Optimal. Leaf size=99 \[ -\frac {c (a+b x)^{1+n}}{d (b c-a d) (c+d x)}-\frac {(a d-b c (1+n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d (b c-a d)^2 (1+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 70}
\begin {gather*} -\frac {(a+b x)^{n+1} (a d-b c (n+1)) \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)^2}-\frac {c (a+b x)^{n+1}}{d (c+d x) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 79
Rubi steps
\begin {align*} \int \frac {x (a+b x)^n}{(c+d x)^2} \, dx &=-\frac {c (a+b x)^{1+n}}{d (b c-a d) (c+d x)}+\frac {(a d-b c (1+n)) \int \frac {(a+b x)^n}{c+d x} \, dx}{d (-b c+a d)}\\ &=-\frac {c (a+b x)^{1+n}}{d (b c-a d) (c+d x)}-\frac {(a d-b c (1+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 (b c-a d)^2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 83, normalized size = 0.84 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {c (-b c+a d)}{c+d x}+\frac {(-a d+b c (1+n)) \, _2F_1\left (1,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{d (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (b x +a \right )^{n}}{\left (d x +c \right )^{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b x\right )^{n}}{\left (c + d x\right )^{2}}\, 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\,{\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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