Optimal. Leaf size=135 \[ -\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {b (7 b B d+2 A b e-9 a B e) (a+b x)^7}{504 e (b d-a e)^3 (d+e x)^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 (a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{504 e (d+e x)^7 (b d-a e)^3}+\frac {(a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{72 e (d+e x)^8 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{9 e (d+e x)^9 (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)^{10}} \, dx &=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{9 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {(b (7 b B d+2 A b e-9 a B e)) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{72 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {b (7 b B d+2 A b e-9 a B e) (a+b x)^7}{504 e (b d-a e)^3 (d+e x)^7}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(603\) vs. \(2(135)=270\).
time = 0.20, size = 603, normalized size = 4.47 \begin {gather*} -\frac {7 a^6 e^6 (8 A e+B (d+9 e x))+6 a^5 b e^5 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 a^2 b^4 e^2 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+6 a b^5 e \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+b^6 \left (2 A e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 B \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{504 e^8 (d+e x)^9} \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.
\(2(129)=258\).
time = 0.09, size = 814, normalized size = 6.03 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. 858 vs.
\(2 (137) = 274\).
time = 0.34, size = 858, normalized size = 6.36 \begin {gather*} -\frac {252 \, B b^{6} x^{7} e^{7} + 7 \, B b^{6} d^{7} + 56 \, A a^{6} e^{7} + 2 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 84 \, {\left (7 \, B b^{6} d e^{6} + 12 \, B a b^{5} e^{7} + 2 \, A b^{6} e^{7}\right )} x^{6} + 3 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 126 \, {\left (7 \, B b^{6} d^{2} e^{5} + 15 \, B a^{2} b^{4} e^{7} + 6 \, A a b^{5} e^{7} + 2 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} + 4 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 126 \, {\left (7 \, B b^{6} d^{3} e^{4} + 16 \, B a^{3} b^{3} e^{7} + 12 \, A a^{2} b^{4} e^{7} + 2 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + 3 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} + 5 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 84 \, {\left (7 \, B b^{6} d^{4} e^{3} + 15 \, B a^{4} b^{2} e^{7} + 20 \, A a^{3} b^{3} e^{7} + 2 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + 3 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + 4 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} + 6 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 36 \, {\left (7 \, B b^{6} d^{5} e^{2} + 12 \, B a^{5} b e^{7} + 30 \, A a^{4} b^{2} e^{7} + 2 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 3 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + 4 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + 5 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} + 7 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 9 \, {\left (7 \, B b^{6} d^{6} e + 7 \, B a^{6} e^{7} + 42 \, A a^{5} b e^{7} + 2 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 3 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + 4 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 5 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + 6 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{504 \, {\left (x^{9} e^{17} + 9 \, d x^{8} e^{16} + 36 \, d^{2} x^{7} e^{15} + 84 \, d^{3} x^{6} e^{14} + 126 \, d^{4} x^{5} e^{13} + 126 \, d^{5} x^{4} e^{12} + 84 \, d^{6} x^{3} e^{11} + 36 \, d^{7} x^{2} e^{10} + 9 \, d^{8} x e^{9} + d^{9} 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. 812 vs.
\(2 (137) = 274\).
time = 0.98, size = 812, normalized size = 6.01 \begin {gather*} -\frac {7 \, B b^{6} d^{7} + {\left (252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} + {\left (588 \, B b^{6} d x^{6} + 252 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} + 336 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} + 180 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 54 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + 3 \, {\left (294 \, B b^{6} d^{2} x^{5} + 84 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 84 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} + 48 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x + 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + {\left (882 \, B b^{6} d^{3} x^{4} + 168 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 108 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 36 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + {\left (588 \, B b^{6} d^{4} x^{3} + 72 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 27 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + 3 \, {\left (84 \, B b^{6} d^{5} x^{2} + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5}\right )} e^{2} + {\left (63 \, B b^{6} d^{6} x + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e}{504 \, {\left (x^{9} e^{17} + 9 \, d x^{8} e^{16} + 36 \, d^{2} x^{7} e^{15} + 84 \, d^{3} x^{6} e^{14} + 126 \, d^{4} x^{5} e^{13} + 126 \, d^{5} x^{4} e^{12} + 84 \, d^{6} x^{3} e^{11} + 36 \, d^{7} x^{2} e^{10} + 9 \, d^{8} x e^{9} + d^{9} 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.
\(2 (137) = 274\).
