3.2344 \(\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx\)
Optimal. Leaf size=400 \[ -\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32768 c^6}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{12288 c^5}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{768 c^4}+\frac{e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{2016 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
[Out]
(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e
))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(32768*c^6) - (5*(b^2 - 4*a*c)*(2*c*d - b*
e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*c*x)*(a + b*x + c*x^
2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d +
3*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(a + b*
x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^2 + 99*b^2*e^2 - 2*c*e*(243*b*d + 32*a*e
) + 154*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(2016*c^3) - (5*(b^2 - 4*a
*c)^3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*ArcTanh[(b
+ 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(65536*c^(13/2))
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Rubi [A] time = 1.0012, antiderivative size = 400, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227
\[ -\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32768 c^6}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{12288 c^5}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{768 c^4}+\frac{e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{2016 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]
[Out]
(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e
))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(32768*c^6) - (5*(b^2 - 4*a*c)*(2*c*d - b*
e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*c*x)*(a + b*x + c*x^
2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d +
3*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(a + b*
x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^2 + 99*b^2*e^2 - 2*c*e*(243*b*d + 32*a*e
) + 154*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(2016*c^3) - (5*(b^2 - 4*a
*c)^3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*ArcTanh[(b
+ 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(65536*c^(13/2))
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Rubi in Sympy [A] time = 98.0643, size = 415, normalized size = 1.04 \[ \frac{e \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9 c} + \frac{e \left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (- 16 a c e^{2} + \frac{99 b^{2} e^{2}}{4} - \frac{243 b c d e}{2} + 160 c^{2} d^{2} - \frac{77 c e x \left (b e - 2 c d\right )}{2}\right )}{504 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 12 a c e^{2} + 11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{768 c^{4}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 12 a c e^{2} + 11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{12288 c^{5}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (- 12 a c e^{2} + 11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{32768 c^{6}} + \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (b e - 2 c d\right ) \left (- 12 a c e^{2} + 11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{65536 c^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)
[Out]
e*(d + e*x)**2*(a + b*x + c*x**2)**(7/2)/(9*c) + e*(a + b*x + c*x**2)**(7/2)*(-1
6*a*c*e**2 + 99*b**2*e**2/4 - 243*b*c*d*e/2 + 160*c**2*d**2 - 77*c*e*x*(b*e - 2*
c*d)/2)/(504*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(5/2)*(-12*a*
c*e**2 + 11*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/(768*c**4) + 5*(b + 2*c*x)*(-
4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(3/2)*(-12*a*c*e**2 + 11*b**2*e*
*2 - 32*b*c*d*e + 32*c**2*d**2)/(12288*c**5) - 5*(b + 2*c*x)*(-4*a*c + b**2)**2*
(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)*(-12*a*c*e**2 + 11*b**2*e**2 - 32*b*c*d*e +
32*c**2*d**2)/(32768*c**6) + 5*(-4*a*c + b**2)**3*(b*e - 2*c*d)*(-12*a*c*e**2 +
