Optimal. Leaf size=14 \[ \frac {(c+d x)^4}{4 d} \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 32}
\begin {gather*} \frac {(c+d x)^4}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^3} \, dx &=\int (c+d x)^3 \, dx\\ &=\frac {(c+d x)^4}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(c+d x)^4}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 13, normalized size = 0.93
method | result | size |
default | \(\frac {\left (d x +c \right )^{4}}{4 d}\) | \(13\) |
gosper | \(\frac {x \left (d^{3} x^{3}+4 c \,d^{2} x^{2}+6 c^{2} d x +4 c^{3}\right )}{4}\) | \(33\) |
risch | \(\frac {d^{3} x^{4}}{4}+c \,d^{2} x^{3}+\frac {3 c^{2} d \,x^{2}}{2}+c^{3} x +\frac {c^{4}}{4 d}\) | \(40\) |
norman | \(\frac {\left (\frac {1}{2} a b \,d^{3}+b^{2} c \,d^{2}\right ) x^{5}+\left (\frac {1}{4} a^{2} d^{3}+2 a b c \,d^{2}+\frac {3}{2} b^{2} c^{2} d \right ) x^{4}+\left (a^{2} c \,d^{2}+3 a b \,c^{2} d +b^{2} c^{3}\right ) x^{3}-\frac {a^{2} \left (3 a^{2} c^{2} d +4 a b \,c^{3}\right )}{2 b^{2}}+\frac {b^{2} d^{3} x^{6}}{4}-\frac {a \left (3 a^{2} c^{2} d +3 a b \,c^{3}\right ) x}{b}}{\left (b x +a \right )^{2}}\) | \(148\) |
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. 31 vs.
\(2 (12) = 24\).
time = 0.28, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \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. 31 vs.
\(2 (12) = 24\).
time = 3.09, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \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. 32 vs.
\(2 (8) = 16\).
time = 0.05, size = 32, normalized size = 2.29 \begin {gather*} c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4} \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. 31 vs.
\(2 (12) = 24\).
time = 0.92, size = 31, normalized size = 2.21 \begin {gather*} \frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} + c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 31, normalized size = 2.21 \begin {gather*} c^3\,x+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3+\frac {d^3\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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