Optimal. Leaf size=85 \[ -\frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a (n+1)} \]
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Rubi [A] time = 0.040364, antiderivative size = 85, normalized size of antiderivative = 1., 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 = {137, 136} \[ -\frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(a+b x)^n (c+d x)^p}{x} \, 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} \, dx\\ &=-\frac{(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,1;2+n;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0616524, size = 88, normalized size = 1.04 \[ \frac{\left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \left (\frac{c}{d x}+1\right )^{-p} (c+d x)^p F_1\left (-n-p;-n,-p;-n-p+1;-\frac{a}{b x},-\frac{c}{d x}\right )}{n+p} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{n} \left (c + d x\right )^{p}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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