11*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a +
b*x + c*x**2)))/(65536*c**(13/2))
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Mathematica [B] time = 2.45202, size = 807, normalized size = 2.02 \[ \frac{315 (b e-2 c d) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \log \left (b+2 c x+2 \sqrt{c} \sqrt{a+x (b+c x)}\right ) \left (b^2-4 a c\right )^3+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-3465 e^3 b^8+210 c e^2 (81 d+11 e x) b^7-84 c e \left (c \left (360 d^2+135 e x d+22 e^2 x^2\right )-485 a e^2\right ) b^6+72 c^2 \left (2 c \left (140 d^3+140 e x d^2+63 e^2 x^2 d+11 e^3 x^3\right )-7 a e^2 (375 d+49 e x)\right ) b^5-16 c^2 \left (10143 a^2 e^3-9 a c \left (2240 d^2+791 e x d+124 e^2 x^2\right ) e+2 c^2 x \left (420 d^3+504 e x d^2+243 e^2 x^2 d+44 e^3 x^3\right )\right ) b^4+32 c^3 \left (9 a^2 (2359 d+293 e x) e^2+8 c^2 x^2 \left (42 d^3+54 e x d^2+27 e^2 x^2 d+5 e^3 x^3\right )-4 a c \left (1680 d^3+1512 e x d^2+639 e^2 x^2 d+107 e^3 x^3\right )\right ) b^3+192 c^3 \left (1221 a^3 e^3-a^2 c \left (5544 d^2+1791 e x d+266 e^2 x^2\right ) e+4 a c^2 x \left (168 d^3+180 e x d^2+81 e^2 x^2 d+14 e^3 x^3\right )+8 c^3 x^3 \left (378 d^3+888 e x d^2+729 e^2 x^2 d+206 e^3 x^3\right )\right ) b^2+128 c^4 \left (16 c^3 \left (420 d^3+1044 e x d^2+891 e^2 x^2 d+259 e^3 x^3\right ) x^4+24 a c^2 \left (546 d^3+1182 e x d^2+921 e^2 x^2 d+251 e^3 x^3\right ) x^2-13 a^3 e^2 (459 d+53 e x)+6 a^2 c \left (924 d^3+684 e x d^2+261 e^2 x^2 d+41 e^3 x^3\right )\right ) b+256 c^4 \left (16 c^4 \left (84 d^3+216 e x d^2+189 e^2 x^2 d+56 e^3 x^3\right ) x^5+8 a c^3 \left (546 d^3+1296 e x d^2+1071 e^2 x^2 d+304 e^3 x^3\right ) x^3+6 a^2 c^2 \left (924 d^3+1728 e x d^2+1239 e^2 x^2 d+320 e^3 x^3\right ) x-256 a^4 e^3+a^3 c e \left (3456 d^2+945 e x d+128 e^2 x^2\right )\right )\right )}{4128768 c^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]
[Out]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^8*e^3 + 210*b^7*c*e^2*(81*d + 11*e*x)
- 84*b^6*c*e*(-485*a*e^2 + c*(360*d^2 + 135*d*e*x + 22*e^2*x^2)) + 72*b^5*c^2*(-
7*a*e^2*(375*d + 49*e*x) + 2*c*(140*d^3 + 140*d^2*e*x + 63*d*e^2*x^2 + 11*e^3*x^
3)) - 16*b^4*c^2*(10143*a^2*e^3 - 9*a*c*e*(2240*d^2 + 791*d*e*x + 124*e^2*x^2) +
2*c^2*x*(420*d^3 + 504*d^2*e*x + 243*d*e^2*x^2 + 44*e^3*x^3)) + 32*b^3*c^3*(9*a
^2*e^2*(2359*d + 293*e*x) + 8*c^2*x^2*(42*d^3 + 54*d^2*e*x + 27*d*e^2*x^2 + 5*e^
3*x^3) - 4*a*c*(1680*d^3 + 1512*d^2*e*x + 639*d*e^2*x^2 + 107*e^3*x^3)) + 192*b^
2*c^3*(1221*a^3*e^3 - a^2*c*e*(5544*d^2 + 1791*d*e*x + 266*e^2*x^2) + 4*a*c^2*x*
(168*d^3 + 180*d^2*e*x + 81*d*e^2*x^2 + 14*e^3*x^3) + 8*c^3*x^3*(378*d^3 + 888*d
^2*e*x + 729*d*e^2*x^2 + 206*e^3*x^3)) + 128*b*c^4*(-13*a^3*e^2*(459*d + 53*e*x)
+ 6*a^2*c*(924*d^3 + 684*d^2*e*x + 261*d*e^2*x^2 + 41*e^3*x^3) + 24*a*c^2*x^2*(
546*d^3 + 1182*d^2*e*x + 921*d*e^2*x^2 + 251*e^3*x^3) + 16*c^3*x^4*(420*d^3 + 10
44*d^2*e*x + 891*d*e^2*x^2 + 259*e^3*x^3)) + 256*c^4*(-256*a^4*e^3 + a^3*c*e*(34
56*d^2 + 945*d*e*x + 128*e^2*x^2) + 16*c^4*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2
*x^2 + 56*e^3*x^3) + 8*a*c^3*x^3*(546*d^3 + 1296*d^2*e*x + 1071*d*e^2*x^2 + 304*
e^3*x^3) + 6*a^2*c^2*x*(924*d^3 + 1728*d^2*e*x + 1239*d*e^2*x^2 + 320*e^3*x^3)))
+ 315*(b^2 - 4*a*c)^3*(-2*c*d + b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d +
3*a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(4128768*c^(13/2))
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Maple [B] time = 0.023, size = 2294, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x)
[Out]
55/65536*e^3*b^9/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/9*e^3*x^
2*(c*x^2+b*x+a)^(7/2)/c-11/768*e^3*b^4/c^4*(c*x^2+b*x+a)^(5/2)+55/12288*e^3*b^6/
c^5*(c*x^2+b*x+a)^(3/2)-5/192*d^3/c^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*d^3*(c*x^2+b*
