Optimal. Leaf size=147 \[ \frac {2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac {2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac {4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \]
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Rubi [A] time = 0.06, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \begin {gather*} \frac {2 (d+e x)^{9/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{9 e^5}+\frac {2 d^2 (d+e x)^{5/2} (c d-b e)^2}{5 e^5}-\frac {4 c (d+e x)^{11/2} (2 c d-b e)}{11 e^5}-\frac {4 d (d+e x)^{7/2} (c d-b e) (2 c d-b e)}{7 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 (d+e x)^{3/2}}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^{5/2}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{9/2}}{e^4}+\frac {c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac {2 d^2 (c d-b e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{7/2}}{7 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 c^2 (d+e x)^{13/2}}{13 e^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 125, normalized size = 0.85 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 164, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 b^2 d^2 e^2-12870 b^2 d e^2 (d+e x)+5005 b^2 e^2 (d+e x)^2-18018 b c d^3 e+38610 b c d^2 e (d+e x)-30030 b c d e (d+e x)^2+8190 b c e (d+e x)^3+9009 c^2 d^4-25740 c^2 d^3 (d+e x)+30030 c^2 d^2 (d+e x)^2-16380 c^2 d (d+e x)^3+3465 c^2 (d+e x)^4\right )}{45045 e^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 214, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (3465 \, c^{2} e^{6} x^{6} + 384 \, c^{2} d^{6} - 1248 \, b c d^{5} e + 1144 \, b^{2} d^{4} e^{2} + 630 \, {\left (7 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 35 \, {\left (3 \, c^{2} d^{2} e^{4} + 312 \, b c d e^{5} + 143 \, b^{2} e^{6}\right )} x^{4} - 10 \, {\left (12 \, c^{2} d^{3} e^{3} - 39 \, b c d^{2} e^{4} - 715 \, b^{2} d e^{5}\right )} x^{3} + 3 \, {\left (48 \, c^{2} d^{4} e^{2} - 156 \, b c d^{3} e^{3} + 143 \, b^{2} d^{2} e^{4}\right )} x^{2} - 4 \, {\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 143 \, b^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 626, normalized size = 4.26 \begin {gather*} \frac {2}{45045} \, {\left (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 )} b^{2} d^{2} e^{\left (-2\right )} + 2574 \, {\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 )} b c d^{2} 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 )} c^{2} d^{2} e^{\left (-4\right )} + 2574 \, {\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 )} b^{2} d e^{\left (-2\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 )} b c d e^{\left (-3\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 )} c^{2} d e^{\left (-4\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 )} b^{2} e^{\left (-2\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 )} b c e^{\left (-3\right )} + 15 \, {\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 )} c^{2} e^{\left (-4\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 141, normalized size = 0.96 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3465 c^{2} x^{4} e^{4}+8190 b c \,e^{4} x^{3}-2520 c^{2} d \,e^{3} x^{3}+5005 b^{2} e^{4} x^{2}-5460 b c d \,e^{3} x^{2}+1680 c^{2} d^{2} e^{2} x^{2}-2860 b^{2} d \,e^{3} x +3120 b c \,d^{2} e^{2} x -960 c^{2} d^{3} e x +1144 b^{2} d^{2} e^{2}-1248 b c \,d^{3} e +384 c^{2} d^{4}\right )}{45045 e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 139, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{2} - 8190 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 138, normalized size = 0.94 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{7\,e^5}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{9\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.23, size = 413, normalized size = 2.81 \begin {gather*} \frac {2 b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {4 b c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {4 b c \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {2 c^{2} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {2 c^{2} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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