Optimal. Leaf size=122 \[ \frac {(a+b x)^{1+n}}{b d^2 (1+n)}+\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}+\frac {c (2 a d-b c (2+n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d^2 (b c-a d)^2 (1+n)} \]
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Rubi [A]
time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 81, 70}
\begin {gather*} \frac {c^2 (a+b x)^{n+1}}{d^2 (c+d x) (b c-a d)}+\frac {c (a+b x)^{n+1} (2 a d-b c (n+2)) \, _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)^2}+\frac {(a+b x)^{n+1}}{b d^2 (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 81
Rule 91
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)^n}{(c+d x)^2} \, dx &=\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}-\frac {\int \frac {(a+b x)^n (-c (a d-b c (1+n))-d (b c-a d) x)}{c+d x} \, dx}{d^2 (b c-a d)}\\ &=\frac {(a+b x)^{1+n}}{b d^2 (1+n)}+\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}+\frac {(c (2 a d-b c (2+n))) \int \frac {(a+b x)^n}{c+d x} \, dx}{d^2 (b c-a d)}\\ &=\frac {(a+b x)^{1+n}}{b d^2 (1+n)}+\frac {c^2 (a+b x)^{1+n}}{d^2 (b c-a d) (c+d x)}+\frac {c (2 a d-b c (2+n)) (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)^2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 115, normalized size = 0.94 \begin {gather*} \frac {(a+b x)^{1+n} \left ((b c-a d) (-a d (c+d x)+b c (c (2+n)+d x))-b c (-2 a d+b c (2+n)) (c+d x) \, _2F_1\left (1,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )\right )}{b d^2 (b c-a d)^2 (1+n) (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (b x +a \right )^{n}}{\left (d x +c \right )^{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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