Optimal. Leaf size=38 \[ \frac {(b c-a d) (a+b x)^4}{4 b^2}+\frac {d (a+b x)^5}{5 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} \frac {(a+b x)^4 (b c-a d)}{4 b^2}+\frac {d (a+b x)^5}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^3 (c+d x) \, dx &=\int \left (\frac {(b c-a d) (a+b x)^3}{b}+\frac {d (a+b x)^4}{b}\right ) \, dx\\ &=\frac {(b c-a d) (a+b x)^4}{4 b^2}+\frac {d (a+b x)^5}{5 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 1.76 \begin {gather*} a^3 c x+\frac {1}{2} a^2 (3 b c+a d) x^2+a b (b c+a d) x^3+\frac {1}{4} b^2 (b c+3 a d) x^4+\frac {1}{5} b^3 d x^5 \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.01, size = 60, normalized size = 1.58 \begin {gather*} x \left (a^3 c+\frac {a^2 x \left (a d+3 b c\right )}{2}+a b x^2 \left (a d+b c\right )+\frac {b^2 x^3 \left (3 a d+b c\right )}{4}+\frac {b^3 d x^4}{5}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs.
\(2(34)=68\).
time = 0.12, size = 73, normalized size = 1.92
method | result | size |
norman | \(\frac {b^{3} d \,x^{5}}{5}+\left (\frac {3}{4} a \,b^{2} d +\frac {1}{4} b^{3} c \right ) x^{4}+\left (a^{2} b d +a \,b^{2} c \right ) x^{3}+\left (\frac {1}{2} a^{3} d +\frac {3}{2} a^{2} b c \right ) x^{2}+a^{3} c x\) | \(70\) |
gosper | \(\frac {1}{5} b^{3} d \,x^{5}+\frac {3}{4} x^{4} a \,b^{2} d +\frac {1}{4} x^{4} b^{3} c +a^{2} b d \,x^{3}+a \,b^{2} c \,x^{3}+\frac {1}{2} x^{2} a^{3} d +\frac {3}{2} a^{2} b c \,x^{2}+a^{3} c x\) | \(73\) |
default | \(\frac {b^{3} d \,x^{5}}{5}+\frac {\left (3 a \,b^{2} d +b^{3} c \right ) x^{4}}{4}+\frac {\left (3 a^{2} b d +3 a \,b^{2} c \right ) x^{3}}{3}+\frac {\left (a^{3} d +3 a^{2} b c \right ) x^{2}}{2}+a^{3} c x\) | \(73\) |
risch | \(\frac {1}{5} b^{3} d \,x^{5}+\frac {3}{4} x^{4} a \,b^{2} d +\frac {1}{4} x^{4} b^{3} c +a^{2} b d \,x^{3}+a \,b^{2} c \,x^{3}+\frac {1}{2} x^{2} a^{3} d +\frac {3}{2} a^{2} b c \,x^{2}+a^{3} c x\) | \(73\) |
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. 69 vs.
\(2 (34) = 68\).
time = 0.26, size = 69, normalized size = 1.82 \begin {gather*} \frac {1}{5} \, b^{3} d x^{5} + a^{3} c x + \frac {1}{4} \, {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{4} + {\left (a b^{2} c + a^{2} b d\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c + a^{3} d\right )} x^{2} \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. 69 vs.
\(2 (34) = 68\).
time = 0.29, size = 69, normalized size = 1.82 \begin {gather*} \frac {1}{5} \, b^{3} d x^{5} + a^{3} c x + \frac {1}{4} \, {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{4} + {\left (a b^{2} c + a^{2} b d\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c + a^{3} d\right )} x^{2} \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. 73 vs.
\(2 (32) = 64\).
time = 0.04, size = 73, normalized size = 1.92 \begin {gather*} a^{3} c x + \frac {b^{3} d x^{5}}{5} + x^{4} \cdot \left (\frac {3 a b^{2} d}{4} + \frac {b^{3} c}{4}\right ) + x^{3} \left (a^{2} b d + a b^{2} c\right ) + x^{2} \left (\frac {a^{3} d}{2} + \frac {3 a^{2} b c}{2}\right ) \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. 72 vs.
\(2 (34) = 68\).
time = 0.00, size = 82, normalized size = 2.16 \begin {gather*} \frac {1}{5} x^{5} b^{3} d+\frac {1}{4} x^{4} b^{3} c+\frac {3}{4} x^{4} b^{2} a d+x^{3} b^{2} a c+x^{3} b a^{2} d+\frac {3}{2} x^{2} b a^{2} c+\frac {1}{2} x^{2} a^{3} d+x a^{3} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 65, normalized size = 1.71 \begin {gather*} x^4\,\left (\frac {c\,b^3}{4}+\frac {3\,a\,d\,b^2}{4}\right )+x^2\,\left (\frac {d\,a^3}{2}+\frac {3\,b\,c\,a^2}{2}\right )+\frac {b^3\,d\,x^5}{5}+a^3\,c\,x+a\,b\,x^3\,\left (a\,d+b\,c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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