Optimal. Leaf size=17 \[ \frac {b^5 (c+d x)^3}{3 d^6} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32}
\begin {gather*} \frac {b^5 (c+d x)^3}{3 d^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (\frac {b c}{d}+b x\right )^5}{(c+d x)^3} \, dx &=\frac {b^5 \int (c+d x)^2 \, dx}{d^5}\\ &=\frac {b^5 (c+d x)^3}{3 d^6}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b^5 (c+d x)^3}{3 d^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 16, normalized size = 0.94
method | result | size |
default | \(\frac {b^{5} \left (d x +c \right )^{3}}{3 d^{6}}\) | \(16\) |
gosper | \(\frac {b^{5} x \left (d^{2} x^{2}+3 c d x +3 c^{2}\right )}{3 d^{5}}\) | \(28\) |
risch | \(\frac {b^{5} x^{3}}{3 d^{3}}+\frac {b^{5} c \,x^{2}}{d^{4}}+\frac {b^{5} c^{2} x}{d^{5}}+\frac {b^{5} c^{3}}{3 d^{6}}\) | \(46\) |
norman | \(\frac {\frac {b^{5} d^{3} x^{5}}{3}+\frac {5 c^{3} b^{5} x^{2}}{2}-\frac {c^{5} b^{5}}{2 d^{2}}+\frac {5 b^{5} c \,d^{2} x^{4}}{3}+\frac {10 b^{5} c^{2} d \,x^{3}}{3}}{d^{4} \left (d x +c \right )^{2}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (15) = 30\).
time = 0.29, size = 35, normalized size = 2.06 \begin {gather*} \frac {b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (15) = 30\).
time = 0.58, size = 35, normalized size = 2.06 \begin {gather*} \frac {b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \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. 34 vs.
\(2 (14) = 28\).
time = 0.04, size = 34, normalized size = 2.00 \begin {gather*} \frac {b^{5} c^{2} x}{d^{5}} + \frac {b^{5} c x^{2}}{d^{4}} + \frac {b^{5} x^{3}}{3 d^{3}} \end {gather*}
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. 35 vs.
\(2 (15) = 30\).
time = 1.87, size = 35, normalized size = 2.06 \begin {gather*} \frac {b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 27, normalized size = 1.59 \begin {gather*} \frac {b^5\,x\,\left (3\,c^2+3\,c\,d\,x+d^2\,x^2\right )}{3\,d^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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