Optimal. Leaf size=143 \[ \frac {(b d-a e)^5 (a+b x)^8}{8 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e)^3 (a+b x)^{10}}{b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^{11}}{11 b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{12}}{12 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6} \]
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Rubi [A]
time = 0.25, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45}
\begin {gather*} \frac {5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac {10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac {e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac {5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac {(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^5 \, dx\\ &=\int \left (\frac {(b d-a e)^5 (a+b x)^7}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^8}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^9}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^{10}}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^{11}}{b^5}+\frac {e^5 (a+b x)^{12}}{b^5}\right ) \, dx\\ &=\frac {(b d-a e)^5 (a+b x)^8}{8 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e)^3 (a+b x)^{10}}{b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^{11}}{11 b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{12}}{12 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(493\) vs. \(2(143)=286\).
time = 0.10, size = 493, normalized size = 3.45 \begin {gather*} \frac {x \left (1716 a^7 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+1716 a^6 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+1287 a^5 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+715 a^4 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+286 a^3 b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+78 a^2 b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )+13 a b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )+b^7 x^7 \left (1287 d^5+5720 d^4 e x+10296 d^3 e^2 x^2+9360 d^2 e^3 x^3+4290 d e^4 x^4+792 e^5 x^5\right )\right )}{10296} \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. \(981\) vs.
\(2(133)=266\).
time = 1.04, size = 982, normalized size = 6.87 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. 570 vs.
\(2 (135) = 270\).
time = 0.28, size = 570, normalized size = 3.99 \begin {gather*} \frac {1}{13} \, b^{7} x^{13} e^{5} + a^{7} d^{5} x + \frac {1}{12} \, {\left (5 \, b^{7} d e^{4} + 7 \, a b^{6} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, b^{7} d^{2} e^{3} + 35 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{7} d^{3} e^{2} + 14 \, a b^{6} d^{2} e^{3} + 21 \, a^{2} b^{5} d e^{4} + 7 \, a^{3} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (b^{7} d^{4} e + 14 \, a b^{6} d^{3} e^{2} + 42 \, a^{2} b^{5} d^{2} e^{3} + 35 \, a^{3} b^{4} d e^{4} + 7 \, a^{4} b^{3} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{5} + 35 \, a b^{6} d^{4} e + 210 \, a^{2} b^{5} d^{3} e^{2} + 350 \, a^{3} b^{4} d^{2} e^{3} + 175 \, a^{4} b^{3} d e^{4} + 21 \, a^{5} b^{2} e^{5}\right )} x^{8} + {\left (a b^{6} d^{5} + 15 \, a^{2} b^{5} d^{4} e + 50 \, a^{3} b^{4} d^{3} e^{2} + 50 \, a^{4} b^{3} d^{2} e^{3} + 15 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (21 \, a^{2} b^{5} d^{5} + 175 \, a^{3} b^{4} d^{4} e + 350 \, a^{4} b^{3} d^{3} e^{2} + 210 \, a^{5} b^{2} d^{2} e^{3} + 35 \, a^{6} b d e^{4} + a^{7} e^{5}\right )} x^{6} + {\left (7 \, a^{3} b^{4} d^{5} + 35 \, a^{4} b^{3} d^{4} e + 42 \, a^{5} b^{2} d^{3} e^{2} + 14 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (7 \, a^{4} b^{3} d^{5} + 21 \, a^{5} b^{2} d^{4} e + 14 \, a^{6} b d^{3} e^{2} + 2 \, a^{7} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{5} + 35 \, a^{6} b d^{4} e + 10 \, a^{7} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{5} + 5 \, a^{7} d^{4} e\right )} x^{2} \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. 594 vs.
