Optimal. Leaf size=158 \[ -\frac {20 b^3 (b d-a e)^3 x}{e^6}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2}{2 e^7}-\frac {2 b^5 (b d-a e) (d+e x)^3}{e^7}+\frac {b^6 (d+e x)^4}{4 e^7}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} -\frac {2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac {15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac {20 b^3 x (b d-a e)^3}{e^6}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {b^6 (d+e x)^4}{4 e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^3} \, dx\\ &=\int \left (-\frac {20 b^3 (b d-a e)^3}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^3}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^2}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^2}{e^6}+\frac {b^6 (d+e x)^3}{e^6}\right ) \, dx\\ &=-\frac {20 b^3 (b d-a e)^3 x}{e^6}-\frac {(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac {6 b (b d-a e)^5}{e^7 (d+e x)}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2}{2 e^7}-\frac {2 b^5 (b d-a e) (d+e x)^3}{e^7}+\frac {b^6 (d+e x)^4}{4 e^7}+\frac {15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 303, normalized size = 1.92 \begin {gather*} \frac {-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 a b^5 e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+b^6 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )+60 b^2 (b d-a e)^4 (d+e x)^2 \log (d+e x)}{4 e^7 (d+e x)^2} \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. \(350\) vs.
\(2(152)=304\).
time = 0.71, size = 351, normalized size = 2.22
method | result | size |
norman | \(\frac {-\frac {a^{6} e^{6}+6 a^{5} b d \,e^{5}-45 a^{4} b^{2} d^{2} e^{4}+180 a^{3} b^{3} d^{3} e^{3}-270 a^{2} b^{4} d^{4} e^{2}+180 a \,b^{5} d^{5} e -45 b^{6} d^{6}}{2 e^{7}}+\frac {b^{6} x^{6}}{4 e}-\frac {2 \left (3 a^{5} b \,e^{5}-15 d \,e^{4} a^{4} b^{2}+60 d^{2} e^{3} a^{3} b^{3}-90 d^{3} e^{2} a^{2} b^{4}+60 d^{4} e a \,b^{5}-15 d^{5} b^{6}\right ) x}{e^{6}}+\frac {5 b^{3} \left (4 e^{3} a^{3}-6 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{3}}{e^{4}}+\frac {5 b^{4} \left (6 a^{2} e^{2}-4 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {b^{5} \left (4 a e -b d \right ) x^{5}}{2 e^{2}}}{\left (e x +d \right )^{2}}+\frac {15 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(343\) |
default | \(\frac {b^{3} \left (\frac {1}{4} b^{3} x^{4} e^{3}+2 a \,b^{2} e^{3} x^{3}-b^{3} d \,e^{2} x^{3}+\frac {15}{2} a^{2} b \,e^{3} x^{2}-9 a \,b^{2} d \,e^{2} x^{2}+3 b^{3} d^{2} e \,x^{2}+20 e^{3} a^{3} x -45 a^{2} b d \,e^{2} x +36 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{e^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{e^{7} \left (e x +d \right )}+\frac {15 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(351\) |
risch | \(\frac {b^{6} x^{4}}{4 e^{3}}+\frac {2 b^{5} a \,x^{3}}{e^{3}}-\frac {b^{6} d \,x^{3}}{e^{4}}+\frac {15 b^{4} a^{2} x^{2}}{2 e^{3}}-\frac {9 b^{5} a d \,x^{2}}{e^{4}}+\frac {3 b^{6} d^{2} x^{2}}{e^{5}}+\frac {20 b^{3} a^{3} x}{e^{3}}-\frac {45 b^{4} a^{2} d x}{e^{4}}+\frac {36 b^{5} a \,d^{2} x}{e^{5}}-\frac {10 b^{6} d^{3} x}{e^{6}}+\frac {\left (-6 a^{5} b \,e^{5}+30 d \,e^{4} a^{4} b^{2}-60 d^{2} e^{3} a^{3} b^{3}+60 d^{3} e^{2} a^{2} b^{4}-30 d^{4} e a \,b^{5}+6 d^{5} b^{6}\right ) x -\frac {a^{6} e^{6}+6 a^{5} b d \,e^{5}-45 a^{4} b^{2} d^{2} e^{4}+100 a^{3} b^{3} d^{3} e^{3}-105 a^{2} b^{4} d^{4} e^{2}+54 a \,b^{5} d^{5} e -11 b^{6} d^{6}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {15 b^{2} \ln \left (e x +d \right ) a^{4}}{e^{3}}-\frac {60 b^{3} \ln \left (e x +d \right ) a^{3} d}{e^{4}}+\frac {90 b^{4} \ln \left (e x +d \right ) a^{2} d^{2}}{e^{5}}-\frac {60 b^{5} \ln \left (e x +d \right ) a \,d^{3}}{e^{6}}+\frac {15 b^{6} \ln \left (e x +d \right ) d^{4}}{e^{7}}\) | \(383\) |
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. 345 vs.
\(2 (157) = 314\).
time = 0.28, size = 345, normalized size = 2.18 \begin {gather*} 15 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{4} \, {\left (b^{6} x^{4} e^{3} - 4 \, {\left (b^{6} d e^{2} - 2 \, a b^{5} e^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{6} d^{2} e - 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{2} - 4 \, {\left (10 \, b^{6} d^{3} - 36 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} - 20 \, a^{3} b^{3} e^{3}\right )} x\right )} e^{\left (-6\right )} + \frac {11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \, {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x}{2 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} 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. 516 vs.
