Optimal. Leaf size=83 \[ -\frac {a^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 a^2 (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a (a+b x)^{3+n}}{b^4 (3+n)}+\frac {(a+b x)^{4+n}}{b^4 (4+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} -\frac {a^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(a+b x)^{n+4}}{b^4 (n+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^3 (a+b x)^n \, dx &=\int \left (-\frac {a^3 (a+b x)^n}{b^3}+\frac {3 a^2 (a+b x)^{1+n}}{b^3}-\frac {3 a (a+b x)^{2+n}}{b^3}+\frac {(a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=-\frac {a^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 a^2 (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a (a+b x)^{3+n}}{b^4 (3+n)}+\frac {(a+b x)^{4+n}}{b^4 (4+n)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 67, normalized size = 0.81 \begin {gather*} \frac {(a+b x)^{1+n} \left (-\frac {a^3}{1+n}+\frac {3 a^2 (a+b x)}{2+n}-\frac {3 a (a+b x)^2}{3+n}+\frac {(a+b x)^3}{4+n}\right )}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 7.73, size = 929, normalized size = 11.19 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x^4 a^n}{4},b\text {==}0\right \},\left \{\frac {a^3 \left (11+6 \text {Log}\left [\frac {a+b x}{b}\right ]\right )+9 a^2 b x \left (3+2 \text {Log}\left [\frac {a+b x}{b}\right ]\right )+18 a b^2 x^2 \left (1+\text {Log}\left [\frac {a+b x}{b}\right ]\right )+6 b^3 x^3 \text {Log}\left [\frac {a+b x}{b}\right ]}{6 b^4 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )},n\text {==}-4\right \},\left \{\frac {\frac {-3 a^3 \left (3+2 \text {Log}\left [\frac {a+b x}{b}\right ]\right )}{2}-6 a^2 b x \left (1+\text {Log}\left [\frac {a+b x}{b}\right ]\right )-3 a b^2 x^2 \text {Log}\left [\frac {a+b x}{b}\right ]+b^3 x^3}{b^4 \left (a^2+2 a b x+b^2 x^2\right )},n\text {==}-3\right \},\left \{\frac {6 a^3 \left (1+\text {Log}\left [\frac {a+b x}{b}\right ]\right )+6 a^2 b x \text {Log}\left [\frac {a+b x}{b}\right ]-3 a b^2 x^2+b^3 x^3}{2 b^4 \left (a+b x\right )},n\text {==}-2\right \},\left \{\frac {-a^3 \text {Log}\left [\frac {a}{b}+x\right ]+a^2 b x-\frac {a b^2 x^2}{2}+\frac {b^3 x^3}{3}}{b^4},n\text {==}-1\right \}\right \},\frac {-6 a^4 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {6 a^3 b n x \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}-\frac {3 a^2 b^2 n x^2 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}-\frac {3 a^2 b^2 n^2 x^2 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {2 a b^3 n x^3 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {3 a b^3 n^2 x^3 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {a b^3 n^3 x^3 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {6 b^4 x^4 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {11 b^4 n x^4 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {6 b^4 n^2 x^4 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}+\frac {b^4 n^3 x^4 \left (a+b x\right )^n}{24 b^4+50 b^4 n+35 b^4 n^2+10 b^4 n^3+b^4 n^4}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 126, normalized size = 1.52
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} n^{3} x^{3}-6 b^{3} n^{2} x^{3}+3 a \,b^{2} n^{2} x^{2}-11 b^{3} n \,x^{3}+9 a \,b^{2} n \,x^{2}-6 b^{3} x^{3}-6 a^{2} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(126\) |
risch | \(-\frac {\left (-b^{4} n^{3} x^{4}-a \,b^{3} n^{3} x^{3}-6 b^{4} n^{2} x^{4}-3 a \,b^{3} n^{2} x^{3}-11 b^{4} n \,x^{4}+3 a^{2} b^{2} n^{2} x^{2}-2 x^{3} a n \,b^{3}-6 b^{4} x^{4}+3 a^{2} n \,x^{2} b^{2}-6 a^{3} b n x +6 a^{4}\right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(146\) |
norman | \(\frac {x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {a n \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {6 a^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {3 a^{2} n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}+\frac {6 n \,a^{3} x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 101, normalized size = 1.22 \begin {gather*} \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 143, normalized size = 1.72 \begin {gather*} \frac {{\left (6 \, a^{3} b n x + {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} + {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.63, size = 1318, normalized size = 15.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs.
\(2 (83) = 166\).
time = 0.00, size = 250, normalized size = 3.01 \begin {gather*} \frac {-6 a^{4} \mathrm {e}^{n \ln \left (a+b x\right )}+6 a^{3} b n x \mathrm {e}^{n \ln \left (a+b x\right )}-3 a^{2} b^{2} n^{2} x^{2} \mathrm {e}^{n \ln \left (a+b x\right )}-3 a^{2} b^{2} n x^{2} \mathrm {e}^{n \ln \left (a+b x\right )}+a b^{3} n^{3} x^{3} \mathrm {e}^{n \ln \left (a+b x\right )}+3 a b^{3} n^{2} x^{3} \mathrm {e}^{n \ln \left (a+b x\right )}+2 a b^{3} n x^{3} \mathrm {e}^{n \ln \left (a+b x\right )}+b^{4} n^{3} x^{4} \mathrm {e}^{n \ln \left (a+b x\right )}+6 b^{4} n^{2} x^{4} \mathrm {e}^{n \ln \left (a+b x\right )}+11 b^{4} n x^{4} \mathrm {e}^{n \ln \left (a+b x\right )}+6 b^{4} x^{4} \mathrm {e}^{n \ln \left (a+b x\right )}}{b^{4} n^{4}+10 b^{4} n^{3}+35 b^{4} n^{2}+50 b^{4} n+24 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 176, normalized size = 2.12 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,n\,x}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,n\,x^2\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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