Optimal. Leaf size=171 \[ \frac {e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac {15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac {2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac {5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac {3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac {(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7} \]
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Rubi [A] time = 0.36, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac {e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac {15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac {2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac {5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac {3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac {(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^6 \, dx\\ &=\int \left (\frac {(b d-a e)^6 (a+b x)^6}{b^6}+\frac {6 e (b d-a e)^5 (a+b x)^7}{b^6}+\frac {15 e^2 (b d-a e)^4 (a+b x)^8}{b^6}+\frac {20 e^3 (b d-a e)^3 (a+b x)^9}{b^6}+\frac {15 e^4 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac {6 e^5 (b d-a e) (a+b x)^{11}}{b^6}+\frac {e^6 (a+b x)^{12}}{b^6}\right ) \, dx\\ &=\frac {(b d-a e)^6 (a+b x)^7}{7 b^7}+\frac {3 e (b d-a e)^5 (a+b x)^8}{4 b^7}+\frac {5 e^2 (b d-a e)^4 (a+b x)^9}{3 b^7}+\frac {2 e^3 (b d-a e)^3 (a+b x)^{10}}{b^7}+\frac {15 e^4 (b d-a e)^2 (a+b x)^{11}}{11 b^7}+\frac {e^5 (b d-a e) (a+b x)^{12}}{2 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 573, normalized size = 3.35 \[ a^6 d^6 x+3 a^5 d^5 x^2 (a e+b d)+\frac {3}{11} b^4 e^4 x^{11} \left (5 a^2 e^2+12 a b d e+5 b^2 d^2\right )+a^4 d^4 x^3 \left (5 a^2 e^2+12 a b d e+5 b^2 d^2\right )+b^3 e^3 x^{10} \left (2 a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+2 b^3 d^3\right )+\frac {5}{2} a^3 d^3 x^4 \left (2 a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+2 b^3 d^3\right )+\frac {5}{3} b^2 e^2 x^9 \left (a^4 e^4+8 a^3 b d e^3+15 a^2 b^2 d^2 e^2+8 a b^3 d^3 e+b^4 d^4\right )+3 a^2 d^2 x^5 \left (a^4 e^4+8 a^3 b d e^3+15 a^2 b^2 d^2 e^2+8 a b^3 d^3 e+b^4 d^4\right )+\frac {3}{4} b e x^8 \left (a^5 e^5+15 a^4 b d e^4+50 a^3 b^2 d^2 e^3+50 a^2 b^3 d^3 e^2+15 a b^4 d^4 e+b^5 d^5\right )+a d x^6 \left (a^5 e^5+15 a^4 b d e^4+50 a^3 b^2 d^2 e^3+50 a^2 b^3 d^3 e^2+15 a b^4 d^4 e+b^5 d^5\right )+\frac {1}{7} x^7 \left (a^6 e^6+36 a^5 b d e^5+225 a^4 b^2 d^2 e^4+400 a^3 b^3 d^3 e^3+225 a^2 b^4 d^4 e^2+36 a b^5 d^5 e+b^6 d^6\right )+\frac {1}{2} b^5 e^5 x^{12} (a e+b d)+\frac {1}{13} b^6 e^6 x^{13} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 689, normalized size = 4.03 \[ \frac {1}{13} x^{13} e^{6} b^{6} + \frac {1}{2} x^{12} e^{5} d b^{6} + \frac {1}{2} x^{12} e^{6} b^{5} a + \frac {15}{11} x^{11} e^{4} d^{2} b^{6} + \frac {36}{11} x^{11} e^{5} d b^{5} a + \frac {15}{11} x^{11} e^{6} b^{4} a^{2} + 2 x^{10} e^{3} d^{3} b^{6} + 9 