Optimal. Leaf size=58 \[ \frac {(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^5}{30 (b d-a e)^2 (d+e x)^5} \]
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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)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac {(a+b x)^5}{6 (d+e x)^6 (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 )^2}{(d+e x)^7} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^7} \, dx\\ &=\frac {(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac {b \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{6 (b d-a e)}\\ &=\frac {(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^5}{30 (b d-a e)^2 (d+e x)^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(58)=116\).
time = 0.03, size = 144, normalized size = 2.48 \begin {gather*} -\frac {5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \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. \(185\) vs.
\(2(54)=108\).
time = 0.63, size = 186, normalized size = 3.21
method | result | size |
risch | \(\frac {-\frac {b^{4} x^{4}}{2 e}-\frac {2 b^{3} \left (2 a e +b d \right ) x^{3}}{3 e^{2}}-\frac {b^{2} \left (3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right ) x^{2}}{2 e^{3}}-\frac {b \left (4 e^{3} a^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{5 e^{4}}-\frac {5 e^{4} a^{4}+4 a^{3} b d \,e^{3}+3 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{6}}\) | \(171\) |
gosper | \(-\frac {15 b^{4} x^{4} e^{4}+40 a \,b^{3} e^{4} x^{3}+20 b^{4} d \,e^{3} x^{3}+45 a^{2} b^{2} e^{4} x^{2}+30 a \,b^{3} d \,e^{3} x^{2}+15 b^{4} d^{2} e^{2} x^{2}+24 a^{3} b \,e^{4} x +18 a^{2} b^{2} d \,e^{3} x +12 a \,b^{3} d^{2} e^{2} x +6 b^{4} d^{3} e x +5 e^{4} a^{4}+4 a^{3} b d \,e^{3}+3 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}}{30 e^{5} \left (e x +d \right )^{6}}\) | \(185\) |
default | \(-\frac {4 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {4 b^{3} \left (a e -b d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {b^{4}}{2 e^{5} \left (e x +d \right )^{2}}-\frac {3 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{5} \left (e x +d \right )^{4}}\) | \(186\) |
norman | \(\frac {-\frac {b^{4} x^{4}}{2 e}-\frac {2 \left (2 e^{2} a \,b^{3}+b^{4} d e \right ) x^{3}}{3 e^{3}}-\frac {\left (3 a^{2} b^{2} e^{3}+2 d \,e^{2} a \,b^{3}+d^{2} e \,b^{4}\right ) x^{2}}{2 e^{4}}-\frac {\left (4 a^{3} b \,e^{4}+3 a^{2} b^{2} d \,e^{3}+2 a \,b^{3} d^{2} e^{2}+b^{4} d^{3} e \right ) x}{5 e^{5}}-\frac {5 a^{4} e^{5}+4 a^{3} b d \,e^{4}+3 a^{2} b^{2} d^{2} e^{3}+2 a \,b^{3} d^{3} e^{2}+b^{4} d^{4} e}{30 e^{6}}}{\left (e x +d \right )^{6}}\) | \(189\) |
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. 219 vs.
\(2 (58) = 116\).
time = 0.30, size = 219, normalized size = 3.78 \begin {gather*} -\frac {15 \, b^{4} x^{4} e^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \, {\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\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. 214 vs.
\(2 (58) = 116\).
time = 3.42, size = 214, normalized size = 3.69 \begin {gather*} -\frac {b^{4} d^{4} + {\left (15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}\right )} e^{4} + 2 \, {\left (10 \, b^{4} d x^{3} + 15 \, a b^{3} d x^{2} + 9 \, a^{2} b^{2} d x + 2 \, a^{3} b d\right )} e^{3} + 3 \, {\left (5 \, b^{4} d^{2} x^{2} + 4 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} + 2 \, {\left (3 \, b^{4} d^{3} x + a b^{3} d^{3}\right )} e}{30 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs.
\(2 (46) = 92\).
time = 22.90, size = 255, normalized size = 4.40 \begin {gather*} \frac {- 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} - 15 b^{4} e^{4} x^{4} + x^{3} \left (- 40 a b^{3} e^{4} - 20 b^{4} d e^{3}\right ) + x^{2} \left (- 45 a^{2} b^{2} e^{4} - 30 a b^{3} d e^{3} - 15 b^{4} d^{2} e^{2}\right ) + x \left (- 24 a^{3} b e^{4} - 18 a^{2} b^{2} d e^{3} - 12 a b^{3} d^{2} e^{2} - 6 b^{4} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \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. 174 vs.
\(2 (58) = 116\).
time = 1.36, size = 174, normalized size = 3.00 \begin {gather*} -\frac {{\left (15 \, b^{4} x^{4} e^{4} + 20 \, b^{4} d x^{3} e^{3} + 15 \, b^{4} d^{2} x^{2} e^{2} + 6 \, b^{4} d^{3} x e + b^{4} d^{4} + 40 \, a b^{3} x^{3} e^{4} + 30 \, a b^{3} d x^{2} e^{3} + 12 \, a b^{3} d^{2} x e^{2} + 2 \, a b^{3} d^{3} e + 45 \, a^{2} b^{2} x^{2} e^{4} + 18 \, a^{2} b^{2} d x e^{3} + 3 \, a^{2} b^{2} d^{2} e^{2} + 24 \, a^{3} b x e^{4} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{30 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 226, normalized size = 3.90 \begin {gather*} -\frac {\frac {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}{30\,e^5}+\frac {b^4\,x^4}{2\,e}+\frac {2\,b^3\,x^3\,\left (2\,a\,e+b\,d\right )}{3\,e^2}+\frac {b\,x\,\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 )}{5\,e^4}+\frac {b^2\,x^2\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2\right )}{2\,e^3}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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