Optimal. Leaf size=252 \[ \frac{8 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^6}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^6}+\frac{4 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac{10 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^6}+\frac{4 c^3 (d+e x)^{15/2}}{15 e^6} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.338585, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{8 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^6}-\frac{2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^6}+\frac{4 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac{10 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^6}+\frac{4 c^3 (d+e x)^{15/2}}{15 e^6} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 69.9227, size = 250, normalized size = 0.99 \[ \frac{4 c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{10 c^{2} \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right )}{13 e^{6}} + \frac{8 c \left (d + e x\right )^{\frac{11}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{11 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{9 e^{6}} + \frac{4 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{5 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)**(3/2)*(c*x**2+b*x+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.522386, size = 291, normalized size = 1.15 \[ \frac{2 (d+e x)^{5/2} \left (-78 c e^2 \left (33 a^2 e^2 (2 d-5 e x)-11 a b e \left (8 d^2-20 d e x+35 e^2 x^2\right )+2 b^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )+143 b e^3 \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+3 c^2 e \left (52 a e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+c^3 \left (-512 d^5+1280 d^4 e x-2240 d^3 e^2 x^2+3360 d^2 e^3 x^3-4620 d e^4 x^4+6006 e^5 x^5\right )\right )}{45045 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.013, size = 359, normalized size = 1.4 \[{\frac{12012\,{c}^{3}{x}^{5}{e}^{5}+34650\,b{c}^{2}{e}^{5}{x}^{4}-9240\,{c}^{3}d{e}^{4}{x}^{4}+32760\,a{c}^{2}{e}^{5}{x}^{3}+32760\,{b}^{2}c{e}^{5}{x}^{3}-25200\,b{c}^{2}d{e}^{4}{x}^{3}+6720\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}+60060\,abc{e}^{5}{x}^{2}-21840\,a{c}^{2}d{e}^{4}{x}^{2}+10010\,{b}^{3}{e}^{5}{x}^{2}-21840\,{b}^{2}cd{e}^{4}{x}^{2}+16800\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}-4480\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+25740\,{a}^{2}c{e}^{5}x+25740\,a{b}^{2}{e}^{5}x-34320\,abcd{e}^{4}x+12480\,a{c}^{2}{d}^{2}{e}^{3}x-5720\,{b}^{3}d{e}^{4}x+12480\,{b}^{2}c{d}^{2}{e}^{3}x-9600\,b{c}^{2}{d}^{3}{e}^{2}x+2560\,{c}^{3}{d}^{4}ex+18018\,{a}^{2}b{e}^{5}-10296\,{a}^{2}cd{e}^{4}-10296\,a{b}^{2}d{e}^{4}+13728\,abc{d}^{2}{e}^{3}-4992\,a{c}^{2}{d}^{3}{e}^{2}+2288\,{b}^{3}{d}^{2}{e}^{3}-4992\,{b}^{2}c{d}^{3}{e}^{2}+3840\,b{c}^{2}{d}^{4}e-1024\,{c}^{3}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.720272, size = 416, normalized size = 1.65 \[ \frac{2 \,{\left (6006 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} - 17325 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 16380 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 12870 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 9009 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.284029, size = 668, normalized size = 2.65 \[ \frac{2 \,{\left (6006 \, c^{3} e^{7} x^{7} - 512 \, c^{3} d^{7} + 1920 \, b c^{2} d^{6} e + 9009 \, a^{2} b d^{2} e^{5} - 2496 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} + 1144 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} - 5148 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 231 \,{\left (32 \, c^{3} d e^{6} + 75 \, b c^{2} e^{7}\right )} x^{6} + 126 \,{\left (c^{3} d^{2} e^{5} + 175 \, b c^{2} d e^{6} + 130 \,{\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} - 35 \,{\left (4 \, c^{3} d^{3} e^{4} - 15 \, b c^{2} d^{2} e^{5} - 624 \,{\left (b^{2} c + a c^{2}\right )} d e^{6} - 143 \,{\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} + 10 \,{\left (16 \, c^{3} d^{4} e^{3} - 60 \, b c^{2} d^{3} e^{4} + 78 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} + 715 \,{\left (b^{3} + 6 \, a b c\right )} d e^{6} + 1287 \,{\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} - 3 \,{\left (64 \, c^{3} d^{5} e^{2} - 240 \, b c^{2} d^{4} e^{3} - 3003 \, a^{2} b e^{7} + 312 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 143 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} - 6864 \,{\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} + 2 \,{\left (128 \, c^{3} d^{6} e - 480 \, b c^{2} d^{5} e^{2} + 9009 \, a^{2} b d e^{6} + 624 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 286 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 1287 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 14.5943, size = 1093, normalized size = 4.34 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)**(3/2)*(c*x**2+b*x+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.286176, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]