Optimal. Leaf size=244 \[ \frac {2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac {6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 d^3 \sqrt {d+e x} (c d-b e)^3}{e^7}-\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]
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Rubi [A] time = 0.10, antiderivative size = 244, 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} \[ \frac {2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac {6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}-\frac {2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac {2 d^3 \sqrt {d+e x} (c d-b e)^3}{e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx &=\int \left (\frac {d^3 (c d-b e)^3}{e^6 \sqrt {d+e x}}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) \sqrt {d+e x}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{5/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{9/2}}{e^6}+\frac {c^3 (d+e x)^{11/2}}{e^6}\right ) \, dx\\ &=\frac {2 d^3 (c d-b e)^3 \sqrt {d+e x}}{e^7}-\frac {2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{3/2}}{e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 206, normalized size = 0.84 \[ \frac {2 \sqrt {d+e x} \left (5005 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-2145 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+9009 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-4095 c^2 (d+e x)^5 (2 c d-b e)+15015 d^3 (c d-b e)^3-15015 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+1155 c^3 (d+e x)^6\right )}{15015 e^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 270, normalized size = 1.11 \[ \frac {2 \, {\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e + 18304 \, b^{2} c d^{4} e^{2} - 6864 \, b^{3} d^{3} e^{3} - 315 \, {\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, b^{2} c e^{6}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, b^{2} c d e^{5} - 429 \, b^{3} e^{6}\right )} x^{3} + 6 \, {\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \, b^{2} c d^{2} e^{4} - 429 \, b^{3} d e^{5}\right )} x^{2} - 8 \, {\left (320 \, c^{3} d^{5} e - 1040 \, b c^{2} d^{4} e^{2} + 1144 \, b^{2} c d^{3} e^{3} - 429 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 305, normalized size = 1.25 \[ \frac {2}{15015} \, {\left (429 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{3} e^{\left (-3\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{2} c e^{\left (-4\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b c^{2} e^{\left (-5\right )} + 5 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} e^{\left (-6\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 286, normalized size = 1.17 \[ -\frac {2 \left (-1155 c^{3} x^{6} e^{6}-4095 b \,c^{2} e^{6} x^{5}+1260 c^{3} d \,e^{5} x^{5}-5005 b^{2} c \,e^{6} x^{4}+4550 b \,c^{2} d \,e^{5} x^{4}-1400 c^{3} d^{2} e^{4} x^{4}-2145 b^{3} e^{6} x^{3}+5720 b^{2} c d \,e^{5} x^{3}-5200 b \,c^{2} d^{2} e^{4} x^{3}+1600 c^{3} d^{3} e^{3} x^{3}+2574 b^{3} d \,e^{5} x^{2}-6864 b^{2} c \,d^{2} e^{4} x^{2}+6240 b \,c^{2} d^{3} e^{3} x^{2}-1920 c^{3} d^{4} e^{2} x^{2}-3432 b^{3} d^{2} e^{4} x +9152 b^{2} c \,d^{3} e^{3} x -8320 b \,c^{2} d^{4} e^{2} x +2560 c^{3} d^{5} e x +6864 b^{3} d^{3} e^{3}-18304 b^{2} c \,d^{4} e^{2}+16640 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right ) \sqrt {e x +d}}{15015 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 288, normalized size = 1.18 \[ \frac {2 \, {\left (\frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 239, normalized size = 0.98 \[ \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{5\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,\sqrt {d+e\,x}}{e^7}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{e^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 90.85, size = 745, normalized size = 3.05 \[ \begin {cases} \frac {- \frac {2 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 {2 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 {6 b^{2} c 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 {6 b^{2} c \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 {6 b c^{2} 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 {6 b c^{2} \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 c^{3} 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 c^{3} \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 {\frac {b^{3} x^{4}}{4} + \frac {3 b^{2} c x^{5}}{5} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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