x^{10} e^{4} d^{2} b^{5} a + 9 x^{10} e^{5} d b^{4} a^{2} + 2 x^{10} e^{6} b^{3} a^{3} + \frac {5}{3} x^{9} e^{2} d^{4} b^{6} + \frac {40}{3} x^{9} e^{3} d^{3} b^{5} a + 25 x^{9} e^{4} d^{2} b^{4} a^{2} + \frac {40}{3} x^{9} e^{5} d b^{3} a^{3} + \frac {5}{3} x^{9} e^{6} b^{2} a^{4} + \frac {3}{4} x^{8} e d^{5} b^{6} + \frac {45}{4} x^{8} e^{2} d^{4} b^{5} a + \frac {75}{2} x^{8} e^{3} d^{3} b^{4} a^{2} + \frac {75}{2} x^{8} e^{4} d^{2} b^{3} a^{3} + \frac {45}{4} x^{8} e^{5} d b^{2} a^{4} + \frac {3}{4} x^{8} e^{6} b a^{5} + \frac {1}{7} x^{7} d^{6} b^{6} + \frac {36}{7} x^{7} e d^{5} b^{5} a + \frac {225}{7} x^{7} e^{2} d^{4} b^{4} a^{2} + \frac {400}{7} x^{7} e^{3} d^{3} b^{3} a^{3} + \frac {225}{7} x^{7} e^{4} d^{2} b^{2} a^{4} + \frac {36}{7} x^{7} e^{5} d b a^{5} + \frac {1}{7} x^{7} e^{6} a^{6} + x^{6} d^{6} b^{5} a + 15 x^{6} e d^{5} b^{4} a^{2} + 50 x^{6} e^{2} d^{4} b^{3} a^{3} + 50 x^{6} e^{3} d^{3} b^{2} a^{4} + 15 x^{6} e^{4} d^{2} b a^{5} + x^{6} e^{5} d a^{6} + 3 x^{5} d^{6} b^{4} a^{2} + 24 x^{5} e d^{5} b^{3} a^{3} + 45 x^{5} e^{2} d^{4} b^{2} a^{4} + 24 x^{5} e^{3} d^{3} b a^{5} + 3 x^{5} e^{4} d^{2} a^{6} + 5 x^{4} d^{6} b^{3} a^{3} + \frac {45}{2} x^{4} e d^{5} b^{2} a^{4} + \frac {45}{2} x^{4} e^{2} d^{4} b a^{5} + 5 x^{4} e^{3} d^{3} a^{6} + 5 x^{3} d^{6} b^{2} a^{4} + 12 x^{3} e d^{5} b a^{5} + 5 x^{3} e^{2} d^{4} a^{6} + 3 x^{2} d^{6} b a^{5} + 3 x^{2} e d^{5} a^{6} + x d^{6} a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 661, normalized size = 3.87 \[ \frac {1}{13} \, b^{6} x^{13} e^{6} + \frac {1}{2} \, b^{6} d x^{12} e^{5} + \frac {15}{11} \, b^{6} d^{2} x^{11} e^{4} + 2 \, b^{6} d^{3} x^{10} e^{3} + \frac {5}{3} \, b^{6} d^{4} x^{9} e^{2} + \frac {3}{4} \, b^{6} d^{5} x^{8} e + \frac {1}{7} \, b^{6} d^{6} x^{7} + \frac {1}{2} \, a b^{5} x^{12} e^{6} + \frac {36}{11} \, a b^{5} d x^{11} e^{5} + 9 \, a b^{5} d^{2} x^{10} e^{4} + \frac {40}{3} \, a b^{5} d^{3} x^{9} e^{3} + \frac {45}{4} \, a b^{5} d^{4} x^{8} e^{2} + \frac {36}{7} \, a b^{5} d^{5} x^{7} e + a b^{5} d^{6} x^{6} + \frac {15}{11} \, a^{2} b^{4} x^{11} e^{6} + 9 \, a^{2} b^{4} d x^{10} e^{5} + 25 \, a^{2} b^{4} d^{2} x^{9} e^{4} + \frac {75}{2} \, a^{2} b^{4} d^{3} x^{8} e^{3} + \frac {225}{7} \, a^{2} b^{4} d^{4} x^{7} e^{2} + 15 \, a^{2} b^{4} d^{5} x^{6} e + 3 \, a^{2} b^{4} d^{6} x^{5} + 2 \, a^{3} b^{3} x^{10} e^{6} + \frac {40}{3} \, a^{3} b^{3} d x^{9} e^{5} + \frac {75}{2} \, a^{3} b^{3} d^{2} x^{8} e^{4} + \frac {400}{7} \, a^{3} b^{3} d^{3} x^{7} e^{3} + 50 \, a^{3} b^{3} d^{4} x^{6} e^{2} + 24 \, a^{3} b^{3} d^{5} x^{5} e + 5 \, a^{3} b^{3} d^{6} x^{4} + \frac {5}{3} \, a^{4} b^{2} x^{9} e^{6} + \frac {45}{4} \, a^{4} b^{2} d x^{8} e^{5} + \frac {225}{7} \, a^{4} b^{2} d^{2} x^{7} e^{4} + 50 \, a^{4} b^{2} d^{3} x^{6} e^{3} + 45 \, a^{4} b^{2} d^{4} x^{5} e^{2} + \frac {45}{2} \, a^{4} b^{2} d^{5} x^{4} e + 5 \, a^{4} b^{2} d^{6} x^{3} + \frac {3}{4} \, a^{5} b x^{8} e^{6} + \frac {36}{7} \, a^{5} b d x^{7} e^{5} + 15 \, a^{5} b d^{2} x^{6} e^{4} + 24 \, a^{5} b d^{3} x^{5} e^{3} + \frac {45}{2} \, a^{5} b d^{4} x^{4} e^{2} + 12 \, a^{5} b d^{5} x^{3} e + 3 \, a^{5} b d^{6} x^{2} + \frac {1}{7} \, a^{6} x^{7} e^{6} + a^{6} d x^{6} e^{5} + 3 \, a^{6} d^{2} x^{5} e^{4} + 5 \, a^{6} d^{3} x^{4} e^{3} + 5 \, a^{6} d^{4} x^{3} e^{2} + 3 \, a^{6} d^{5} x^{2} e + a^{6} d^{6} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 615, normalized size = 3.60 \[ \frac {b^{6} e^{6} x^{13}}{13}+a^{6} d^{6} x +\frac {\left (6 e^{6} a \,b^{5}+6 d \,e^{5} b^{6}\right ) x^{12}}{12}+\frac {\left (15 e^{6} a^{2} b^{4}+36 d \,e^{5} a \,b^{5}+15 d^{2} e^{4} b^{6}\right ) x^{11}}{11}+\frac {\left (20 e^{6} a^{3} b^{3}+90 d \,e^{5} a^{2} b^{4}+90 d^{2} e^{4} a \,b^{5}+20 d^{3} e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (15 e^{6} a^{4} b^{2}+120 d \,e^{5} a^{3} b^{3}+225 d^{2} e^{4} a^{2} b^{4}+120 d^{3} e^{3} a \,b^{5}+15 d^{4} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (6 e^{6} a^{5} b +90 d \,e^{5} a^{4} b^{2}+300 d^{2} e^{4} a^{3} b^{3}+300 d^{3} e^{3} a^{2} b^{4}+90 d^{4} e^{2} a \,b^{5}+6 d^{5} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (e^{6} a^{6}+36 d \,e^{5} a^{5} b +225 d^{2} e^{4} a^{4} b^{2}+400 d^{3} e^{3} a^{3} b^{3}+225 d^{4} e^{2} a^{2} b^{4}+36 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) x^{7}}{7}+\frac {\left (6 d \,e^{5} a^{6}+90 d^{2} e^{4} a^{5} b +300 d^{3} e^{3} a^{4} b^{2}+300 d^{4} e^{2} a^{3} b^{3}+90 d^{5} e \,a^{2} b^{4}+6 d^{6} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (15 d^{2} e^{4} a^{6}+120 d^{3} e^{3} a^{5} b +225 d^{4} e^{2} a^{4} b^{2}+120 d^{5} e \,a^{3} b^{3}+15 d^{6} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (20 d^{3} e^{3} a^{6}+90 d^{4} e^{2} a^{5} b +90 d^{5} e \,a^{4} b^{2}+20 d^{6} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (15 d^{4} e^{2} a^{6}+36 d^{5} e \,a^{5} b +15 d^{6} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (6 d^{5} e \,a^{6}+6 d^{6} a^{5} b \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.44, size = 599, normalized size = 3.50 \[ \frac {1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac {1}{2} \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} + {\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac {5}{3} \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} + {\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \, {\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \, {\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 579, normalized size = 3.39 \[ x^6\,\left (a^6\,d\,e^5+15\,a^5\,b\,d^2\,e^4+50\,a^4\,b^2\,d^3\,e^3+50\,a^3\,b^3\,d^4\,e^2+15\,a^2\,b^4\,d^5\,e+a\,b^5\,d^6\right )+x^8\,\left (\frac {3\,a^5\,b\,e^6}{4}+\frac {45\,a^4\,b^2\,d\,e^5}{4}+\frac {75\,a^3\,b^3\,d^2\,e^4}{2}+\frac {75\,a^2\,b^4\,d^3\,e^3}{2}+\frac {45\,a\,b^5\,d^4\,e^2}{4}+\frac {3\,b^6\,d^5\,e}{4}\right )+x^5\,\left (3\,a^6\,d^2\,e^4+24\,a^5\,b\,d^3\,e^3+45\,a^4\,b^2\,d^4\,e^2+24\,a^3\,b^3\,d^5\,e+3\,a^2\,b^4\,d^6\right )+x^9\,\left (\frac {5\,a^4\,b^2\,e^6}{3}+\frac {40\,a^3\,b^3\,d\,e^5}{3}+25\,a^2\,b^4\,d^2\,e^4+\frac {40\,a\,b^5\,d^3\,e^3}{3}+\frac {5\,b^6\,d^4\,e^2}{3}\right )+x^7\,\left (\frac {a^6\,e^6}{7}+\frac {36\,a^5\,b\,d\,e^5}{7}+\frac {225\,a^4\,b^2\,d^2\,e^4}{7}+\frac {400\,a^3\,b^3\,d^3\,e^3}{7}+\frac {225\,a^2\,b^4\,d^4\,e^2}{7}+\frac {36\,a\,b^5\,d^5\,e}{7}+\frac {b^6\,d^6}{7}\right )+a^6\,d^6\,x+\frac {b^6\,e^6\,x^{13}}{13}+\frac {5\,a^3\,d^3\,x^4\,\left (2\,a^3\,e^3+9\,a^2\,b\,d\,e^2+9\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{2}+b^3\,e^3\,x^{10}\,\left (2\,a^3\,e^3+9\,a^2\,b\,d\,e^2+9\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )+3\,a^5\,d^5\,x^2\,\left (a\,e+b\,d\right )+\frac {b^5\,e^5\,x^{12}\,\left (a\,e+b\,d\right )}{2}+a^4\,d^4\,x^3\,\left (5\,a^2\,e^2+12\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {3\,b^4\,e^4\,x^{11}\,\left (5\,a^2\,e^2+12\,a\,b\,d\,e+5\,b^2\,d^2\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 677, normalized size = 3.96 \[ a^{6} d^{6} x + \frac {b^{6} e^{6} x^{13}}{13} + x^{12} \left (\frac {a b^{5} e^{6}}{2} + \frac {b^{6} d e^{5}}{2}\right ) + x^{11} \left (\frac {15 a^{2} b^{4} e^{6}}{11} + \frac {36 a b^{5} d e^{5}}{11} + \frac {15 b^{6} d^{2} e^{4}}{11}\right ) + x^{10} \left (2 a^{3} b^{3} e^{6} + 9 a^{2} b^{4} d e^{5} + 9 a b^{5} d^{2} e^{4} + 2 b^{6} d^{3} e^{3}\right ) + x^{9} \left (\frac {5 a^{4} b^{2} e^{6}}{3} + \frac {40 a^{3} b^{3} d e^{5}}{3} + 25 a^{2} b^{4} d^{2} e^{4} + \frac {40 a b^{5} d^{3} e^{3}}{3} + \frac {5 b^{6} d^{4} e^{2}}{3}\right ) + x^{8} \left (\frac {3 a^{5} b e^{6}}{4} + \frac {45 a^{4} b^{2} d e^{5}}{4} + \frac {75 a^{3} b^{3} d^{2} e^{4}}{2} + \frac {75 a^{2} b^{4} d^{3} e^{3}}{2} + \frac {45 a b^{5} d^{4} e^{2}}{4} + \frac {3 b^{6} d^{5} e}{4}\right ) + x^{7} \left (\frac {a^{6} e^{6}}{7} + \frac {36 a^{5} b d e^{5}}{7} + \frac {225 a^{4} b^{2} d^{2} e^{4}}{7} + \frac {400 a^{3} b^{3} d^{3} e^{3}}{7} + \frac {225 a^{2} b^{4} d^{4} e^{2}}{7} + \frac {36 a b^{5} d^{5} e}{7} + \frac {b^{6} d^{6}}{7}\right ) + x^{6} \left (a^{6} d e^{5} + 15 a^{5} b d^{2} e^{4} + 50 a^{4} b^{2} d^{3} e^{3} + 50 a^{3} b^{3} d^{4} e^{2} + 15 a^{2} b^{4} d^{5} e + a b^{5} d^{6}\right ) + x^{5} \left (3 a^{6} d^{2} e^{4} + 24 a^{5} b d^{3} e^{3} + 45 a^{4} b^{2} d^{4} e^{2} + 24 a^{3} b^{3} d^{5} e + 3 a^{2} b^{4} d^{6}\right ) + x^{4} \left (5 a^{6} d^{3} e^{3} + \frac {45 a^{5} b d^{4} e^{2}}{2} + \frac {45 a^{4} b^{2} d^{5} e}{2} + 5 a^{3} b^{3} d^{6}\right ) + x^{3} \left (5 a^{6} d^{4} e^{2} + 12 a^{5} b d^{5} e + 5 a^{4} b^{2} d^{6}\right ) + x^{2} \left (3 a^{6} d^{5} e + 3 a^{5} b d^{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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