Optimal. Leaf size=14 \[ \frac {(c+d x)^3}{3 d} \]
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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)^3}{3 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 )^2}{(a+b x)^2} \, dx &=\int (c+d x)^2 \, dx\\ &=\frac {(c+d x)^3}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(c+d x)^3}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 13, normalized size = 0.93
method | result | size |
default | \(\frac {\left (d x +c \right )^{3}}{3 d}\) | \(13\) |
gosper | \(\frac {x \left (d^{2} x^{2}+3 c d x +3 c^{2}\right )}{3}\) | \(22\) |
risch | \(\frac {d^{2} x^{3}}{3}+c d \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}\) | \(29\) |
norman | \(\frac {\left (\frac {1}{3} a \,d^{2}+b c d \right ) x^{3}+\left (a c d +b \,c^{2}\right ) x^{2}+a \,c^{2} x +\frac {b \,d^{2} x^{4}}{3}}{b x +a}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 20, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.87, size = 20, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} 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. 19 vs.
\(2 (8) = 16\).
time = 0.04, size = 19, normalized size = 1.36 \begin {gather*} c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{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. 84 vs.
\(2 (12) = 24\).
time = 0.59, size = 84, normalized size = 6.00 \begin {gather*} \frac {{\left (\frac {3 \, b^{2} c^{2}}{{\left (b x + a\right )}^{2}} + \frac {3 \, b c d}{b x + a} - \frac {6 \, a b c d}{{\left (b x + a\right )}^{2}} - \frac {3 \, a d^{2}}{b x + a} + \frac {3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + d^{2}\right )} {\left (b x + a\right )}^{3}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 1.43 \begin {gather*} c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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