Optimal. Leaf size=108 \[ -\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d) (1+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {90, 70}
\begin {gather*} -\frac {(a d+b c) (a+b x)^{n+1}}{b^2 d^2 (n+1)}+\frac {(a+b x)^{n+2}}{b^2 d (n+2)}+\frac {c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (n+1) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 90
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)^n}{c+d x} \, dx &=\int \left (\frac {(-b c-a d) (a+b x)^n}{b d^2}+\frac {(a+b x)^{1+n}}{b d}+\frac {c^2 (a+b x)^n}{d^2 (c+d x)}\right ) \, dx\\ &=-\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 \int \frac {(a+b x)^n}{c+d x} \, dx}{d^2}\\ &=-\frac {(b c+a d) (a+b x)^{1+n}}{b^2 d^2 (1+n)}+\frac {(a+b x)^{2+n}}{b^2 d (2+n)}+\frac {c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d) (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 100, normalized size = 0.93 \begin {gather*} \frac {(a+b x)^{1+n} \left (-((b c-a d) (a d+b c (2+n)-b d (1+n) x))+b^2 c^2 (2+n) \, _2F_1\left (1,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )\right )}{b^2 d^2 (b c-a d) (1+n) (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (b x +a \right )^{n}}{d x +c}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {x^{2} \left (a + b x\right )^{n}}{c + d x}\, dx \end {gather*}
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*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {x^2\,{\left (a+b\,x\right )}^n}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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