Optimal. Leaf size=65 \[ -\frac {b (c+d x)^4 (b c-a d)}{2 d^3}+\frac {(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac {b^2 (c+d x)^5}{5 d^3} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {610, 43} \begin {gather*} -\frac {b (c+d x)^4 (b c-a d)}{2 d^3}+\frac {(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac {b^2 (c+d x)^5}{5 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 610
Rubi steps
\begin {align*} \int \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\frac {\int (b c+b d x)^2 (a d+b d x)^2 \, dx}{b^2 d^2}\\ &=\frac {\int \left ((b c-a d)^2 (b c+b d x)^2-2 (b c-a d) (b c+b d x)^3+(b c+b d x)^4\right ) \, dx}{b^2 d^2}\\ &=\frac {(b c-a d)^2 (c+d x)^3}{3 d^3}-\frac {b (b c-a d) (c+d x)^4}{2 d^3}+\frac {b^2 (c+d x)^5}{5 d^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 79, normalized size = 1.22 \begin {gather*} \frac {1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac {1}{2} b d x^4 (a d+b c)+a c x^2 (a d+b c)+\frac {1}{5} b^2 d^2 x^5 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 89, normalized size = 1.37 \begin {gather*} \frac {1}{5} x^{5} d^{2} b^{2} + \frac {1}{2} x^{4} d c b^{2} + \frac {1}{2} x^{4} d^{2} b a + \frac {1}{3} x^{3} c^{2} b^{2} + \frac {4}{3} x^{3} d c b a + \frac {1}{3} x^{3} d^{2} a^{2} + x^{2} c^{2} b a + x^{2} d c a^{2} + x c^{2} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 89, normalized size = 1.37 \begin {gather*} \frac {1}{5} \, b^{2} d^{2} x^{5} + \frac {1}{2} \, b^{2} c d x^{4} + \frac {1}{2} \, a b d^{2} x^{4} + \frac {1}{3} \, b^{2} c^{2} x^{3} + \frac {4}{3} \, a b c d x^{3} + \frac {1}{3} \, a^{2} d^{2} x^{3} + a b c^{2} x^{2} + a^{2} c d x^{2} + a^{2} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 69, normalized size = 1.06 \begin {gather*} \frac {b^{2} d^{2} x^{5}}{5}+\frac {\left (a d +b c \right ) b d \,x^{4}}{2}+a^{2} c^{2} x +\left (a d +b c \right ) a c \,x^{2}+\frac {\left (2 a b c d +\left (a d +b c \right )^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 72, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, b^{2} d^{2} x^{5} + \frac {1}{2} \, {\left (b c + a d\right )} b d x^{4} + a^{2} c^{2} x + \frac {1}{3} \, {\left (b c + a d\right )}^{2} x^{3} + \frac {1}{3} \, {\left (2 \, b d x^{3} + 3 \, {\left (b c + a d\right )} x^{2}\right )} a c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 74, normalized size = 1.14 \begin {gather*} x^3\,\left (\frac {a^2\,d^2}{3}+\frac {4\,a\,b\,c\,d}{3}+\frac {b^2\,c^2}{3}\right )+a^2\,c^2\,x+\frac {b^2\,d^2\,x^5}{5}+a\,c\,x^2\,\left (a\,d+b\,c\right )+\frac {b\,d\,x^4\,\left (a\,d+b\,c\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 87, normalized size = 1.34 \begin {gather*} a^{2} c^{2} x + \frac {b^{2} d^{2} x^{5}}{5} + x^{4} \left (\frac {a b d^{2}}{2} + \frac {b^{2} c d}{2}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3} + \frac {4 a b c d}{3} + \frac {b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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