Optimal. Leaf size=427 \[ -\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{3 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{5 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{13/2}}{13 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{15/2}}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8} \]
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Rubi [A]
time = 0.15, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {785}
\begin {gather*} \frac {2 (d+e x)^{9/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^8}+\frac {6 c^2 (d+e x)^{13/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{13 e^8}-\frac {10 c (d+e x)^{11/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac {14 c^3 (d+e x)^{15/2} (2 c d-b e)}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}{e^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{3/2}}{e^7}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{5/2}}{e^7}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{7/2}}{e^7}+\frac {5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{e^7}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{11/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{13/2}}{e^7}+\frac {2 c^4 (d+e x)^{15/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{3 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{5 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{13/2}}{13 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{15/2}}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 601, normalized size = 1.41 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-14 c^4 \left (2048 d^7-3072 d^6 e x+3840 d^5 e^2 x^2-4480 d^4 e^3 x^3+5040 d^3 e^4 x^4-5544 d^2 e^5 x^5+6006 d e^6 x^6-6435 e^7 x^7\right )+2431 b e^4 \left (105 a^3 e^3+63 a^2 b e^2 (-2 d+3 e x)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )+221 c e^3 \left (462 a^3 e^3 (-2 d+3 e x)+297 a^2 b e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+132 a b^2 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b^3 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-51 c^2 e^2 \left (286 a^2 e^2 \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-65 a b e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+15 b^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )+17 c^3 e \left (30 a e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+7 b \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )\right )}{765765 e^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 713, normalized size = 1.67 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 680, normalized size = 1.59 \begin {gather*} \frac {2}{765765} \, {\left (90090 \, {\left (x e + d\right )}^{\frac {17}{2}} c^{4} - 357357 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (x e + d\right )}^{\frac {15}{2}} + 176715 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + 3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 348075 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e - b^{3} c e^{3} - 3 \, a b c^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 85085 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{2} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 328185 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e - a b^{3} e^{5} - 3 \, a^{2} b c e^{5} + 10 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{3} - 10 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{2} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 153153 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{4} + 3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6} - 20 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{3} + 3 \, {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{2} - 6 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 255255 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e + 3 \, {\left (3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} d^{5} - 5 \, {\left (b^{3} c e^{3} + 3 \, a b c^{2} e^{3}\right )} d^{4} - a^{3} b e^{7} + {\left (b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} d^{3} - 3 \, {\left (a b^{3} e^{5} + 3 \, a^{2} b c e^{5}\right )} d^{2} + {\left (3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.25, size = 767, normalized size = 1.80 \begin {gather*} -\frac {2}{765765} \, {\left (28672 \, c^{4} d^{8} - {\left (90090 \, c^{4} x^{8} + 357357 \, b c^{3} x^{7} + 176715 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} + 348075 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} + 255255 \, a^{3} b x + 85085 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} + 328185 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} + 153153 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}\right )} e^{8} - {\left (6006 \, c^{4} d x^{7} + 27489 \, b c^{3} d x^{6} + 16065 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} + 38675 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + 255255 \, a^{3} b d + 12155 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 65637 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} + 51051 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x\right )} e^{7} + 2 \, {\left (3234 \, c^{4} d^{2} x^{6} + 14994 \, b c^{3} d^{2} x^{5} + 8925 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{4} + 22100 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{3} + 7293 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{2} + 43758 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x + 51051 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} e^{6} - 8 \, {\left (882 \, c^{4} d^{3} x^{5} + 4165 \, b c^{3} d^{3} x^{4} + 2550 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} x^{3} + 6630 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} x^{2} + 2431 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} x + 21879 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3}\right )} e^{5} + 16 \, {\left (490 \, c^{4} d^{4} x^{4} + 2380 \, b c^{3} d^{4} x^{3} + 1530 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2} + 4420 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} x + 2431 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4}\right )} e^{4} - 128 \, {\left (70 \, c^{4} d^{5} x^{3} + 357 \, b c^{3} d^{5} x^{2} + 255 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x + 1105 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5}\right )} e^{3} + 256 \, {\left (42 \, c^{4} d^{6} x^{2} + 238 \, b c^{3} d^{6} x + 255 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6}\right )} e^{2} - 7168 \, {\left (2 \, c^{4} d^{7} x + 17 \, b c^{3} d^{7}\right )} e\right )} \sqrt {x e + d} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.71, size = 843, normalized size = 1.97 \begin {gather*} \frac {2 \cdot \left (\frac {2 c^{4} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (7 b c^{3} e - 14 c^{4} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 42 b c^{3} d e + 42 c^{4} d^{2}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (15 a b c^{2} e^{3} - 30 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 45 b^{2} c^{2} d e^{2} + 105 b c^{3} d^{2} e - 70 c^{4} d^{3}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (9 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 90 a b c^{2} d^{2} e^{3} - 60 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 30 b^{3} c d^{2} e^{3} - 90 b^{2} c^{2} d^{3} e^{2} + 105 b c^{3} d^{4} e - 42 c^{4} d^{5}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a^{3} c e^{6} + 3 a^{2} b^{2} e^{6} - 18 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 6 a b^{3} d e^{5} + 36 a b^{2} c d^{2} e^{4} - 60 a b c^{2} d^{3} e^{3} + 30 a c^{3} d^{4} e^{2} + 3 b^{4} d^{2} e^{4} - 20 b^{3} c d^{3} e^{3} + 45 b^{2} c^{2} d^{4} e^{2} - 42 b c^{3} d^{5} e + 14 c^{4} d^{6}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} + 9 a^{2} b c d^{2} e^{5} - 6 a^{2} c^{2} d^{3} e^{4} + 3 a b^{3} d^{2} e^{5} - 12 a b^{2} c d^{3} e^{4} + 15 a b c^{2} d^{4} e^{3} - 6 a c^{3} d^{5} e^{2} - b^{4} d^{3} e^{4} + 5 b^{3} c d^{4} e^{3} - 9 b^{2} c^{2} d^{5} e^{2} + 7 b c^{3} d^{6} e - 2 c^{4} d^{7}\right )}{3 e^{7}}\right )}{e} \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. 1876 vs.
\(2 (403) = 806\).
time = 1.61, size = 1876, normalized size = 4.39 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 444, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{13\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{9\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{3\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{7\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{5\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{11\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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