\(2 (135) = 270\).
time = 2.22, size = 594, normalized size = 4.15 \begin {gather*} \frac {1}{8} \, b^{7} d^{5} x^{8} + a b^{6} d^{5} x^{7} + \frac {7}{2} \, a^{2} b^{5} d^{5} x^{6} + 7 \, a^{3} b^{4} d^{5} x^{5} + \frac {35}{4} \, a^{4} b^{3} d^{5} x^{4} + 7 \, a^{5} b^{2} d^{5} x^{3} + \frac {7}{2} \, a^{6} b d^{5} x^{2} + a^{7} d^{5} x + \frac {1}{10296} \, {\left (792 \, b^{7} x^{13} + 6006 \, a b^{6} x^{12} + 19656 \, a^{2} b^{5} x^{11} + 36036 \, a^{3} b^{4} x^{10} + 40040 \, a^{4} b^{3} x^{9} + 27027 \, a^{5} b^{2} x^{8} + 10296 \, a^{6} b x^{7} + 1716 \, a^{7} x^{6}\right )} e^{5} + \frac {1}{792} \, {\left (330 \, b^{7} d x^{12} + 2520 \, a b^{6} d x^{11} + 8316 \, a^{2} b^{5} d x^{10} + 15400 \, a^{3} b^{4} d x^{9} + 17325 \, a^{4} b^{3} d x^{8} + 11880 \, a^{5} b^{2} d x^{7} + 4620 \, a^{6} b d x^{6} + 792 \, a^{7} d x^{5}\right )} e^{4} + \frac {1}{132} \, {\left (120 \, b^{7} d^{2} x^{11} + 924 \, a b^{6} d^{2} x^{10} + 3080 \, a^{2} b^{5} d^{2} x^{9} + 5775 \, a^{3} b^{4} d^{2} x^{8} + 6600 \, a^{4} b^{3} d^{2} x^{7} + 4620 \, a^{5} b^{2} d^{2} x^{6} + 1848 \, a^{6} b d^{2} x^{5} + 330 \, a^{7} d^{2} x^{4}\right )} e^{3} + \frac {1}{36} \, {\left (36 \, b^{7} d^{3} x^{10} + 280 \, a b^{6} d^{3} x^{9} + 945 \, a^{2} b^{5} d^{3} x^{8} + 1800 \, a^{3} b^{4} d^{3} x^{7} + 2100 \, a^{4} b^{3} d^{3} x^{6} + 1512 \, a^{5} b^{2} d^{3} x^{5} + 630 \, a^{6} b d^{3} x^{4} + 120 \, a^{7} d^{3} x^{3}\right )} e^{2} + \frac {5}{72} \, {\left (8 \, b^{7} d^{4} x^{9} + 63 \, a b^{6} d^{4} x^{8} + 216 \, a^{2} b^{5} d^{4} x^{7} + 420 \, a^{3} b^{4} d^{4} x^{6} + 504 \, a^{4} b^{3} d^{4} x^{5} + 378 \, a^{5} b^{2} d^{4} x^{4} + 168 \, a^{6} b d^{4} x^{3} + 36 \, a^{7} d^{4} x^{2}\right )} e \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. 673 vs.
\(2 (129) = 258\).
