Optimal. Leaf size=65 \[ \frac {(b d-a e)^2 (d+e x)^4}{4 e^3}-\frac {2 b (b d-a e) (d+e x)^5}{5 e^3}+\frac {b^2 (d+e x)^6}{6 e^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45}
\begin {gather*} -\frac {2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac {(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac {b^2 (d+e x)^6}{6 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^3 \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (d+e x)^3}{e^2}-\frac {2 b (b d-a e) (d+e x)^4}{e^2}+\frac {b^2 (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (d+e x)^4}{4 e^3}-\frac {2 b (b d-a e) (d+e x)^5}{5 e^3}+\frac {b^2 (d+e x)^6}{6 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 122, normalized size = 1.88 \begin {gather*} a^2 d^3 x+\frac {1}{2} a d^2 (2 b d+3 a e) x^2+\frac {1}{3} d \left (b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^3+\frac {1}{4} e \left (3 b^2 d^2+6 a b d e+a^2 e^2\right ) x^4+\frac {1}{5} b e^2 (3 b d+2 a e) x^5+\frac {1}{6} b^2 e^3 x^6 \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. \(124\) vs.
\(2(59)=118\).
time = 0.48, size = 125, normalized size = 1.92
method | result | size |
norman | \(\frac {b^{2} e^{3} x^{6}}{6}+\left (\frac {2}{5} a b \,e^{3}+\frac {3}{5} b^{2} d \,e^{2}\right ) x^{5}+\left (\frac {1}{4} a^{2} e^{3}+\frac {3}{2} a b d \,e^{2}+\frac {3}{4} b^{2} d^{2} e \right ) x^{4}+\left (a^{2} d \,e^{2}+2 a b \,d^{2} e +\frac {1}{3} b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} d^{2} e +a b \,d^{3}\right ) x^{2}+a^{2} d^{3} x\) | \(121\) |
default | \(\frac {b^{2} e^{3} x^{6}}{6}+\frac {\left (2 a b \,e^{3}+3 b^{2} d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 a b d \,e^{2}+3 b^{2} d^{2} e \right ) x^{4}}{4}+\frac {\left (3 a^{2} d \,e^{2}+6 a b \,d^{2} e +b^{2} d^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{2} d^{2} e +2 a b \,d^{3}\right ) x^{2}}{2}+a^{2} d^{3} x\) | \(125\) |
risch | \(\frac {1}{6} b^{2} e^{3} x^{6}+\frac {2}{5} x^{5} a b \,e^{3}+\frac {3}{5} x^{5} b^{2} d \,e^{2}+\frac {1}{4} x^{4} a^{2} e^{3}+\frac {3}{2} x^{4} a b d \,e^{2}+\frac {3}{4} x^{4} b^{2} d^{2} e +x^{3} a^{2} d \,e^{2}+2 x^{3} a b \,d^{2} e +\frac {1}{3} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{2} x^{2}+x^{2} a b \,d^{3}+a^{2} d^{3} x\) | \(131\) |
gosper | \(\frac {x \left (10 b^{2} e^{3} x^{5}+24 x^{4} a b \,e^{3}+36 x^{4} b^{2} d \,e^{2}+15 x^{3} a^{2} e^{3}+90 x^{3} a b d \,e^{2}+45 x^{3} b^{2} d^{2} e +60 x^{2} a^{2} d \,e^{2}+120 x^{2} a b \,d^{2} e +20 x^{2} b^{2} d^{3}+90 x \,a^{2} d^{2} e +60 x a b \,d^{3}+60 a^{2} d^{3}\right )}{60}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 121, normalized size = 1.86 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} e^{3} + a^{2} d^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\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. 125 vs.
\(2 (61) = 122\).
time = 2.37, size = 125, normalized size = 1.92 \begin {gather*} \frac {1}{3} \, b^{2} d^{3} x^{3} + a b d^{3} x^{2} + a^{2} d^{3} x + \frac {1}{60} \, {\left (10 \, b^{2} x^{6} + 24 \, a b x^{5} + 15 \, a^{2} x^{4}\right )} e^{3} + \frac {1}{10} \, {\left (6 \, b^{2} d x^{5} + 15 \, a b d x^{4} + 10 \, a^{2} d x^{3}\right )} e^{2} + \frac {1}{4} \, {\left (3 \, b^{2} d^{2} x^{4} + 8 \, a b d^{2} x^{3} + 6 \, a^{2} d^{2} x^{2}\right )} e \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. 133 vs.
\(2 (56) = 112\).
time = 0.02, size = 133, normalized size = 2.05 \begin {gather*} a^{2} d^{3} x + \frac {b^{2} e^{3} x^{6}}{6} + x^{5} \cdot \left (\frac {2 a b e^{3}}{5} + \frac {3 b^{2} d e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a b d e^{2}}{2} + \frac {3 b^{2} d^{2} e}{4}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac {b^{2} d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{2} e}{2} + a b d^{3}\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. 127 vs.
\(2 (61) = 122\).
time = 1.73, size = 127, normalized size = 1.95 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} e^{3} + \frac {3}{5} \, b^{2} d x^{5} e^{2} + \frac {3}{4} \, b^{2} d^{2} x^{4} e + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {2}{5} \, a b x^{5} e^{3} + \frac {3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.51, size = 115, normalized size = 1.77 \begin {gather*} x^3\,\left (a^2\,d\,e^2+2\,a\,b\,d^2\,e+\frac {b^2\,d^3}{3}\right )+x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,a\,b\,d\,e^2}{2}+\frac {3\,b^2\,d^2\,e}{4}\right )+a^2\,d^3\,x+\frac {b^2\,e^3\,x^6}{6}+\frac {a\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )}{2}+\frac {b\,e^2\,x^5\,\left (2\,a\,e+3\,b\,d\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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