Optimal. Leaf size=253 \[ -\frac {(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac {(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac {(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )} \]
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Rubi [A] time = 0.59, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6725, 68} \[ -\frac {(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac {(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac {(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 6725
Rubi steps
\begin {align*} \int \frac {x^2 (c+d x)^{1+n}}{a+b x^3} \, dx &=\int \left (\frac {(c+d x)^{1+n}}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(c+d x)^{1+n}}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(c+d x)^{1+n}}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx\\ &=\frac {\int \frac {(c+d x)^{1+n}}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {(c+d x)^{1+n}}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {(c+d x)^{1+n}}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}\\ &=-\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right ) (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 213, normalized size = 0.84 \[ \frac {(c+d x)^{n+2} \left (-\frac {\, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}-\frac {\, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}-\frac {\, _2F_1\left (1,n+2;n+3;\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}\,{d x} \]
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 \[ \int \frac {x^{2} \left (d x +c \right )^{n +1}}{b \,x^{3}+a}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (c+d\,x\right )}^{n+1}}{b\,x^3+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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