Optimal. Leaf size=265 \[ -\frac {2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac {2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac {2 d^2 \sqrt {d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac {2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \]
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Rubi [A] time = 0.16, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac {2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac {2 d^2 \sqrt {d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac {2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 \sqrt {d+e x}}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{7/2}}{e^5}+\frac {B c^2 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 \sqrt {d+e x}}{e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{3/2}}{3 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{9/2}}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 273, normalized size = 1.03 \begin {gather*} \frac {2 \sqrt {d+e x} \left (11 A e \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^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 )\right )+B \left (99 b^2 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+22 b c e \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 )-5 c^2 \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 )\right )\right )}{3465 e^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 399, normalized size = 1.51 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3465 A b^2 d^2 e^3-2310 A b^2 d e^3 (d+e x)+693 A b^2 e^3 (d+e x)^2-6930 A b c d^3 e^2+6930 A b c d^2 e^2 (d+e x)-4158 A b c d e^2 (d+e x)^2+990 A b c e^2 (d+e x)^3+3465 A c^2 d^4 e-4620 A c^2 d^3 e (d+e x)+4158 A c^2 d^2 e (d+e x)^2-1980 A c^2 d e (d+e x)^3+385 A c^2 e (d+e x)^4-3465 b^2 B d^3 e^2+3465 b^2 B d^2 e^2 (d+e x)-2079 b^2 B d e^2 (d+e x)^2+495 b^2 B e^2 (d+e x)^3+6930 b B c d^4 e-9240 b B c d^3 e (d+e x)+8316 b B c d^2 e (d+e x)^2-3960 b B c d e (d+e x)^3+770 b B c e (d+e x)^4-3465 B c^2 d^5+5775 B c^2 d^4 (d+e x)-6930 B c^2 d^3 (d+e x)^2+4950 B c^2 d^2 (d+e x)^3-1925 B c^2 d (d+e x)^4+315 B c^2 (d+e x)^5\right )}{3465 e^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 290, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1848 \, A b^{2} d^{2} e^{3} + 1408 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 1584 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 35 \, {\left (10 \, B c^{2} d e^{4} - 11 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{2} e^{3} - 88 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 99 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{3} e^{2} - 231 \, A b^{2} e^{5} - 176 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 198 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{4} e - 231 \, A b^{2} d e^{4} - 176 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 198 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 378, normalized size = 1.43 \begin {gather*} \frac {2}{3465} \, {\left (231 \, {\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 b^{2} e^{\left (-2\right )} + 99 \, {\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 b^{2} e^{\left (-3\right )} + 198 \, {\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 c e^{\left (-3\right )} + 22 \, {\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 b c e^{\left (-4\right )} + 11 \, {\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 c^{2} e^{\left (-4\right )} + 5 \, {\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 )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 341, normalized size = 1.29 \begin {gather*} \frac {2 \left (315 B \,c^{2} x^{5} e^{5}+385 A \,c^{2} e^{5} x^{4}+770 B b c \,e^{5} x^{4}-350 B \,c^{2} d \,e^{4} x^{4}+990 A b c \,e^{5} x^{3}-440 A \,c^{2} d \,e^{4} x^{3}+495 B \,b^{2} e^{5} x^{3}-880 B b c d \,e^{4} x^{3}+400 B \,c^{2} d^{2} e^{3} x^{3}+693 A \,b^{2} e^{5} x^{2}-1188 A b c d \,e^{4} x^{2}+528 A \,c^{2} d^{2} e^{3} x^{2}-594 B \,b^{2} d \,e^{4} x^{2}+1056 B b c \,d^{2} e^{3} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}-924 A \,b^{2} d \,e^{4} x +1584 A b c \,d^{2} e^{3} x -704 A \,c^{2} d^{3} e^{2} x +792 B \,b^{2} d^{2} e^{3} x -1408 B b c \,d^{3} e^{2} x +640 B \,c^{2} d^{4} e x +1848 A \,b^{2} d^{2} e^{3}-3168 A b c \,d^{3} e^{2}+1408 A \,c^{2} d^{4} e -1584 B \,b^{2} d^{3} e^{2}+2816 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 291, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c^{2} - 385 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \sqrt {e x + d}\right )}}{3465 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 254, normalized size = 0.96 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{9\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{5\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{7\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{3\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}}{e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 106.60, size = 944, normalized size = 3.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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