Optimal. Leaf size=38 \[ -\frac {b^2 (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {67}
\begin {gather*} -\frac {b^2 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{x^3} \, dx &=-\frac {b^2 (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} -\frac {b^2 (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 11.58, size = 150, normalized size = 3.95 \begin {gather*} \frac {\left (-a^3-a^2 b n x+a b^2 x^2-a b^2 n x^2+a b^2 n x^2 \text {LerchPhi}\left [\frac {a+b x}{a},1,1+n\right ]-a b^2 n^2 x^2 \text {LerchPhi}\left [\frac {a+b x}{a},1,1+n\right ]+b^3 n x^3 \text {LerchPhi}\left [\frac {a+b x}{a},1,1+n\right ]-b^3 n^2 x^3 \text {LerchPhi}\left [\frac {a+b x}{a},1,1+n\right ]\right ) \left (\frac {a+b x}{b}\right )^n b^n}{2 a^3 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n}}{x^{3}}\, dx\]
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 = 0.33, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 918 vs.
\(2 (31) = 62\).
time = 1.99, size = 918, normalized size = 24.16
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 = -0.03 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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