Optimal. Leaf size=54 \[ -\frac {d^2 (c+d x)^{1+n} \, _2F_1\left (3,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^3 (1+n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 = {70}
\begin {gather*} -\frac {d^2 (c+d x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rubi steps
\begin {align*} \int \frac {(c+d x)^n}{(a+b x)^3} \, dx &=-\frac {d^2 (c+d x)^{1+n} \, _2F_1\left (3,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^3 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 54, normalized size = 1.00 \begin {gather*} \frac {d^2 (c+d x)^{1+n} \, _2F_1\left (3,1+n;2+n;-\frac {b (c+d x)}{-b c+a d}\right )}{(-b c+a d)^3 (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{n}}{\left (b x +a \right )^{3}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{n}}{\left (a + b x\right )^{3}}\, 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 = -0.02 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^n}{{\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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