, 292, 31, 634, 617, 204, 628} \[ -\frac {2 c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac {c d \log \left (\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {2 c d \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {c^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac {d^2 \sqrt [3]{a+b x^3}}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 245
Rule 246
Rule 261
Rule 292
Rule 331
Rule 617
Rule 628
Rule 634
Rule 1886
Rule 1893
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{\left (a+b x^3\right )^{2/3}} \, dx &=d^2 \int \frac {x^2}{\left (a+b x^3\right )^{2/3}} \, dx+\int \frac {c^2+2 c d x}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}+\int \left (\frac {c^2}{\left (a+b x^3\right )^{2/3}}+\frac {2 c d x}{\left (a+b x^3\right )^{2/3}}\right ) \, dx\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}+c^2 \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx+(2 c d) \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}+(2 c d) \operatorname {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {\left (c^2 \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}+\frac {c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}+\frac {(2 c d) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}-\frac {(2 c d) \operatorname {Subst}\left (\int \frac {1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}+\frac {c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac {2 c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac {(c d) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}-\frac {(c d) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}+\frac {c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac {2 c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac {c d \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 )}{3 b^{2/3}}+\frac {(2 c d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{b^{2/3}}\\ &=\frac {d^2 \sqrt [3]{a+b x^3}}{b}-\frac {2 c d \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {c^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}-\frac {2 c d \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3}}+\frac {c d \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 )}{3 b^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 95, normalized size = 0.67 \[ \frac {b c^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+d \left (b c x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {b x^3}{b x^3+a}\right )+d \left (a+b x^3\right )\right )}{b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 54.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{2} x^{2} + 2 \, c d x + c^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {\left (d x +c \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^2}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.33, size = 109, normalized size = 0.77 \[ d^{2} \left (\begin {cases} \frac {x^{3}}{3 a^{\frac {2}{3}}} & \text {for}\: b = 0 \\\frac {\sqrt [3]{a + b x^{3}}}{b} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {2 c d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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