Optimal. Leaf size=89 \[ \frac {(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^7}{36 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^7}{252 (b d-a e)^3 (d+e x)^7} \]
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Rubi [A]
time = 0.01, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, 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^2 (a+b x)^7}{252 (d+e x)^7 (b d-a e)^3}+\frac {b (a+b x)^7}{36 (d+e x)^8 (b d-a e)^2}+\frac {(a+b x)^7}{9 (d+e x)^9 (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)^{10}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx\\ &=\frac {(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac {(2 b) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{9 (b d-a e)}\\ &=\frac {(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^7}{36 (b d-a e)^2 (d+e x)^8}+\frac {b^2 \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{36 (b d-a e)^2}\\ &=\frac {(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^7}{36 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^7}{252 (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. \(277\) vs. \(2(89)=178\).
time = 0.06, size = 277, normalized size = 3.11 \begin {gather*} -\frac {28 a^6 e^6+21 a^5 b e^5 (d+9 e x)+15 a^4 b^2 e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+6 a^2 b^4 e^2 \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 )+3 a b^5 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 )+b^6 \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 )}{252 e^7 (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. \(356\) vs.
\(2(83)=166\).
time = 0.66, size = 357, normalized size = 4.01
method | result | size |
risch | \(\frac {-\frac {b^{6} x^{6}}{3 e}-\frac {b^{5} \left (3 a e +b d \right ) x^{5}}{2 e^{2}}-\frac {b^{4} \left (6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {b^{3} \left (10 e^{3} a^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {b^{2} \left (15 e^{4} a^{4}+10 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+3 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{7 e^{5}}-\frac {b \left (21 a^{5} e^{5}+15 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+6 a^{2} b^{3} d^{3} e^{2}+3 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{28 e^{6}}-\frac {28 a^{6} e^{6}+21 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+6 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}}{252 e^{7}}}{\left (e x +d \right )^{9}}\) | \(335\) |
default | \(-\frac {15 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{9 e^{7} \left (e x +d \right )^{9}}-\frac {b^{6}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{5}}-\frac {10 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{7} \left (e x +d \right )^{6}}-\frac {3 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{4 e^{7} \left (e x +d \right )^{8}}-\frac {3 b^{5} \left (a e -b d \right )}{2 e^{7} \left (e x +d \right )^{4}}\) | \(357\) |
norman | \(\frac {-\frac {b^{6} x^{6}}{3 e}-\frac {\left (3 e^{3} a \,b^{5}+d \,e^{2} b^{6}\right ) x^{5}}{2 e^{4}}-\frac {\left (6 e^{4} a^{2} b^{4}+3 d \,e^{3} a \,b^{5}+d^{2} e^{2} b^{6}\right ) x^{4}}{2 e^{5}}-\frac {\left (10 e^{5} a^{3} b^{3}+6 a^{2} b^{4} d \,e^{4}+3 d^{2} e^{3} a \,b^{5}+d^{3} e^{2} b^{6}\right ) x^{3}}{3 e^{6}}-\frac {\left (15 a^{4} b^{2} e^{6}+10 a^{3} b^{3} d \,e^{5}+6 a^{2} b^{4} d^{2} e^{4}+3 a \,b^{5} d^{3} e^{3}+b^{6} d^{4} e^{2}\right ) x^{2}}{7 e^{7}}-\frac {\left (21 a^{5} b \,e^{7}+15 a^{4} b^{2} d \,e^{6}+10 a^{3} b^{3} d^{2} e^{5}+6 a^{2} b^{4} d^{3} e^{4}+3 a \,b^{5} e^{3} d^{4}+b^{6} d^{5} e^{2}\right ) x}{28 e^{8}}-\frac {28 a^{6} e^{8}+21 a^{5} b d \,e^{7}+15 a^{4} b^{2} d^{2} e^{6}+10 a^{3} b^{3} d^{3} e^{5}+6 a^{2} b^{4} d^{4} e^{4}+3 a \,b^{5} d^{5} e^{3}+b^{6} d^{6} e^{2}}{252 e^{9}}}{\left (e x +d \right )^{9}}\) | \(375\) |
gosper | \(-\frac {84 b^{6} x^{6} e^{6}+378 a \,b^{5} e^{6} x^{5}+126 b^{6} d \,e^{5} x^{5}+756 a^{2} b^{4} e^{6} x^{4}+378 a \,b^{5} d \,e^{5} x^{4}+126 b^{6} d^{2} e^{4} x^{4}+840 a^{3} b^{3} e^{6} x^{3}+504 a^{2} b^{4} d \,e^{5} x^{3}+252 a \,b^{5} d^{2} e^{4} x^{3}+84 b^{6} d^{3} e^{3} x^{3}+540 a^{4} b^{2} e^{6} x^{2}+360 a^{3} b^{3} d \,e^{5} x^{2}+216 a^{2} b^{4} d^{2} e^{4} x^{2}+108 a \,b^{5} d^{3} e^{3} x^{2}+36 b^{6} d^{4} e^{2} x^{2}+189 a^{5} b \,e^{6} x +135 a^{4} b^{2} d \,e^{5} x +90 a^{3} b^{3} d^{2} e^{4} x +54 a^{2} b^{4} d^{3} e^{3} x +27 a \,b^{5} d^{4} e^{2} x +9 b^{6} d^{5} e x +28 a^{6} e^{6}+21 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+6 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}}{252 e^{7} \left (e x +d \right )^{9}}\) | \(376\) |
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. 408 vs.
