Optimal. Leaf size=248 \[ \frac {8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac {4 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt {d+e x}}-\frac {10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac {4 c^3 (d+e x)^{9/2}}{9 e^6} \]
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Rubi [A] time = 0.13, antiderivative size = 248, 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 = {771} \begin {gather*} \frac {8 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6}+\frac {4 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt {d+e x}}-\frac {10 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^6}+\frac {4 c^3 (d+e x)^{9/2}}{9 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^{3/2}}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^5 \sqrt {d+e x}}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) \sqrt {d+e x}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{5/2}}{e^5}+\frac {2 c^3 (d+e x)^{7/2}}{e^5}\right ) \, dx\\ &=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 \sqrt {d+e x}}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt {d+e x}}{e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3/2}}{3 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^6}+\frac {4 c^3 (d+e x)^{9/2}}{9 e^6}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 287, normalized size = 1.16 \begin {gather*} \frac {252 c e^2 \left (5 a^2 e^2 (2 d+e x)+5 a b e \left (-8 d^2-4 d e x+e^2 x^2\right )+2 b^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-210 b e^3 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )-18 c^2 e \left (5 b \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-28 a e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )+20 c^3 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )}{315 e^6 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 425, normalized size = 1.71 \begin {gather*} \frac {2 \left (-315 a^2 b e^5+630 a^2 c e^4 (d+e x)+630 a^2 c d e^4+630 a b^2 e^4 (d+e x)+630 a b^2 d e^4-1890 a b c d^2 e^3-3780 a b c d e^3 (d+e x)+630 a b c e^3 (d+e x)^2+1260 a c^2 d^3 e^2+3780 a c^2 d^2 e^2 (d+e x)-1260 a c^2 d e^2 (d+e x)^2+252 a c^2 e^2 (d+e x)^3-315 b^3 d^2 e^3-630 b^3 d e^3 (d+e x)+105 b^3 e^3 (d+e x)^2+1260 b^2 c d^3 e^2+3780 b^2 c d^2 e^2 (d+e x)-1260 b^2 c d e^2 (d+e x)^2+252 b^2 c e^2 (d+e x)^3-1575 b c^2 d^4 e-6300 b c^2 d^3 e (d+e x)+3150 b c^2 d^2 e (d+e x)^2-1260 b c^2 d e (d+e x)^3+225 b c^2 e (d+e x)^4+630 c^3 d^5+3150 c^3 d^4 (d+e x)-2100 c^3 d^3 (d+e x)^2+1260 c^3 d^2 (d+e x)^3-450 c^3 d (d+e x)^4+70 c^3 (d+e x)^5\right )}{315 e^6 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 316, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (70 \, c^{3} e^{5} x^{5} + 2560 \, c^{3} d^{5} - 5760 \, b c^{2} d^{4} e - 315 \, a^{2} b e^{5} + 4032 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 840 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 1260 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 25 \, {\left (4 \, c^{3} d e^{4} - 9 \, b c^{2} e^{5}\right )} x^{4} + 4 \, {\left (40 \, c^{3} d^{2} e^{3} - 90 \, b c^{2} d e^{4} + 63 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - {\left (320 \, c^{3} d^{3} e^{2} - 720 \, b c^{2} d^{2} e^{3} + 504 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - 105 \, {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \, {\left (640 \, c^{3} d^{4} e - 1440 \, b c^{2} d^{3} e^{2} + 1008 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 210 \, {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 315 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 460, normalized size = 1.85 \begin {gather*} \frac {2}{315} \, {\left (70 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} e^{48} - 450 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e^{48} + 1260 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e^{48} - 2100 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{48} + 3150 \, \sqrt {x e + d} c^{3} d^{4} e^{48} + 225 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{49} - 1260 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{49} + 3150 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{49} - 6300 \, \sqrt {x e + d} b c^{2} d^{3} e^{49} + 252 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{50} + 252 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} e^{50} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{50} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d e^{50} + 3780 \, \sqrt {x e + d} b^{2} c d^{2} e^{50} + 3780 \, \sqrt {x e + d} a c^{2} d^{2} e^{50} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{51} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a b c e^{51} - 630 \, \sqrt {x e + d} b^{3} d e^{51} - 3780 \, \sqrt {x e + d} a b c d e^{51} + 630 \, \sqrt {x e + d} a b^{2} e^{52} + 630 \, \sqrt {x e + d} a^{2} c e^{52}\right )} e^{\left (-54\right )} + \frac {2 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-6\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 359, normalized size = 1.45 \begin {gather*} -\frac {2 \left (-70 c^{3} e^{5} x^{5}-225 b \,c^{2} e^{5} x^{4}+100 c^{3} d \,e^{4} x^{4}-252 a \,c^{2} e^{5} x^{3}-252 b^{2} c \,e^{5} x^{3}+360 b \,c^{2} d \,e^{4} x^{3}-160 c^{3} d^{2} e^{3} x^{3}-630 a b c \,e^{5} x^{2}+504 a \,c^{2} d \,e^{4} x^{2}-105 b^{3} e^{5} x^{2}+504 b^{2} c d \,e^{4} x^{2}-720 b \,c^{2} d^{2} e^{3} x^{2}+320 c^{3} d^{3} e^{2} x^{2}-630 a^{2} c \,e^{5} x -630 a \,b^{2} e^{5} x +2520 a b c d \,e^{4} x -2016 a \,c^{2} d^{2} e^{3} x +420 b^{3} d \,e^{4} x -2016 b^{2} c \,d^{2} e^{3} x +2880 b \,c^{2} d^{3} e^{2} x -1280 c^{3} d^{4} e x +315 a^{2} b \,e^{5}-1260 a^{2} c d \,e^{4}-1260 a \,b^{2} d \,e^{4}+5040 a b c \,d^{2} e^{3}-4032 a \,c^{2} d^{3} e^{2}+840 b^{3} d^{2} e^{3}-4032 b^{2} c \,d^{3} e^{2}+5760 b \,c^{2} d^{4} e -2560 c^{3} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 316, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (\frac {70 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} - 225 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 252 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 630 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{315 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 333, normalized size = 1.34 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{5\,e^6}+\frac {-2\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4-12\,a\,b\,c\,d^2\,e^3+8\,a\,c^2\,d^3\,e^2-2\,b^3\,d^2\,e^3+8\,b^2\,c\,d^3\,e^2-10\,b\,c^2\,d^4\,e+4\,c^3\,d^5}{e^6\,\sqrt {d+e\,x}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{3\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 90.27, size = 316, normalized size = 1.27 \begin {gather*} \frac {4 c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (10 b c^{2} e - 20 c^{3} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (8 a c^{2} e^{2} + 8 b^{2} c e^{2} - 40 b c^{2} d e + 40 c^{3} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 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 )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (4 a^{2} c e^{4} + 4 a b^{2} e^{4} - 24 a b c d e^{3} + 24 a c^{2} d^{2} e^{2} - 4 b^{3} d e^{3} + 24 b^{2} c d^{2} e^{2} - 40 b c^{2} d^{3} e + 20 c^{3} d^{4}\right )}{e^{6}} - \frac {2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6} \sqrt {d + e x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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