Optimal. Leaf size=181 \[ \frac {2 (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2}}{3 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2}}{11 e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45}
\begin {gather*} -\frac {12 b^5 (d+e x)^{11/2} (b d-a e)}{11 e^7}+\frac {10 b^4 (d+e x)^{9/2} (b d-a e)^2}{3 e^7}-\frac {40 b^3 (d+e x)^{7/2} (b d-a e)^3}{7 e^7}+\frac {6 b^2 (d+e x)^{5/2} (b d-a e)^4}{e^7}-\frac {4 b (d+e x)^{3/2} (b d-a e)^5}{e^7}+\frac {2 \sqrt {d+e x} (b d-a e)^6}{e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 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}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^6}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 \sqrt {d+e x}}-\frac {6 b (b d-a e)^5 \sqrt {d+e x}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{7/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{9/2}}{e^6}+\frac {b^6 (d+e x)^{11/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {6 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{7/2}}{7 e^7}+\frac {10 b^4 (b d-a e)^2 (d+e x)^{9/2}}{3 e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{11/2}}{11 e^7}+\frac {2 b^6 (d+e x)^{13/2}}{13 e^7}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 290, normalized size = 1.60 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3003 a^6 e^6+6006 a^5 b e^5 (-2 d+e x)+3003 a^4 b^2 e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+1716 a^3 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 a^2 b^4 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+26 a b^5 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+b^6 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{3003 e^7} \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. \(454\) vs.
\(2(159)=318\).
time = 0.63, size = 455, normalized size = 2.51 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.
\(2 (165) = 330\).
time = 0.32, size = 570, normalized size = 3.15 \begin {gather*} \frac {2}{15015} \, {\left (3432 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{3} b^{3} e^{\left (-3\right )} + 572 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} b^{4} e^{\left (-4\right )} + 130 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b^{5} e^{\left (-5\right )} + 5 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{6} e^{\left (-6\right )} + 15015 \, \sqrt {x e + d} a^{6} + 3003 \, {\left (10 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b e^{\left (-1\right )} + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )}\right )} a^{4} + 143 \, {\left (84 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b^{2} e^{\left (-2\right )} + 36 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{3} e^{\left (-3\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{4} e^{\left (-4\right )}\right )} a^{2}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.38, size = 327, normalized size = 1.81 \begin {gather*} \frac {2}{3003} \, {\left (1024 \, b^{6} d^{6} + {\left (231 \, b^{6} x^{6} + 1638 \, a b^{5} x^{5} + 5005 \, a^{2} b^{4} x^{4} + 8580 \, a^{3} b^{3} x^{3} + 9009 \, a^{4} b^{2} x^{2} + 6006 \, a^{5} b x + 3003 \, a^{6}\right )} e^{6} - 4 \, {\left (63 \, b^{6} d x^{5} + 455 \, a b^{5} d x^{4} + 1430 \, a^{2} b^{4} d x^{3} + 2574 \, a^{3} b^{3} d x^{2} + 3003 \, a^{4} b^{2} d x + 3003 \, a^{5} b d\right )} e^{5} + 8 \, {\left (35 \, b^{6} d^{2} x^{4} + 260 \, a b^{5} d^{2} x^{3} + 858 \, a^{2} b^{4} d^{2} x^{2} + 1716 \, a^{3} b^{3} d^{2} x + 3003 \, a^{4} b^{2} d^{2}\right )} e^{4} - 64 \, {\left (5 \, b^{6} d^{3} x^{3} + 39 \, a b^{5} d^{3} x^{2} + 143 \, a^{2} b^{4} d^{3} x + 429 \, a^{3} b^{3} d^{3}\right )} e^{3} + 128 \, {\left (3 \, b^{6} d^{4} x^{2} + 26 \, a b^{5} d^{4} x + 143 \, a^{2} b^{4} d^{4}\right )} e^{2} - 512 \, {\left (b^{6} d^{5} x + 13 \, a b^{5} d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-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. 1003 vs.
\(2 (168) = 336\).
time = 52.49, size = 1003, normalized size = 5.54 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{6} d}{\sqrt {d + e x}} - 2 a^{6} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {12 a^{5} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {12 a^{5} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {30 a^{4} b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {30 a^{4} b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {40 a^{3} b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {40 a^{3} b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {30 a^{2} b^{4} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {30 a^{2} b^{4} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} - \frac {12 a b^{5} d \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{5}} - \frac {12 a b^{5} \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} - \frac {2 b^{6} d \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{6}} - \frac {2 b^{6} \left (- \frac {d^{7}}{\sqrt {d + e x}} - 7 d^{6} \sqrt {d + e x} + 7 d^{5} \left (d + e x\right )^{\frac {3}{2}} - 7 d^{4} \left (d + e x\right )^{\frac {5}{2}} + 5 d^{3} \left (d + e x\right )^{\frac {7}{2}} - \frac {7 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{3} + \frac {7 d \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{6}}}{e} & \text {for}\: e \neq 0 \\\frac {a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \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. 395 vs.
\(2 (165) = 330\).
time = 0.94, size = 395, normalized size = 2.18 \begin {gather*} \frac {2}{3003} \, {\left (6006 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{5} b e^{\left (-1\right )} + 3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{4} b^{2} e^{\left (-2\right )} + 1716 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{3} b^{3} e^{\left (-3\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} b^{4} e^{\left (-4\right )} + 26 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b^{5} e^{\left (-5\right )} + {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{6} e^{\left (-6\right )} + 3003 \, \sqrt {x e + d} a^{6}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 162, normalized size = 0.90 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,\sqrt {d+e\,x}}{e^7}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {10\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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