Optimal. Leaf size=185 \[ -\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {b (7 b B d+3 A b e-10 a B e) (a+b x)^7}{360 e (b d-a e)^3 (d+e x)^8}+\frac {b^2 (7 b B d+3 A b e-10 a B e) (a+b x)^7}{2520 e (b d-a e)^4 (d+e x)^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\begin {gather*} \frac {b^2 (a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{2520 e (d+e x)^7 (b d-a e)^4}+\frac {b (a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{360 e (d+e x)^8 (b d-a e)^3}+\frac {(a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{90 e (d+e x)^9 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{10 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {(b (7 b B d+3 A b e-10 a B e)) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{45 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {b (7 b B d+3 A b e-10 a B e) (a+b x)^7}{360 e (b d-a e)^3 (d+e x)^8}+\frac {\left (b^2 (7 b B d+3 A b e-10 a B e)\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{360 e (b d-a e)^3}\\ &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {b (7 b B d+3 A b e-10 a B e) (a+b x)^7}{360 e (b d-a e)^3 (d+e x)^8}+\frac {b^2 (7 b B d+3 A b e-10 a B e) (a+b x)^7}{2520 e (b d-a e)^4 (d+e x)^7}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(185)=370\).
time = 0.18, size = 602, normalized size = 3.25 \begin {gather*} -\frac {28 a^6 e^6 (9 A e+B (d+10 e x))+42 a^5 b e^5 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+6 a b^5 e \left (2 A e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+b^6 \left (3 A e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 B \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(813\) vs.
\(2(177)=354\).
time = 0.09, size = 814, normalized size = 4.40 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 868 vs.
\(2 (188) = 376\).
time = 0.36, size = 868, normalized size = 4.69 \begin {gather*} -\frac {840 \, B b^{6} x^{7} e^{7} + 7 \, B b^{6} d^{7} + 252 \, A a^{6} e^{7} + 3 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 210 \, {\left (7 \, B b^{6} d e^{6} + 18 \, B a b^{5} e^{7} + 3 \, A b^{6} e^{7}\right )} x^{6} + 6 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 252 \, {\left (7 \, B b^{6} d^{2} e^{5} + 30 \, B a^{2} b^{4} e^{7} + 12 \, A a b^{5} e^{7} + 3 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} + 10 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 210 \, {\left (7 \, B b^{6} d^{3} e^{4} + 40 \, B a^{3} b^{3} e^{7} + 30 \, A a^{2} b^{4} e^{7} + 3 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + 6 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} + 15 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 120 \, {\left (7 \, B b^{6} d^{4} e^{3} + 45 \, B a^{4} b^{2} e^{7} + 60 \, A a^{3} b^{3} e^{7} + 3 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + 6 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + 10 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} + 21 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 45 \, {\left (7 \, B b^{6} d^{5} e^{2} + 42 \, B a^{5} b e^{7} + 105 \, A a^{4} b^{2} e^{7} + 3 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 6 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + 10 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + 15 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} + 28 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 10 \, {\left (7 \, B b^{6} d^{6} e + 28 \, B a^{6} e^{7} + 168 \, A a^{5} b e^{7} + 3 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 6 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + 10 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 15 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + 21 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{2520 \, {\left (x^{10} e^{18} + 10 \, d x^{9} e^{17} + 45 \, d^{2} x^{8} e^{16} + 120 \, d^{3} x^{7} e^{15} + 210 \, d^{4} x^{6} e^{14} + 252 \, d^{5} x^{5} e^{13} + 210 \, d^{6} x^{4} e^{12} + 120 \, d^{7} x^{3} e^{11} + 45 \, d^{8} x^{2} e^{10} + 10 \, d^{9} x e^{9} + d^{10} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 825 vs.
\(2 (188) = 376\).
time = 1.01, size = 825, normalized size = 4.46 \begin {gather*} -\frac {7 \, B b^{6} d^{7} + {\left (840 \, B b^{6} x^{7} + 252 \, A a^{6} + 630 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 1512 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 2100 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1800 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 945 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 280 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} + {\left (1470 \, B b^{6} d x^{6} + 756 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 1260 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} + 1200 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} + 675 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 210 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 3 \, {\left (588 \, B b^{6} d^{2} x^{5} + 210 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 240 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} + 150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 50 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x + 7 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + 5 \, {\left (294 \, B b^{6} d^{3} x^{4} + 72 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 54 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x + 3 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + 5 \, {\left (168 \, B b^{6} d^{4} x^{3} + 27 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (105 \, B b^{6} d^{5} x^{2} + 10 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + 2 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + {\left (70 \, B b^{6} d^{6} x + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e}{2520 \, {\left (x^{10} e^{18} + 10 \, d x^{9} e^{17} + 45 \, d^{2} x^{8} e^{16} + 120 \, d^{3} x^{7} e^{15} + 210 \, d^{4} x^{6} e^{14} + 252 \, d^{5} x^{5} e^{13} + 210 \, d^{6} x^{4} e^{12} + 120 \, d^{7} x^{3} e^{11} + 45 \, d^{8} x^{2} e^{10} + 10 \, d^{9} x e^{9} + d^{10} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 856 vs.
