Optimal. Leaf size=58 \[ \frac {(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^7}{56 (b d-a e)^2 (d+e x)^7} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 47, 37}
\begin {gather*} \frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^9} \, dx\\ &=\frac {(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac {b \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{8 (b d-a e)}\\ &=\frac {(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^7}{56 (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. \(277\) vs. \(2(58)=116\).
time = 0.06, size = 277, normalized size = 4.78 \begin {gather*} -\frac {7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a^3 b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a^2 b^4 e^2 \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 )+2 a b^5 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 )+b^6 \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 )}{56 e^7 (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. \(356\) vs.
\(2(54)=108\).
time = 0.64, size = 357, normalized size = 6.16
method | result | size |
risch | \(\frac {-\frac {b^{6} x^{6}}{2 e}-\frac {b^{5} \left (2 a e +b d \right ) x^{5}}{e^{2}}-\frac {5 b^{4} \left (3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (4 e^{3} a^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {b^{2} \left (5 e^{4} a^{4}+4 a^{3} b d \,e^{3}+3 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{2 e^{5}}-\frac {b \left (6 a^{5} e^{5}+5 a^{4} b d \,e^{4}+4 a^{3} b^{2} d^{2} e^{3}+3 a^{2} b^{3} d^{3} e^{2}+2 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{7 e^{6}}-\frac {7 a^{6} e^{6}+6 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+4 a^{3} b^{3} d^{3} e^{3}+3 a^{2} b^{4} d^{4} e^{2}+2 a \,b^{5} d^{5} e +b^{6} d^{6}}{56 e^{7}}}{\left (e x +d \right )^{8}}\) | \(335\) |
default | \(-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {b^{6}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {2 b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )^{3}}-\frac {4 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{7} \left (e x +d \right )^{5}}-\frac {5 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}\) | \(357\) |
norman | \(\frac {-\frac {b^{6} x^{6}}{2 e}-\frac {\left (2 e^{2} a \,b^{5}+d e \,b^{6}\right ) x^{5}}{e^{3}}-\frac {5 \left (3 e^{3} a^{2} b^{4}+2 d \,e^{2} a \,b^{5}+d^{2} e \,b^{6}\right ) x^{4}}{4 e^{4}}-\frac {\left (4 e^{4} a^{3} b^{3}+3 d \,e^{3} a^{2} b^{4}+2 d^{2} e^{2} a \,b^{5}+d^{3} e \,b^{6}\right ) x^{3}}{e^{5}}-\frac {\left (5 e^{5} a^{4} b^{2}+4 d \,e^{4} a^{3} b^{3}+3 d^{2} e^{3} a^{2} b^{4}+2 d^{3} e^{2} a \,b^{5}+d^{4} e \,b^{6}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 a^{5} b \,e^{6}+5 a^{4} b^{2} d \,e^{5}+4 a^{3} b^{3} d^{2} e^{4}+3 a^{2} b^{4} d^{3} e^{3}+2 a \,b^{5} d^{4} e^{2}+b^{6} d^{5} e \right ) x}{7 e^{7}}-\frac {7 a^{6} e^{7}+6 a^{5} b d \,e^{6}+5 a^{4} b^{2} d^{2} e^{5}+4 a^{3} b^{3} d^{3} e^{4}+3 a^{2} b^{4} e^{3} d^{4}+2 a \,b^{5} d^{5} e^{2}+b^{6} d^{6} e}{56 e^{8}}}{\left (e x +d \right )^{8}}\) | \(363\) |
gosper | \(-\frac {28 b^{6} x^{6} e^{6}+112 a \,b^{5} e^{6} x^{5}+56 b^{6} d \,e^{5} x^{5}+210 a^{2} b^{4} e^{6} x^{4}+140 a \,b^{5} d \,e^{5} x^{4}+70 b^{6} d^{2} e^{4} x^{4}+224 a^{3} b^{3} e^{6} x^{3}+168 a^{2} b^{4} d \,e^{5} x^{3}+112 a \,b^{5} d^{2} e^{4} x^{3}+56 b^{6} d^{3} e^{3} x^{3}+140 a^{4} b^{2} e^{6} x^{2}+112 a^{3} b^{3} d \,e^{5} x^{2}+84 a^{2} b^{4} d^{2} e^{4} x^{2}+56 a \,b^{5} d^{3} e^{3} x^{2}+28 b^{6} d^{4} e^{2} x^{2}+48 a^{5} b \,e^{6} x +40 a^{4} b^{2} d \,e^{5} x +32 a^{3} b^{3} d^{2} e^{4} x +24 a^{2} b^{4} d^{3} e^{3} x +16 a \,b^{5} d^{4} e^{2} x +8 b^{6} d^{5} e x +7 a^{6} e^{6}+6 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+4 a^{3} b^{3} d^{3} e^{3}+3 a^{2} b^{4} d^{4} e^{2}+2 a \,b^{5} d^{5} e +b^{6} d^{6}}{56 e^{7} \left (e x +d \right )^{8}}\) | \(376\) |
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. 398 vs.
