Optimal. Leaf size=124 \[ \frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}-\frac {(a d (1-n)+b c (1+n)) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (1+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 120, normalized size of antiderivative = 0.97, 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 = {81, 72, 71}
\begin {gather*} \frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (\frac {a (1-n)}{n+1}+\frac {b c}{d}\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 81
Rubi steps
\begin {align*} \int x (a+b x)^n (c+d x)^{-n} \, dx &=\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}+\frac {1}{2} \left (-\frac {a (1-n)}{b}-\frac {c (1+n)}{d}\right ) \int (a+b x)^n (c+d x)^{-n} \, dx\\ &=\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}+\frac {1}{2} \left (\left (-\frac {a (1-n)}{b}-\frac {c (1+n)}{d}\right ) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx\\ &=\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 b d}-\frac {\left (\frac {b c}{d}+\frac {a (1-n)}{1+n}\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 101, normalized size = 0.81 \begin {gather*} \frac {(a+b x)^{1+n} (c+d x)^{-n} \left (b (c+d x)-\frac {(-a d (-1+n)+b c (1+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{2 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\, dx\]
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\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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