Optimal. Leaf size=85 \[ \frac {b (a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} F_1\left (1+n;-p,2;2+n;-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a^2 (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {142, 141}
\begin {gather*} \frac {b (a+b x)^{n+1} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a^2 (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^p}{x^2} \, dx &=\left ((c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p}\right ) \int \frac {(a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^p}{x^2} \, dx\\ &=\frac {b (a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} F_1\left (1+n;-p,2;2+n;-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a^2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 93, normalized size = 1.09 \begin {gather*} \frac {\left (1+\frac {a}{b x}\right )^{-n} \left (1+\frac {c}{d x}\right )^{-p} (a+b x)^n (c+d x)^p F_1\left (1-n-p;-n,-p;2-n-p;-\frac {a}{b x},-\frac {c}{d x}\right )}{(-1+n+p) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{p}}{x^{2}}\, dx\]
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 {\left (a + b x\right )^{n} \left (c + d x\right )^{p}}{x^{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 {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^p}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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