Optimal. Leaf size=65 \[ \frac {d (a+b x)^6 (b c-a d)}{3 b^3}+\frac {(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \]
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Rubi [A] time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \[ \frac {d (a+b x)^6 (b c-a d)}{3 b^3}+\frac {(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^4 (c+d x)^2 \, dx\\ &=\int \left (\frac {(b c-a d)^2 (a+b x)^4}{b^2}+\frac {2 d (b c-a d) (a+b x)^5}{b^2}+\frac {d^2 (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac {d (b c-a d) (a+b x)^6}{3 b^3}+\frac {d^2 (a+b x)^7}{7 b^3}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 148, normalized size = 2.28 \[ a^4 c^2 x+a^3 c x^2 (a d+2 b c)+\frac {1}{5} b^2 x^5 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a b x^4 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac {1}{3} a^2 x^3 \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac {1}{3} b^3 d x^6 (2 a d+b c)+\frac {1}{7} b^4 d^2 x^7 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 170, normalized size = 2.62 \[ \frac {1}{7} x^{7} d^{2} b^{4} + \frac {1}{3} x^{6} d c b^{4} + \frac {2}{3} x^{6} d^{2} b^{3} a + \frac {1}{5} x^{5} c^{2} b^{4} + \frac {8}{5} x^{5} d c b^{3} a + \frac {6}{5} x^{5} d^{2} b^{2} a^{2} + x^{4} c^{2} b^{3} a + 3 x^{4} d c b^{2} a^{2} + x^{4} d^{2} b a^{3} + 2 x^{3} c^{2} b^{2} a^{2} + \frac {8}{3} x^{3} d c b a^{3} + \frac {1}{3} x^{3} d^{2} a^{4} + 2 x^{2} c^{2} b a^{3} + x^{2} d c a^{4} + x c^{2} a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 170, normalized size = 2.62 \[ \frac {1}{7} \, b^{4} d^{2} x^{7} + \frac {1}{3} \, b^{4} c d x^{6} + \frac {2}{3} \, a b^{3} d^{2} x^{6} + \frac {1}{5} \, b^{4} c^{2} x^{5} + \frac {8}{5} \, a b^{3} c d x^{5} + \frac {6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac {8}{3} \, a^{3} b c d x^{3} + \frac {1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 231, normalized size = 3.55 \[ \frac {b^{4} d^{2} x^{7}}{7}+a^{4} c^{2} x +\frac {\left (2 a \,b^{3} d^{2}+2 \left (a d +b c \right ) b^{3} d \right ) x^{6}}{6}+\frac {\left (a^{2} b^{2} d^{2}+4 \left (a d +b c \right ) a \,b^{2} d +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) b^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (a d +b c \right ) a^{2} b d +2 \left (a d +b c \right ) a \,b^{2} c +2 \left (2 a b c d +\left (a d +b c \right )^{2}\right ) a b \right ) x^{4}}{4}+\frac {\left (a^{2} b^{2} c^{2}+4 \left (a d +b c \right ) a^{2} b c +\left (2 a b c d +\left (a d +b c \right )^{2}\right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} b \,c^{2}+2 \left (a d +b c \right ) a^{3} c \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 156, normalized size = 2.40 \[ \frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 144, normalized size = 2.22 \[ x^3\,\left (\frac {a^4\,d^2}{3}+\frac {8\,a^3\,b\,c\,d}{3}+2\,a^2\,b^2\,c^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,d^2}{5}+\frac {8\,a\,b^3\,c\,d}{5}+\frac {b^4\,c^2}{5}\right )+a^4\,c^2\,x+\frac {b^4\,d^2\,x^7}{7}+a^3\,c\,x^2\,\left (a\,d+2\,b\,c\right )+\frac {b^3\,d\,x^6\,\left (2\,a\,d+b\,c\right )}{3}+a\,b\,x^4\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 168, normalized size = 2.58 \[ a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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