Optimal. Leaf size=92 \[ \frac {d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac {3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac {(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac {d^3 (a+b x)^7}{7 b^4} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac {3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac {(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac {d^3 (a+b x)^7}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^3 (c+d x)^3 \, dx &=\int \left (\frac {(b c-a d)^3 (a+b x)^3}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^4}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^5}{b^3}+\frac {d^3 (a+b x)^6}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^4}{4 b^4}+\frac {3 d (b c-a d)^2 (a+b x)^5}{5 b^4}+\frac {d^2 (b c-a d) (a+b x)^6}{2 b^4}+\frac {d^3 (a+b x)^7}{7 b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 161, normalized size = 1.75 \[ a^3 c^3 x+\frac {3}{5} b d x^5 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c x^3 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac {3}{2} a^2 c^2 x^2 (a d+b c)+\frac {1}{4} x^4 \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )+\frac {1}{2} b^2 d^2 x^6 (a d+b c)+\frac {1}{7} b^3 d^3 x^7 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.37, size = 188, normalized size = 2.04 \[ \frac {1}{7} x^{7} d^{3} b^{3} + \frac {1}{2} x^{6} d^{2} c b^{3} + \frac {1}{2} x^{6} d^{3} b^{2} a + \frac {3}{5} x^{5} d c^{2} b^{3} + \frac {9}{5} x^{5} d^{2} c b^{2} a + \frac {3}{5} x^{5} d^{3} b a^{2} + \frac {1}{4} x^{4} c^{3} b^{3} + \frac {9}{4} x^{4} d c^{2} b^{2} a + \frac {9}{4} x^{4} d^{2} c b a^{2} + \frac {1}{4} x^{4} d^{3} a^{3} + x^{3} c^{3} b^{2} a + 3 x^{3} d c^{2} b a^{2} + x^{3} d^{2} c a^{3} + \frac {3}{2} x^{2} c^{3} b a^{2} + \frac {3}{2} x^{2} d c^{2} a^{3} + x c^{3} a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 188, normalized size = 2.04 \[ \frac {1}{7} \, b^{3} d^{3} x^{7} + \frac {1}{2} \, b^{3} c d^{2} x^{6} + \frac {1}{2} \, a b^{2} d^{3} x^{6} + \frac {3}{5} \, b^{3} c^{2} d x^{5} + \frac {9}{5} \, a b^{2} c d^{2} x^{5} + \frac {3}{5} \, a^{2} b d^{3} x^{5} + \frac {1}{4} \, b^{3} c^{3} x^{4} + \frac {9}{4} \, a b^{2} c^{2} d x^{4} + \frac {9}{4} \, a^{2} b c d^{2} x^{4} + \frac {1}{4} \, a^{3} d^{3} x^{4} + a b^{2} c^{3} x^{3} + 3 \, a^{2} b c^{2} d x^{3} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{3} x^{2} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 177, normalized size = 1.92 \[ \frac {b^{3} d^{3} x^{7}}{7}+a^{3} c^{3} x +\frac {\left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right ) x^{6}}{6}+\frac {\left (3 a^{2} b \,d^{3}+9 a \,b^{2} c \,d^{2}+3 b^{3} c^{2} d \right ) x^{5}}{5}+\frac {\left (a^{3} d^{3}+9 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{3} c \,d^{2}+9 a^{2} b \,c^{2} d +3 a \,b^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{3} c^{2} d +3 a^{2} b \,c^{3}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 167, normalized size = 1.82 \[ \frac {1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac {1}{2} \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} + {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 152, normalized size = 1.65 \[ x^4\,\left (\frac {a^3\,d^3}{4}+\frac {9\,a^2\,b\,c\,d^2}{4}+\frac {9\,a\,b^2\,c^2\,d}{4}+\frac {b^3\,c^3}{4}\right )+a^3\,c^3\,x+\frac {b^3\,d^3\,x^7}{7}+a\,c\,x^3\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )+\frac {3\,b\,d\,x^5\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5}+\frac {3\,a^2\,c^2\,x^2\,\left (a\,d+b\,c\right )}{2}+\frac {b^2\,d^2\,x^6\,\left (a\,d+b\,c\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 190, normalized size = 2.07 \[ a^{3} c^{3} x + \frac {b^{3} d^{3} x^{7}}{7} + x^{6} \left (\frac {a b^{2} d^{3}}{2} + \frac {b^{3} c d^{2}}{2}\right ) + x^{5} \left (\frac {3 a^{2} b d^{3}}{5} + \frac {9 a b^{2} c d^{2}}{5} + \frac {3 b^{3} c^{2} d}{5}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + \frac {9 a^{2} b c d^{2}}{4} + \frac {9 a b^{2} c^{2} d}{4} + \frac {b^{3} c^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 3 a^{2} b c^{2} d + a b^{2} c^{3}\right ) + x^{2} \left (\frac {3 a^{3} c^{2} d}{2} + \frac {3 a^{2} b c^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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