Optimal. Leaf size=86 \[ -\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37}
\begin {gather*} \frac {(a+b x)^7 (-8 a B e+A b e+7 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^9} \, dx &=-\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{8 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{8 e (b d-a e) (d+e x)^8}+\frac {(7 b B d+A b e-8 a B e) (a+b x)^7}{56 e (b d-a e)^2 (d+e x)^7}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(597\) vs. \(2(86)=172\).
time = 0.21, size = 597, normalized size = 6.94 \begin {gather*} -\frac {a^6 e^6 (7 A e+B (d+8 e x))+2 a^5 b e^5 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a^4 b^2 e^4 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a^2 b^4 e^2 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+2 a b^5 e \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+b^6 \left (A e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 B \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{56 e^8 (d+e x)^8} \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.
\(2(82)=164\).
time = 0.07, size = 814, normalized size = 9.47 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. 825 vs.
\(2 (87) = 174\).
time = 0.33, size = 825, normalized size = 9.59 \begin {gather*} -\frac {56 \, B b^{6} x^{7} e^{7} + 7 \, B b^{6} d^{7} + 7 \, A a^{6} e^{7} + {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 28 \, {\left (7 \, B b^{6} d e^{6} + 6 \, B a b^{5} e^{7} + A b^{6} e^{7}\right )} x^{6} + {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 56 \, {\left (7 \, B b^{6} d^{2} e^{5} + 5 \, B a^{2} b^{4} e^{7} + 2 \, A a b^{5} e^{7} + {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{5} + {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 70 \, {\left (7 \, B b^{6} d^{3} e^{4} + 4 \, B a^{3} b^{3} e^{7} + 3 \, A a^{2} b^{4} e^{7} + {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{4} + {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 56 \, {\left (7 \, B b^{6} d^{4} e^{3} + 3 \, B a^{4} b^{2} e^{7} + 4 \, A a^{3} b^{3} e^{7} + {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{3} + {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 28 \, {\left (7 \, B b^{6} d^{5} e^{2} + 2 \, B a^{5} b e^{7} + 5 \, A a^{4} b^{2} e^{7} + {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{2} + {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d + 8 \, {\left (7 \, B b^{6} d^{6} e + B a^{6} e^{7} + 6 \, A a^{5} b e^{7} + {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x}{56 \, {\left (x^{8} e^{16} + 8 \, d x^{7} e^{15} + 28 \, d^{2} x^{6} e^{14} + 56 \, d^{3} x^{5} e^{13} + 70 \, d^{4} x^{4} e^{12} + 56 \, d^{5} x^{3} e^{11} + 28 \, d^{6} x^{2} e^{10} + 8 \, d^{7} x e^{9} + d^{8} 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. 795 vs.
\(2 (87) = 174\).
time = 1.34, size = 795, normalized size = 9.24 \begin {gather*} -\frac {7 \, B b^{6} d^{7} + {\left (56 \, B b^{6} x^{7} + 7 \, A a^{6} + 28 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 56 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 70 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 56 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 28 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} e^{7} + {\left (196 \, B b^{6} d x^{6} + 56 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{5} + 70 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{4} + 56 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{3} + 28 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{2} + 8 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x + {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} e^{6} + {\left (392 \, B b^{6} d^{2} x^{5} + 70 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{4} + 56 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{3} + 28 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{2} + 8 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2}\right )} e^{5} + {\left (490 \, B b^{6} d^{3} x^{4} + 56 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{3} + 28 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{2} + 8 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3}\right )} e^{4} + {\left (392 \, B b^{6} d^{4} x^{3} + 28 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{2} + 8 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4}\right )} e^{3} + {\left (196 \, B b^{6} d^{5} x^{2} + 8 \, {\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 (56 \, B b^{6} d^{6} x + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6}\right )} e}{56 \, {\left (x^{8} e^{16} + 8 \, d x^{7} e^{15} + 28 \, d^{2} x^{6} e^{14} + 56 \, d^{3} x^{5} e^{13} + 70 \, d^{4} x^{4} e^{12} + 56 \, d^{5} x^{3} e^{11} + 28 \, d^{6} x^{2} e^{10} + 8 \, d^{7} x e^{9} + d^{8} 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. 854 vs.
\(2 (87) = 174\).
