Optimal. Leaf size=267 \[ -\frac {2 (d+e x)^{9/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{9 e^6}+\frac {2 (d+e x)^{7/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{7 e^6}-\frac {2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac {2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac {2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \]
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Rubi [A] time = 0.15, antiderivative size = 267, 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)^{9/2} \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 )}{9 e^6}+\frac {2 (d+e x)^{7/2} \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 )}{7 e^6}-\frac {2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac {2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac {2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2 \sqrt {d+e x}}{e^5}+\frac {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}}{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)^{5/2}}{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)^{7/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{9/2}}{e^5}+\frac {B c^2 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 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)^{9/2}}{9 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{11/2}}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 273, normalized size = 1.02 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (13 A e \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \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 )+B \left (143 b^2 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+26 b c 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 )-5 c^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 )\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 399, normalized size = 1.49 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 A b^2 d^2 e^3-18018 A b^2 d e^3 (d+e x)+6435 A b^2 e^3 (d+e x)^2-30030 A b c d^3 e^2+54054 A b c d^2 e^2 (d+e x)-38610 A b c d e^2 (d+e x)^2+10010 A b c e^2 (d+e x)^3+15015 A c^2 d^4 e-36036 A c^2 d^3 e (d+e x)+38610 A c^2 d^2 e (d+e x)^2-20020 A c^2 d e (d+e x)^3+4095 A c^2 e (d+e x)^4-15015 b^2 B d^3 e^2+27027 b^2 B d^2 e^2 (d+e x)-19305 b^2 B d e^2 (d+e x)^2+5005 b^2 B e^2 (d+e x)^3+30030 b B c d^4 e-72072 b B c d^3 e (d+e x)+77220 b B c d^2 e (d+e x)^2-40040 b B c d e (d+e x)^3+8190 b B c e (d+e x)^4-15015 B c^2 d^5+45045 B c^2 d^4 (d+e x)-64350 B c^2 d^3 (d+e x)^2+50050 B c^2 d^2 (d+e x)^3-20475 B c^2 d (d+e x)^4+3465 B c^2 (d+e x)^5\right )}{45045 e^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 357, normalized size = 1.34 \begin {gather*} \frac {2 \, {\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 3432 \, A b^{2} d^{3} e^{3} + 1664 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e - 2288 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{2} + 315 \, {\left (B c^{2} d e^{5} + 13 \, {\left (2 \, B b c + A c^{2}\right )} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{2} e^{4} - 13 \, {\left (2 \, B b c + A c^{2}\right )} d e^{5} - 143 \, {\left (B b^{2} + 2 \, A b c\right )} e^{6}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{3} e^{3} + 1287 \, A b^{2} e^{6} - 104 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{4} + 143 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{4} e^{2} - 429 \, A b^{2} d e^{5} - 208 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{3} + 286 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{5} e - 429 \, A b^{2} d^{2} e^{4} - 208 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{2} + 286 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 835, normalized size = 3.13
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 341, normalized size = 1.28 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5} e^{5}+4095 A \,c^{2} e^{5} x^{4}+8190 B b c \,e^{5} x^{4}-3150 B \,c^{2} d \,e^{4} x^{4}+10010 A b c \,e^{5} x^{3}-3640 A \,c^{2} d \,e^{4} x^{3}+5005 B \,b^{2} e^{5} x^{3}-7280 B b c d \,e^{4} x^{3}+2800 B \,c^{2} d^{2} e^{3} x^{3}+6435 A \,b^{2} e^{5} x^{2}-8580 A b c d \,e^{4} x^{2}+3120 A \,c^{2} d^{2} e^{3} x^{2}-4290 B \,b^{2} d \,e^{4} x^{2}+6240 B b c \,d^{2} e^{3} x^{2}-2400 B \,c^{2} d^{3} e^{2} x^{2}-5148 A \,b^{2} d \,e^{4} x +6864 A b c \,d^{2} e^{3} x -2496 A \,c^{2} d^{3} e^{2} x +3432 B \,b^{2} d^{2} e^{3} x -4992 B b c \,d^{3} e^{2} x +1920 B \,c^{2} d^{4} e x +3432 A \,b^{2} d^{2} e^{3}-4576 A b c \,d^{3} e^{2}+1664 A \,c^{2} d^{4} e -2288 B \,b^{2} d^{3} e^{2}+3328 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{45045 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 291, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{2} - 4095 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\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 {9}{2}} - 6435 \, {\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 {7}{2}} + 9009 \, {\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 {5}{2}} - 15015 \, {\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 )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 254, normalized size = 0.95 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\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 )}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\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 )}{9\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{5\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.01, size = 377, normalized size = 1.41 \begin {gather*} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A b^{2} d^{2} e^{3} - 2 A b c d^{3} e^{2} + A c^{2} d^{4} e - B b^{2} d^{3} e^{2} + 2 B b c d^{4} e - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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