Optimal. Leaf size=54 \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]
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Rubi [A] time = 0.0131582, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {68} \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^m}{(c+d x)^3} \, dx &=\frac{b^2 (a+b x)^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0138603, size = 54, normalized size = 1. \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.071, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{3}}}\, 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 )}^{m}}{{\left (d x + c\right )}^{3}}\,{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 )}^{m}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, 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 )^{m}}{\left (c + d x\right )^{3}}\, 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 )}^{m}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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