time = 1.10, size = 854, normalized size = 9.93 \begin {gather*} -\frac {{\left (56 \, B b^{6} x^{7} e^{7} + 196 \, B b^{6} d x^{6} e^{6} + 392 \, B b^{6} d^{2} x^{5} e^{5} + 490 \, B b^{6} d^{3} x^{4} e^{4} + 392 \, B b^{6} d^{4} x^{3} e^{3} + 196 \, B b^{6} d^{5} x^{2} e^{2} + 56 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 168 \, B a b^{5} x^{6} e^{7} + 28 \, A b^{6} x^{6} e^{7} + 336 \, B a b^{5} d x^{5} e^{6} + 56 \, A b^{6} d x^{5} e^{6} + 420 \, B a b^{5} d^{2} x^{4} e^{5} + 70 \, A b^{6} d^{2} x^{4} e^{5} + 336 \, B a b^{5} d^{3} x^{3} e^{4} + 56 \, A b^{6} d^{3} x^{3} e^{4} + 168 \, B a b^{5} d^{4} x^{2} e^{3} + 28 \, A b^{6} d^{4} x^{2} e^{3} + 48 \, B a b^{5} d^{5} x e^{2} + 8 \, A b^{6} d^{5} x e^{2} + 6 \, B a b^{5} d^{6} e + A b^{6} d^{6} e + 280 \, B a^{2} b^{4} x^{5} e^{7} + 112 \, A a b^{5} x^{5} e^{7} + 350 \, B a^{2} b^{4} d x^{4} e^{6} + 140 \, A a b^{5} d x^{4} e^{6} + 280 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 112 \, A a b^{5} d^{2} x^{3} e^{5} + 140 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 56 \, A a b^{5} d^{3} x^{2} e^{4} + 40 \, B a^{2} b^{4} d^{4} x e^{3} + 16 \, A a b^{5} d^{4} x e^{3} + 5 \, B a^{2} b^{4} d^{5} e^{2} + 2 \, A a b^{5} d^{5} e^{2} + 280 \, B a^{3} b^{3} x^{4} e^{7} + 210 \, A a^{2} b^{4} x^{4} e^{7} + 224 \, B a^{3} b^{3} d x^{3} e^{6} + 168 \, A a^{2} b^{4} d x^{3} e^{6} + 112 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 84 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 32 \, B a^{3} b^{3} d^{3} x e^{4} + 24 \, A a^{2} b^{4} d^{3} x e^{4} + 4 \, B a^{3} b^{3} d^{4} e^{3} + 3 \, A a^{2} b^{4} d^{4} e^{3} + 168 \, B a^{4} b^{2} x^{3} e^{7} + 224 \, A a^{3} b^{3} x^{3} e^{7} + 84 \, B a^{4} b^{2} d x^{2} e^{6} + 112 \, A a^{3} b^{3} d x^{2} e^{6} + 24 \, B a^{4} b^{2} d^{2} x e^{5} + 32 \, A a^{3} b^{3} d^{2} x e^{5} + 3 \, B a^{4} b^{2} d^{3} e^{4} + 4 \, A a^{3} b^{3} d^{3} e^{4} + 56 \, B a^{5} b x^{2} e^{7} + 140 \, A a^{4} b^{2} x^{2} e^{7} + 16 \, B a^{5} b d x e^{6} + 40 \, A a^{4} b^{2} d x e^{6} + 2 \, B a^{5} b d^{2} e^{5} + 5 \, A a^{4} b^{2} d^{2} e^{5} + 8 \, B a^{6} x e^{7} + 48 \, A a^{5} b x e^{7} + B a^{6} d e^{6} + 6 \, A a^{5} b d e^{6} + 7 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{56 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 854, normalized size = 9.93 \begin {gather*} -\frac {\frac {B\,a^6\,d\,e^6+7\,A\,a^6\,e^7+2\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+3\,B\,a^4\,b^2\,d^3\,e^4+5\,A\,a^4\,b^2\,d^2\,e^5+4\,B\,a^3\,b^3\,d^4\,e^3+4\,A\,a^3\,b^3\,d^3\,e^4+5\,B\,a^2\,b^4\,d^5\,e^2+3\,A\,a^2\,b^4\,d^4\,e^3+6\,B\,a\,b^5\,d^6\,e+2\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+A\,b^6\,d^6\,e}{56\,e^8}+\frac {x\,\left (B\,a^6\,e^6+2\,B\,a^5\,b\,d\,e^5+6\,A\,a^5\,b\,e^6+3\,B\,a^4\,b^2\,d^2\,e^4+5\,A\,a^4\,b^2\,d\,e^5+4\,B\,a^3\,b^3\,d^3\,e^3+4\,A\,a^3\,b^3\,d^2\,e^4+5\,B\,a^2\,b^4\,d^4\,e^2+3\,A\,a^2\,b^4\,d^3\,e^3+6\,B\,a\,b^5\,d^5\,e+2\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+A\,b^6\,d^5\,e\right )}{7\,e^7}+\frac {5\,b^3\,x^4\,\left (4\,B\,a^3\,e^3+5\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+6\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (A\,b\,e+6\,B\,a\,e+7\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^5\,e^5+3\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5+4\,B\,a^3\,b^2\,d^2\,e^3+4\,A\,a^3\,b^2\,d\,e^4+5\,B\,a^2\,b^3\,d^3\,e^2+3\,A\,a^2\,b^3\,d^2\,e^3+6\,B\,a\,b^4\,d^4\,e+2\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+A\,b^5\,d^4\,e\right )}{2\,e^6}+\frac {b^2\,x^3\,\left (3\,B\,a^4\,e^4+4\,B\,a^3\,b\,d\,e^3+4\,A\,a^3\,b\,e^4+5\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3+6\,B\,a\,b^3\,d^3\,e+2\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{e^5}+\frac {b^4\,x^5\,\left (5\,B\,a^2\,e^2+6\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{e^3}+\frac {B\,b^6\,x^7}{e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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