Optimal. Leaf size=38 \[ \frac {(a+b x)^5 (b d-a e)}{5 b^2}+\frac {e (a+b x)^6}{6 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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, 43} \begin {gather*} \frac {(a+b x)^5 (b d-a e)}{5 b^2}+\frac {e (a+b x)^6}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x) \, dx\\ &=\int \left (\frac {(b d-a e) (a+b x)^4}{b}+\frac {e (a+b x)^5}{b}\right ) \, dx\\ &=\frac {(b d-a e) (a+b x)^5}{5 b^2}+\frac {e (a+b x)^6}{6 b^2}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 84, normalized size = 2.21 \begin {gather*} \frac {1}{30} x \left (15 a^4 (2 d+e x)+20 a^3 b x (3 d+2 e x)+15 a^2 b^2 x^2 (4 d+3 e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.36, size = 97, normalized size = 2.55 \begin {gather*} \frac {1}{6} x^{6} e b^{4} + \frac {1}{5} x^{5} d b^{4} + \frac {4}{5} x^{5} e b^{3} a + x^{4} d b^{3} a + \frac {3}{2} x^{4} e b^{2} a^{2} + 2 x^{3} d b^{2} a^{2} + \frac {4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac {1}{2} x^{2} e a^{4} + x d a^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 102, normalized size = 2.68 \begin {gather*} \frac {1}{6} \, b^{4} x^{6} e + \frac {1}{5} \, b^{4} d x^{5} + \frac {4}{5} \, a b^{3} x^{5} e + a b^{3} d x^{4} + \frac {3}{2} \, a^{2} b^{2} x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac {4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac {1}{2} \, a^{4} x^{2} e + a^{4} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 97, normalized size = 2.55 \begin {gather*} \frac {b^{4} e \,x^{6}}{6}+a^{4} d x +\frac {\left (4 e a \,b^{3}+d \,b^{4}\right ) x^{5}}{5}+\frac {\left (6 e \,b^{2} a^{2}+4 d a \,b^{3}\right ) x^{4}}{4}+\frac {\left (4 e \,a^{3} b +6 d \,b^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (e \,a^{4}+4 d \,a^{3} b \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 96, normalized size = 2.53 \begin {gather*} \frac {1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac {1}{5} \, {\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 88, normalized size = 2.32 \begin {gather*} x^2\,\left (\frac {e\,a^4}{2}+2\,b\,d\,a^3\right )+x^5\,\left (\frac {d\,b^4}{5}+\frac {4\,a\,e\,b^3}{5}\right )+\frac {b^4\,e\,x^6}{6}+a^4\,d\,x+\frac {2\,a^2\,b\,x^3\,\left (2\,a\,e+3\,b\,d\right )}{3}+\frac {a\,b^2\,x^4\,\left (3\,a\,e+2\,b\,d\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 100, normalized size = 2.63 \begin {gather*} a^{4} d x + \frac {b^{4} e x^{6}}{6} + x^{5} \left (\frac {4 a b^{3} e}{5} + \frac {b^{4} d}{5}\right ) + x^{4} \left (\frac {3 a^{2} b^{2} e}{2} + a b^{3} d\right ) + x^{3} \left (\frac {4 a^{3} b e}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac {a^{4} e}{2} + 2 a^{3} b d\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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