\(2 (157) = 314\).
time = 3.71, size = 516, normalized size = 3.27 \begin {gather*} \frac {22 \, b^{6} d^{6} + {\left (b^{6} x^{6} + 8 \, a b^{5} x^{5} + 30 \, a^{2} b^{4} x^{4} + 80 \, a^{3} b^{3} x^{3} - 24 \, a^{5} b x - 2 \, a^{6}\right )} e^{6} - 2 \, {\left (b^{6} d x^{5} + 10 \, a b^{5} d x^{4} + 60 \, a^{2} b^{4} d x^{3} - 80 \, a^{3} b^{3} d x^{2} - 60 \, a^{4} b^{2} d x + 6 \, a^{5} b d\right )} e^{5} + 5 \, {\left (b^{6} d^{2} x^{4} + 16 \, a b^{5} d^{2} x^{3} - 66 \, a^{2} b^{4} d^{2} x^{2} - 32 \, a^{3} b^{3} d^{2} x + 18 \, a^{4} b^{2} d^{2}\right )} e^{4} - 4 \, {\left (5 \, b^{6} d^{3} x^{3} - 63 \, a b^{5} d^{3} x^{2} - 15 \, a^{2} b^{4} d^{3} x + 50 \, a^{3} b^{3} d^{3}\right )} e^{3} - 2 \, {\left (34 \, b^{6} d^{4} x^{2} - 12 \, a b^{5} d^{4} x - 105 \, a^{2} b^{4} d^{4}\right )} e^{2} - 4 \, {\left (4 \, b^{6} d^{5} x + 27 \, a b^{5} d^{5}\right )} e + 60 \, {\left (b^{6} d^{6} + a^{4} b^{2} x^{2} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{2} - a^{4} b^{2} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{2} - 8 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{2} - 3 \, a^{2} b^{4} d^{3} x + a^{3} b^{3} d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{2} - 8 \, a b^{5} d^{4} x + 6 \, a^{2} b^{4} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x - 2 \, a b^{5} d^{5}\right )} e\right )} \log \left (x e + d\right )}{4 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\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. 340 vs.
\(2 (144) = 288\).
time = 1.28, size = 340, normalized size = 2.15 \begin {gather*} \frac {b^{6} x^{4}}{4 e^{3}} + \frac {15 b^{2} \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{7}} + x^{3} \cdot \left (\frac {2 a b^{5}}{e^{3}} - \frac {b^{6} d}{e^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{2} b^{4}}{2 e^{3}} - \frac {9 a b^{5} d}{e^{4}} + \frac {3 b^{6} d^{2}}{e^{5}}\right ) + x \left (\frac {20 a^{3} b^{3}}{e^{3}} - \frac {45 a^{2} b^{4} d}{e^{4}} + \frac {36 a b^{5} d^{2}}{e^{5}} - \frac {10 b^{6} d^{3}}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 6 a^{5} b d e^{5} + 45 a^{4} b^{2} d^{2} e^{4} - 100 a^{3} b^{3} d^{3} e^{3} + 105 a^{2} b^{4} d^{4} e^{2} - 54 a b^{5} d^{5} e + 11 b^{6} d^{6} + x \left (- 12 a^{5} b e^{6} + 60 a^{4} b^{2} d e^{5} - 120 a^{3} b^{3} d^{2} e^{4} + 120 a^{2} b^{4} d^{3} e^{3} - 60 a b^{5} d^{4} e^{2} + 12 b^{6} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \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. 341 vs.
\(2 (157) = 314\).
time = 2.27, size = 341, normalized size = 2.16 \begin {gather*} 15 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{4} \, {\left (b^{6} x^{4} e^{9} - 4 \, b^{6} d x^{3} e^{8} + 12 \, b^{6} d^{2} x^{2} e^{7} - 40 \, b^{6} d^{3} x e^{6} + 8 \, a b^{5} x^{3} e^{9} - 36 \, a b^{5} d x^{2} e^{8} + 144 \, a b^{5} d^{2} x e^{7} + 30 \, a^{2} b^{4} x^{2} e^{9} - 180 \, a^{2} b^{4} d x e^{8} + 80 \, a^{3} b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac {{\left (11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \, {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.57, size = 441, normalized size = 2.79 \begin {gather*} x\,\left (\frac {20\,a^3\,b^3}{e^3}-\frac {b^6\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{e}-\frac {15\,a^2\,b^4}{e^3}+\frac {3\,b^6\,d^2}{e^5}\right )}{e}-\frac {3\,d^2\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{e^2}\right )-\frac {\frac {a^6\,e^6+6\,a^5\,b\,d\,e^5-45\,a^4\,b^2\,d^2\,e^4+100\,a^3\,b^3\,d^3\,e^3-105\,a^2\,b^4\,d^4\,e^2+54\,a\,b^5\,d^5\,e-11\,b^6\,d^6}{2\,e}-x\,\left (-6\,a^5\,b\,e^5+30\,a^4\,b^2\,d\,e^4-60\,a^3\,b^3\,d^2\,e^3+60\,a^2\,b^4\,d^3\,e^2-30\,a\,b^5\,d^4\,e+6\,b^6\,d^5\right )}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x^3\,\left (\frac {2\,a\,b^5}{e^3}-\frac {b^6\,d}{e^4}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {6\,a\,b^5}{e^3}-\frac {3\,b^6\,d}{e^4}\right )}{2\,e}-\frac {15\,a^2\,b^4}{2\,e^3}+\frac {3\,b^6\,d^2}{2\,e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (15\,a^4\,b^2\,e^4-60\,a^3\,b^3\,d\,e^3+90\,a^2\,b^4\,d^2\,e^2-60\,a\,b^5\,d^3\,e+15\,b^6\,d^4\right )}{e^7}+\frac {b^6\,x^4}{4\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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