Optimal. Leaf size=117 \[ -\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 a c x^2}-\frac {(b c-a d) (a d (1+n)+b (c-c n)) (a+b x)^{1+n} (c+d x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {98, 133}
\begin {gather*} -\frac {(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} (a d (n+1)+b (c-c n)) \, _2F_1\left (2,n+1;n+2;\frac {c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (n+1)}-\frac {(a+b x)^{n+1} (c+d x)^{1-n}}{2 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 98
Rule 133
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^{-n}}{x^3} \, dx &=-\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 a c x^2}-\frac {(a d (1+n)+b (c-c n)) \int \frac {(a+b x)^n (c+d x)^{-n}}{x^2} \, dx}{2 a c}\\ &=-\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{2 a c x^2}-\frac {(b c-a d) (a d (1+n)+b (c-c n)) (a+b x)^{1+n} (c+d x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 99, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{1+n} (c+d x)^{-1-n} \left (-\frac {a^2 (c+d x)^2}{x^2}+\frac {(b c-a d) (b c (-1+n)-a d (1+n)) \, _2F_1\left (2,1+n;2+n;\frac {c (a+b x)}{a (c+d x)}\right )}{1+n}\right )}{2 a^3 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{-n}}{x^{3}}\, dx\]
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 {{\left (a+b\,x\right )}^n}{x^3\,{\left (c+d\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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