\(2 (58) = 116\).
time = 0.33, size = 398, normalized size = 6.86 \begin {gather*} -\frac {28 \, b^{6} x^{6} e^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (x^{8} e^{15} + 8 \, d x^{7} e^{14} + 28 \, d^{2} x^{6} e^{13} + 56 \, d^{3} x^{5} e^{12} + 70 \, d^{4} x^{4} e^{11} + 56 \, d^{5} x^{3} e^{10} + 28 \, d^{6} x^{2} e^{9} + 8 \, d^{7} x e^{8} + d^{8} e^{7}\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. 394 vs.
\(2 (58) = 116\).
time = 2.33, size = 394, normalized size = 6.79 \begin {gather*} -\frac {b^{6} d^{6} + {\left (28 \, b^{6} x^{6} + 112 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 224 \, a^{3} b^{3} x^{3} + 140 \, a^{4} b^{2} x^{2} + 48 \, a^{5} b x + 7 \, a^{6}\right )} e^{6} + 2 \, {\left (28 \, b^{6} d x^{5} + 70 \, a b^{5} d x^{4} + 84 \, a^{2} b^{4} d x^{3} + 56 \, a^{3} b^{3} d x^{2} + 20 \, a^{4} b^{2} d x + 3 \, a^{5} b d\right )} e^{5} + {\left (70 \, b^{6} d^{2} x^{4} + 112 \, a b^{5} d^{2} x^{3} + 84 \, a^{2} b^{4} d^{2} x^{2} + 32 \, a^{3} b^{3} d^{2} x + 5 \, a^{4} b^{2} d^{2}\right )} e^{4} + 4 \, {\left (14 \, b^{6} d^{3} x^{3} + 14 \, a b^{5} d^{3} x^{2} + 6 \, a^{2} b^{4} d^{3} x + a^{3} b^{3} d^{3}\right )} e^{3} + {\left (28 \, b^{6} d^{4} x^{2} + 16 \, a b^{5} d^{4} x + 3 \, a^{2} b^{4} d^{4}\right )} e^{2} + 2 \, {\left (4 \, b^{6} d^{5} x + a b^{5} d^{5}\right )} e}{56 \, {\left (x^{8} e^{15} + 8 \, d x^{7} e^{14} + 28 \, d^{2} x^{6} e^{13} + 56 \, d^{3} x^{5} e^{12} + 70 \, d^{4} x^{4} e^{11} + 56 \, d^{5} x^{3} e^{10} + 28 \, d^{6} x^{2} e^{9} + 8 \, d^{7} x e^{8} + d^{8} e^{7}\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. 352 vs.
\(2 (58) = 116\).
time = 1.45, size = 352, normalized size = 6.07 \begin {gather*} -\frac {{\left (28 \, b^{6} x^{6} e^{6} + 56 \, b^{6} d x^{5} e^{5} + 70 \, b^{6} d^{2} x^{4} e^{4} + 56 \, b^{6} d^{3} x^{3} e^{3} + 28 \, b^{6} d^{4} x^{2} e^{2} + 8 \, b^{6} d^{5} x e + b^{6} d^{6} + 112 \, a b^{5} x^{5} e^{6} + 140 \, a b^{5} d x^{4} e^{5} + 112 \, a b^{5} d^{2} x^{3} e^{4} + 56 \, a b^{5} d^{3} x^{2} e^{3} + 16 \, a b^{5} d^{4} x e^{2} + 2 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} x^{4} e^{6} + 168 \, a^{2} b^{4} d x^{3} e^{5} + 84 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 24 \, a^{2} b^{4} d^{3} x e^{3} + 3 \, a^{2} b^{4} d^{4} e^{2} + 224 \, a^{3} b^{3} x^{3} e^{6} + 112 \, a^{3} b^{3} d x^{2} e^{5} + 32 \, a^{3} b^{3} d^{2} x e^{4} + 4 \, a^{3} b^{3} d^{3} e^{3} + 140 \, a^{4} b^{2} x^{2} e^{6} + 40 \, a^{4} b^{2} d x e^{5} + 5 \, a^{4} b^{2} d^{2} e^{4} + 48 \, a^{5} b x e^{6} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6}\right )} e^{\left (-7\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 = 0.14, size = 410, normalized size = 7.07 \begin {gather*} -\frac {\frac {7\,a^6\,e^6+6\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+4\,a^3\,b^3\,d^3\,e^3+3\,a^2\,b^4\,d^4\,e^2+2\,a\,b^5\,d^5\,e+b^6\,d^6}{56\,e^7}+\frac {b^6\,x^6}{2\,e}+\frac {b^3\,x^3\,\left (4\,a^3\,e^3+3\,a^2\,b\,d\,e^2+2\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {b\,x\,\left (6\,a^5\,e^5+5\,a^4\,b\,d\,e^4+4\,a^3\,b^2\,d^2\,e^3+3\,a^2\,b^3\,d^3\,e^2+2\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{7\,e^6}+\frac {b^5\,x^5\,\left (2\,a\,e+b\,d\right )}{e^2}+\frac {b^2\,x^2\,\left (5\,a^4\,e^4+4\,a^3\,b\,d\,e^3+3\,a^2\,b^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{2\,e^5}+\frac {5\,b^4\,x^4\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2\right )}{4\,e^3}}{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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