time = 0.06, size = 673, normalized size = 4.71 \begin {gather*} a^{7} d^{5} x + \frac {b^{7} e^{5} x^{13}}{13} + x^{12} \cdot \left (\frac {7 a b^{6} e^{5}}{12} + \frac {5 b^{7} d e^{4}}{12}\right ) + x^{11} \cdot \left (\frac {21 a^{2} b^{5} e^{5}}{11} + \frac {35 a b^{6} d e^{4}}{11} + \frac {10 b^{7} d^{2} e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {7 a^{3} b^{4} e^{5}}{2} + \frac {21 a^{2} b^{5} d e^{4}}{2} + 7 a b^{6} d^{2} e^{3} + b^{7} d^{3} e^{2}\right ) + x^{9} \cdot \left (\frac {35 a^{4} b^{3} e^{5}}{9} + \frac {175 a^{3} b^{4} d e^{4}}{9} + \frac {70 a^{2} b^{5} d^{2} e^{3}}{3} + \frac {70 a b^{6} d^{3} e^{2}}{9} + \frac {5 b^{7} d^{4} e}{9}\right ) + x^{8} \cdot \left (\frac {21 a^{5} b^{2} e^{5}}{8} + \frac {175 a^{4} b^{3} d e^{4}}{8} + \frac {175 a^{3} b^{4} d^{2} e^{3}}{4} + \frac {105 a^{2} b^{5} d^{3} e^{2}}{4} + \frac {35 a b^{6} d^{4} e}{8} + \frac {b^{7} d^{5}}{8}\right ) + x^{7} \left (a^{6} b e^{5} + 15 a^{5} b^{2} d e^{4} + 50 a^{4} b^{3} d^{2} e^{3} + 50 a^{3} b^{4} d^{3} e^{2} + 15 a^{2} b^{5} d^{4} e + a b^{6} d^{5}\right ) + x^{6} \left (\frac {a^{7} e^{5}}{6} + \frac {35 a^{6} b d e^{4}}{6} + 35 a^{5} b^{2} d^{2} e^{3} + \frac {175 a^{4} b^{3} d^{3} e^{2}}{3} + \frac {175 a^{3} b^{4} d^{4} e}{6} + \frac {7 a^{2} b^{5} d^{5}}{2}\right ) + x^{5} \left (a^{7} d e^{4} + 14 a^{6} b d^{2} e^{3} + 42 a^{5} b^{2} d^{3} e^{2} + 35 a^{4} b^{3} d^{4} e + 7 a^{3} b^{4} d^{5}\right ) + x^{4} \cdot \left (\frac {5 a^{7} d^{2} e^{3}}{2} + \frac {35 a^{6} b d^{3} e^{2}}{2} + \frac {105 a^{5} b^{2} d^{4} e}{4} + \frac {35 a^{4} b^{3} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{7} d^{3} e^{2}}{3} + \frac {35 a^{6} b d^{4} e}{3} + 7 a^{5} b^{2} d^{5}\right ) + x^{2} \cdot \left (\frac {5 a^{7} d^{4} e}{2} + \frac {7 a^{6} b d^{5}}{2}\right ) \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. 646 vs.
\(2 (135) = 270\).
time = 1.06, size = 646, normalized size = 4.52 \begin {gather*} \frac {1}{13} \, b^{7} x^{13} e^{5} + \frac {5}{12} \, b^{7} d x^{12} e^{4} + \frac {10}{11} \, b^{7} d^{2} x^{11} e^{3} + b^{7} d^{3} x^{10} e^{2} + \frac {5}{9} \, b^{7} d^{4} x^{9} e + \frac {1}{8} \, b^{7} d^{5} x^{8} + \frac {7}{12} \, a b^{6} x^{12} e^{5} + \frac {35}{11} \, a b^{6} d x^{11} e^{4} + 7 \, a b^{6} d^{2} x^{10} e^{3} + \frac {70}{9} \, a b^{6} d^{3} x^{9} e^{2} + \frac {35}{8} \, a b^{6} d^{4} x^{8} e + a b^{6} d^{5} x^{7} + \frac {21}{11} \, a^{2} b^{5} x^{11} e^{5} + \frac {21}{2} \, a^{2} b^{5} d x^{10} e^{4} + \frac {70}{3} \, a^{2} b^{5} d^{2} x^{9} e^{3} + \frac {105}{4} \, a^{2} b^{5} d^{3} x^{8} e^{2} + 15 \, a^{2} b^{5} d^{4} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{5} x^{6} + \frac {7}{2} \, a^{3} b^{4} x^{10} e^{5} + \frac {175}{9} \, a^{3} b^{4} d x^{9} e^{4} + \frac {175}{4} \, a^{3} b^{4} d^{2} x^{8} e^{3} + 50 \, a^{3} b^{4} d^{3} x^{7} e^{2} + \frac {175}{6} \, a^{3} b^{4} d^{4} x^{6} e + 7 \, a^{3} b^{4} d^{5} x^{5} + \frac {35}{9} \, a^{4} b^{3} x^{9} e^{5} + \frac {175}{8} \, a^{4} b^{3} d x^{8} e^{4} + 50 \, a^{4} b^{3} d^{2} x^{7} e^{3} + \frac {175}{3} \, a^{4} b^{3} d^{3} x^{6} e^{2} + 35 \, a^{4} b^{3} d^{4} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{5} x^{4} + \frac {21}{8} \, a^{5} b^{2} x^{8} e^{5} + 15 \, a^{5} b^{2} d x^{7} e^{4} + 35 \, a^{5} b^{2} d^{2} x^{6} e^{3} + 42 \, a^{5} b^{2} d^{3} x^{5} e^{2} + \frac {105}{4} \, a^{5} b^{2} d^{4} x^{4} e + 7 \, a^{5} b^{2} d^{5} x^{3} + a^{6} b x^{7} e^{5} + \frac {35}{6} \, a^{6} b d x^{6} e^{4} + 14 \, a^{6} b d^{2} x^{5} e^{3} + \frac {35}{2} \, a^{6} b d^{3} x^{4} e^{2} + \frac {35}{3} \, a^{6} b d^{4} x^{3} e + \frac {7}{2} \, a^{6} b d^{5} x^{2} + \frac {1}{6} \, a^{7} x^{6} e^{5} + a^{7} d x^{5} e^{4} + \frac {5}{2} \, a^{7} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{7} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{7} d^{4} x^{2} e + a^{7} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.17, size = 570, normalized size = 3.99 \begin {gather*} x^5\,\left (a^7\,d\,e^4+14\,a^6\,b\,d^2\,e^3+42\,a^5\,b^2\,d^3\,e^2+35\,a^4\,b^3\,d^4\,e+7\,a^3\,b^4\,d^5\right )+x^9\,\left (\frac {35\,a^4\,b^3\,e^5}{9}+\frac {175\,a^3\,b^4\,d\,e^4}{9}+\frac {70\,a^2\,b^5\,d^2\,e^3}{3}+\frac {70\,a\,b^6\,d^3\,e^2}{9}+\frac {5\,b^7\,d^4\,e}{9}\right )+x^7\,\left (a^6\,b\,e^5+15\,a^5\,b^2\,d\,e^4+50\,a^4\,b^3\,d^2\,e^3+50\,a^3\,b^4\,d^3\,e^2+15\,a^2\,b^5\,d^4\,e+a\,b^6\,d^5\right )+x^6\,\left (\frac {a^7\,e^5}{6}+\frac {35\,a^6\,b\,d\,e^4}{6}+35\,a^5\,b^2\,d^2\,e^3+\frac {175\,a^4\,b^3\,d^3\,e^2}{3}+\frac {175\,a^3\,b^4\,d^4\,e}{6}+\frac {7\,a^2\,b^5\,d^5}{2}\right )+x^8\,\left (\frac {21\,a^5\,b^2\,e^5}{8}+\frac {175\,a^4\,b^3\,d\,e^4}{8}+\frac {175\,a^3\,b^4\,d^2\,e^3}{4}+\frac {105\,a^2\,b^5\,d^3\,e^2}{4}+\frac {35\,a\,b^6\,d^4\,e}{8}+\frac {b^7\,d^5}{8}\right )+a^7\,d^5\,x+\frac {b^7\,e^5\,x^{13}}{13}+\frac {5\,a^4\,d^2\,x^4\,\left (2\,a^3\,e^3+14\,a^2\,b\,d\,e^2+21\,a\,b^2\,d^2\,e+7\,b^3\,d^3\right )}{4}+\frac {b^4\,e^2\,x^{10}\,\left (7\,a^3\,e^3+21\,a^2\,b\,d\,e^2+14\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{2}+\frac {a^6\,d^4\,x^2\,\left (5\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^4\,x^{12}\,\left (7\,a\,e+5\,b\,d\right )}{12}+\frac {a^5\,d^3\,x^3\,\left (10\,a^2\,e^2+35\,a\,b\,d\,e+21\,b^2\,d^2\right )}{3}+\frac {b^5\,e^3\,x^{11}\,\left (21\,a^2\,e^2+35\,a\,b\,d\,e+10\,b^2\,d^2\right )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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