time = 2.40, size = 856, normalized size = 6.34 \begin {gather*} -\frac {{\left (252 \, B b^{6} x^{7} e^{7} + 588 \, B b^{6} d x^{6} e^{6} + 882 \, B b^{6} d^{2} x^{5} e^{5} + 882 \, B b^{6} d^{3} x^{4} e^{4} + 588 \, B b^{6} d^{4} x^{3} e^{3} + 252 \, B b^{6} d^{5} x^{2} e^{2} + 63 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 1008 \, B a b^{5} x^{6} e^{7} + 168 \, A b^{6} x^{6} e^{7} + 1512 \, B a b^{5} d x^{5} e^{6} + 252 \, A b^{6} d x^{5} e^{6} + 1512 \, B a b^{5} d^{2} x^{4} e^{5} + 252 \, A b^{6} d^{2} x^{4} e^{5} + 1008 \, B a b^{5} d^{3} x^{3} e^{4} + 168 \, A b^{6} d^{3} x^{3} e^{4} + 432 \, B a b^{5} d^{4} x^{2} e^{3} + 72 \, A b^{6} d^{4} x^{2} e^{3} + 108 \, B a b^{5} d^{5} x e^{2} + 18 \, A b^{6} d^{5} x e^{2} + 12 \, B a b^{5} d^{6} e + 2 \, A b^{6} d^{6} e + 1890 \, B a^{2} b^{4} x^{5} e^{7} + 756 \, A a b^{5} x^{5} e^{7} + 1890 \, B a^{2} b^{4} d x^{4} e^{6} + 756 \, A a b^{5} d x^{4} e^{6} + 1260 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 504 \, A a b^{5} d^{2} x^{3} e^{5} + 540 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 216 \, A a b^{5} d^{3} x^{2} e^{4} + 135 \, B a^{2} b^{4} d^{4} x e^{3} + 54 \, A a b^{5} d^{4} x e^{3} + 15 \, B a^{2} b^{4} d^{5} e^{2} + 6 \, A a b^{5} d^{5} e^{2} + 2016 \, B a^{3} b^{3} x^{4} e^{7} + 1512 \, A a^{2} b^{4} x^{4} e^{7} + 1344 \, B a^{3} b^{3} d x^{3} e^{6} + 1008 \, A a^{2} b^{4} d x^{3} e^{6} + 576 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 432 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 144 \, B a^{3} b^{3} d^{3} x e^{4} + 108 \, A a^{2} b^{4} d^{3} x e^{4} + 16 \, B a^{3} b^{3} d^{4} e^{3} + 12 \, A a^{2} b^{4} d^{4} e^{3} + 1260 \, B a^{4} b^{2} x^{3} e^{7} + 1680 \, A a^{3} b^{3} x^{3} e^{7} + 540 \, B a^{4} b^{2} d x^{2} e^{6} + 720 \, A a^{3} b^{3} d x^{2} e^{6} + 135 \, B a^{4} b^{2} d^{2} x e^{5} + 180 \, A a^{3} b^{3} d^{2} x e^{5} + 15 \, B a^{4} b^{2} d^{3} e^{4} + 20 \, A a^{3} b^{3} d^{3} e^{4} + 432 \, B a^{5} b x^{2} e^{7} + 1080 \, A a^{4} b^{2} x^{2} e^{7} + 108 \, B a^{5} b d x e^{6} + 270 \, A a^{4} b^{2} d x e^{6} + 12 \, B a^{5} b d^{2} e^{5} + 30 \, A a^{4} b^{2} d^{2} e^{5} + 63 \, B a^{6} x e^{7} + 378 \, A a^{5} b x e^{7} + 7 \, B a^{6} d e^{6} + 42 \, A a^{5} b d e^{6} + 56 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{504 \, {\left (x e + d\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 877, normalized size = 6.50 \begin {gather*} -\frac {\frac {7\,B\,a^6\,d\,e^6+56\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5+42\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+16\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4+15\,B\,a^2\,b^4\,d^5\,e^2+12\,A\,a^2\,b^4\,d^4\,e^3+12\,B\,a\,b^5\,d^6\,e+6\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+2\,A\,b^6\,d^6\,e}{504\,e^8}+\frac {x\,\left (7\,B\,a^6\,e^6+12\,B\,a^5\,b\,d\,e^5+42\,A\,a^5\,b\,e^6+15\,B\,a^4\,b^2\,d^2\,e^4+30\,A\,a^4\,b^2\,d\,e^5+16\,B\,a^3\,b^3\,d^3\,e^3+20\,A\,a^3\,b^3\,d^2\,e^4+15\,B\,a^2\,b^4\,d^4\,e^2+12\,A\,a^2\,b^4\,d^3\,e^3+12\,B\,a\,b^5\,d^5\,e+6\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+2\,A\,b^6\,d^5\,e\right )}{56\,e^7}+\frac {b^3\,x^4\,\left (16\,B\,a^3\,e^3+15\,B\,a^2\,b\,d\,e^2+12\,A\,a^2\,b\,e^3+12\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+2\,A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (2\,A\,b\,e+12\,B\,a\,e+7\,B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (12\,B\,a^5\,e^5+15\,B\,a^4\,b\,d\,e^4+30\,A\,a^4\,b\,e^5+16\,B\,a^3\,b^2\,d^2\,e^3+20\,A\,a^3\,b^2\,d\,e^4+15\,B\,a^2\,b^3\,d^3\,e^2+12\,A\,a^2\,b^3\,d^2\,e^3+12\,B\,a\,b^4\,d^4\,e+6\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+2\,A\,b^5\,d^4\,e\right )}{14\,e^6}+\frac {b^2\,x^3\,\left (15\,B\,a^4\,e^4+16\,B\,a^3\,b\,d\,e^3+20\,A\,a^3\,b\,e^4+15\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+12\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+2\,A\,b^4\,d^3\,e\right )}{6\,e^5}+\frac {b^4\,x^5\,\left (15\,B\,a^2\,e^2+12\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^6\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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