\(2 (89) = 178\).
time = 0.33, size = 408, normalized size = 4.58 \begin {gather*} -\frac {84 \, b^{6} x^{6} e^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \, {\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \, {\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \, {\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + d^{9} 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. 407 vs.
\(2 (89) = 178\).
time = 2.79, size = 407, normalized size = 4.57 \begin {gather*} -\frac {b^{6} d^{6} + {\left (84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}\right )} e^{6} + 3 \, {\left (42 \, b^{6} d x^{5} + 126 \, a b^{5} d x^{4} + 168 \, a^{2} b^{4} d x^{3} + 120 \, a^{3} b^{3} d x^{2} + 45 \, a^{4} b^{2} d x + 7 \, a^{5} b d\right )} e^{5} + 3 \, {\left (42 \, b^{6} d^{2} x^{4} + 84 \, a b^{5} d^{2} x^{3} + 72 \, a^{2} b^{4} d^{2} x^{2} + 30 \, a^{3} b^{3} d^{2} x + 5 \, a^{4} b^{2} d^{2}\right )} e^{4} + 2 \, {\left (42 \, b^{6} d^{3} x^{3} + 54 \, a b^{5} d^{3} x^{2} + 27 \, a^{2} b^{4} d^{3} x + 5 \, a^{3} b^{3} d^{3}\right )} e^{3} + 3 \, {\left (12 \, b^{6} d^{4} x^{2} + 9 \, a b^{5} d^{4} x + 2 \, a^{2} b^{4} d^{4}\right )} e^{2} + 3 \, {\left (3 \, b^{6} d^{5} x + a b^{5} d^{5}\right )} e}{252 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + d^{9} 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.
\(2 (89) = 178\).
time = 1.80, size = 352, normalized size = 3.96 \begin {gather*} -\frac {{\left (84 \, b^{6} x^{6} e^{6} + 126 \, b^{6} d x^{5} e^{5} + 126 \, b^{6} d^{2} x^{4} e^{4} + 84 \, b^{6} d^{3} x^{3} e^{3} + 36 \, b^{6} d^{4} x^{2} e^{2} + 9 \, b^{6} d^{5} x e + b^{6} d^{6} + 378 \, a b^{5} x^{5} e^{6} + 378 \, a b^{5} d x^{4} e^{5} + 252 \, a b^{5} d^{2} x^{3} e^{4} + 108 \, a b^{5} d^{3} x^{2} e^{3} + 27 \, a b^{5} d^{4} x e^{2} + 3 \, a b^{5} d^{5} e + 756 \, a^{2} b^{4} x^{4} e^{6} + 504 \, a^{2} b^{4} d x^{3} e^{5} + 216 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 54 \, a^{2} b^{4} d^{3} x e^{3} + 6 \, a^{2} b^{4} d^{4} e^{2} + 840 \, a^{3} b^{3} x^{3} e^{6} + 360 \, a^{3} b^{3} d x^{2} e^{5} + 90 \, a^{3} b^{3} d^{2} x e^{4} + 10 \, a^{3} b^{3} d^{3} e^{3} + 540 \, a^{4} b^{2} x^{2} e^{6} + 135 \, a^{4} b^{2} d x e^{5} + 15 \, a^{4} b^{2} d^{2} e^{4} + 189 \, a^{5} b x e^{6} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{252 \, {\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 = 0.61, size = 423, normalized size = 4.75 \begin {gather*} -\frac {\frac {28\,a^6\,e^6+21\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+6\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+b^6\,d^6}{252\,e^7}+\frac {b^6\,x^6}{3\,e}+\frac {b^3\,x^3\,\left (10\,a^3\,e^3+6\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{3\,e^4}+\frac {b\,x\,\left (21\,a^5\,e^5+15\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3+6\,a^2\,b^3\,d^3\,e^2+3\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{28\,e^6}+\frac {b^5\,x^5\,\left (3\,a\,e+b\,d\right )}{2\,e^2}+\frac {b^2\,x^2\,\left (15\,a^4\,e^4+10\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+3\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{7\,e^5}+\frac {b^4\,x^4\,\left (6\,a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )}{2\,e^3}}{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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