\(2 (188) = 376\).
time = 1.35, size = 856, normalized size = 4.63 \begin {gather*} -\frac {{\left (840 \, B b^{6} x^{7} e^{7} + 1470 \, B b^{6} d x^{6} e^{6} + 1764 \, B b^{6} d^{2} x^{5} e^{5} + 1470 \, B b^{6} d^{3} x^{4} e^{4} + 840 \, B b^{6} d^{4} x^{3} e^{3} + 315 \, B b^{6} d^{5} x^{2} e^{2} + 70 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 3780 \, B a b^{5} x^{6} e^{7} + 630 \, A b^{6} x^{6} e^{7} + 4536 \, B a b^{5} d x^{5} e^{6} + 756 \, A b^{6} d x^{5} e^{6} + 3780 \, B a b^{5} d^{2} x^{4} e^{5} + 630 \, A b^{6} d^{2} x^{4} e^{5} + 2160 \, B a b^{5} d^{3} x^{3} e^{4} + 360 \, A b^{6} d^{3} x^{3} e^{4} + 810 \, B a b^{5} d^{4} x^{2} e^{3} + 135 \, A b^{6} d^{4} x^{2} e^{3} + 180 \, B a b^{5} d^{5} x e^{2} + 30 \, A b^{6} d^{5} x e^{2} + 18 \, B a b^{5} d^{6} e + 3 \, A b^{6} d^{6} e + 7560 \, B a^{2} b^{4} x^{5} e^{7} + 3024 \, A a b^{5} x^{5} e^{7} + 6300 \, B a^{2} b^{4} d x^{4} e^{6} + 2520 \, A a b^{5} d x^{4} e^{6} + 3600 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 1440 \, A a b^{5} d^{2} x^{3} e^{5} + 1350 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 540 \, A a b^{5} d^{3} x^{2} e^{4} + 300 \, B a^{2} b^{4} d^{4} x e^{3} + 120 \, A a b^{5} d^{4} x e^{3} + 30 \, B a^{2} b^{4} d^{5} e^{2} + 12 \, A a b^{5} d^{5} e^{2} + 8400 \, B a^{3} b^{3} x^{4} e^{7} + 6300 \, A a^{2} b^{4} x^{4} e^{7} + 4800 \, B a^{3} b^{3} d x^{3} e^{6} + 3600 \, A a^{2} b^{4} d x^{3} e^{6} + 1800 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 1350 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 400 \, B a^{3} b^{3} d^{3} x e^{4} + 300 \, A a^{2} b^{4} d^{3} x e^{4} + 40 \, B a^{3} b^{3} d^{4} e^{3} + 30 \, A a^{2} b^{4} d^{4} e^{3} + 5400 \, B a^{4} b^{2} x^{3} e^{7} + 7200 \, A a^{3} b^{3} x^{3} e^{7} + 2025 \, B a^{4} b^{2} d x^{2} e^{6} + 2700 \, A a^{3} b^{3} d x^{2} e^{6} + 450 \, B a^{4} b^{2} d^{2} x e^{5} + 600 \, A a^{3} b^{3} d^{2} x e^{5} + 45 \, B a^{4} b^{2} d^{3} e^{4} + 60 \, A a^{3} b^{3} d^{3} e^{4} + 1890 \, B a^{5} b x^{2} e^{7} + 4725 \, A a^{4} b^{2} x^{2} e^{7} + 420 \, B a^{5} b d x e^{6} + 1050 \, A a^{4} b^{2} d x e^{6} + 42 \, B a^{5} b d^{2} e^{5} + 105 \, A a^{4} b^{2} d^{2} e^{5} + 280 \, B a^{6} x e^{7} + 1680 \, A a^{5} b x e^{7} + 28 \, B a^{6} d e^{6} + 168 \, A a^{5} b d e^{6} + 252 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{2520 \, {\left (x e + d\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 888, normalized size = 4.80 \begin {gather*} -\frac {\frac {28\,B\,a^6\,d\,e^6+252\,A\,a^6\,e^7+42\,B\,a^5\,b\,d^2\,e^5+168\,A\,a^5\,b\,d\,e^6+45\,B\,a^4\,b^2\,d^3\,e^4+105\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3+60\,A\,a^3\,b^3\,d^3\,e^4+30\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3+18\,B\,a\,b^5\,d^6\,e+12\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+3\,A\,b^6\,d^6\,e}{2520\,e^8}+\frac {x\,\left (28\,B\,a^6\,e^6+42\,B\,a^5\,b\,d\,e^5+168\,A\,a^5\,b\,e^6+45\,B\,a^4\,b^2\,d^2\,e^4+105\,A\,a^4\,b^2\,d\,e^5+40\,B\,a^3\,b^3\,d^3\,e^3+60\,A\,a^3\,b^3\,d^2\,e^4+30\,B\,a^2\,b^4\,d^4\,e^2+30\,A\,a^2\,b^4\,d^3\,e^3+18\,B\,a\,b^5\,d^5\,e+12\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+3\,A\,b^6\,d^5\,e\right )}{252\,e^7}+\frac {b^3\,x^4\,\left (40\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+18\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{12\,e^4}+\frac {b^5\,x^6\,\left (3\,A\,b\,e+18\,B\,a\,e+7\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (42\,B\,a^5\,e^5+45\,B\,a^4\,b\,d\,e^4+105\,A\,a^4\,b\,e^5+40\,B\,a^3\,b^2\,d^2\,e^3+60\,A\,a^3\,b^2\,d\,e^4+30\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3+18\,B\,a\,b^4\,d^4\,e+12\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+3\,A\,b^5\,d^4\,e\right )}{56\,e^6}+\frac {b^2\,x^3\,\left (45\,B\,a^4\,e^4+40\,B\,a^3\,b\,d\,e^3+60\,A\,a^3\,b\,e^4+30\,B\,a^2\,b^2\,d^2\,e^2+30\,A\,a^2\,b^2\,d\,e^3+18\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+3\,A\,b^4\,d^3\,e\right )}{21\,e^5}+\frac {b^4\,x^5\,\left (30\,B\,a^2\,e^2+18\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{10\,e^3}+\frac {B\,b^6\,x^7}{3\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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