x+a)^(1/2)*x*a^2+5/512*d^3/c^3*(c*x^2+b*x+a)^(1/2)*b^5+5/16*d^3/c^(1/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-5/1024*d^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*b^6+5/24*d^3*(c*x^2+b*x+a)^(3/2)*x*a-2/63*e^3*a/c^2*(c*x^
2+b*x+a)^(7/2)+1/6*d^3*(c*x^2+b*x+a)^(5/2)*x+5/128*d^2*e*b^4/c^3*(c*x^2+b*x+a)^(
3/2)-5/16*d^2*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a-15/32*d^2*e*b/c*(c*x^2+b*x+a)^(1/2)*
x*a^2+15/64*d^2*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a+25/128*d*e^2*b^2/c^2*(c*x^2+b*
x+a)^(3/2)*x*a+165/512*d*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2-285/2048*d*e^2*b^
4/c^3*(c*x^2+b*x+a)^(1/2)*x*a+15/256*e^3*b/c^(5/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))-45/2048*d*e^2*b^5/c^4*(c*x^2+b*x+a)^(3/2)+135/16384*d*e^2*b^7
/c^5*(c*x^2+b*x+a)^(1/2)+5/48*d^3/c*(c*x^2+b*x+a)^(3/2)*b*a-5/64*d^3/c^2*(c*x^2+
b*x+a)^(1/2)*b^3*a-15/64*d^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*b^2*a^2+5/256*e^3*b^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)+15/512*e^3*b^2/c^3*a^3*(c*x^2
+b*x+a)^(1/2)-5/96*d^3/c*(c*x^2+b*x+a)^(3/2)*x*b^2+15/256*d^3/c^(5/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-15/1024*d^2*e*b^6/c^4*(c*x^2+b*x+a)^(1/2
)+15/2048*d^2*e*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/8*d^2*
e*b^2/c^2*(c*x^2+b*x+a)^(5/2)-45/4096*e^3*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a-11/144*e^3*b/c^2*x*(c*x^2+b*x+a)^(7/2)+3/8*d*e^2*x*(c*x^2+b
*x+a)^(7/2)/c-15/128*d*e^2*a^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))-135/32768*d*e^2*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-27/1
12*d*e^2*b/c^2*(c*x^2+b*x+a)^(7/2)+9/128*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(5/2)+15/25
6*e^3*b/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x-85/1024*e^3*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*
a^2+125/8192*e^3*b^6/c^5*(c*x^2+b*x+a)^(1/2)*a+105/2048*e^3*b^5/c^(9/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-25/256*e^3*b^3/c^(7/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+5/256*d^3/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32*d^3/
c*(c*x^2+b*x+a)^(1/2)*b*a^2-15/128*d*e^2*a^3/c*(c*x^2+b*x+a)^(1/2)*x+25/256*d*e^
2*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a+105/2048*d*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*a+9/64*d*e^2*b^2/c^2*(c*x^2+b*x+a)^(5/2)*x-45/1024*d*e^2*
b^4/c^3*(c*x^2+b*x+a)^(3/2)*x+165/1024*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a^2-5/6
4*d*e^2*a^2/c*(c*x^2+b*x+a)^(3/2)*x+45/128*d*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a^3-225/1024*d*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*a^2-1/32*d*e^2*a/c^2*(c*x^2+b*x+a)^(5/2)*b-15/256*d*e^2*a^3/
c^2*(c*x^2+b*x+a)^(1/2)*b-1/4*d^2*e*b/c*(c*x^2+b*x+a)^(5/2)*x+5/64*d^2*e*b^3/c^2
*(c*x^2+b*x+a)^(3/2)*x-5/32*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a-15/512*d^2*e*b^5
/c^3*(c*x^2+b*x+a)^(1/2)*x-15/64*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a^2+15/128*d^
2*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a+5/128*e^3*b/c^2*a^2*(c*x^2+b*x+a)^(3/2)*x+1/32
*e^3*b/c^2*a*(c*x^2+b*x+a)^(5/2)*x-35/768*e^3*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x*a-5/
32*d^3/c*(c*x^2+b*x+a)^(1/2)*x*a*b^2-1/16*d*e^2*a/c*(c*x^2+b*x+a)^(5/2)*x-285/40
96*d*e^2*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a+45/128*d^2*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-45/512*d^2*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))*a+125/4096*e^3*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x*a-15/32*d^2*e*
b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+135/8192*d*e^2*b^6/c^4
*(c*x^2+b*x+a)^(1/2)*x-5/128*d*e^2*a^2/c^2*(c*x^2+b*x+a)^(3/2)*b+1/64*e^3*b^2/c^
3*a*(c*x^2+b*x+a)^(5/2)-11/384*e^3*b^3/c^3*(c*x^2+b*x+a)^(5/2)*x+55/6144*e^3*b^5
/c^4*(c*x^2+b*x+a)^(3/2)*x-35/1536*e^3*b^4/c^4*(c*x^2+b*x+a)^(3/2)*a-55/16384*e^
3*b^7/c^5*(c*x^2+b*x+a)^(1/2)*x-85/2048*e^3*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a^2+1/12
*d^3/c*(c*x^2+b*x+a)^(5/2)*b+3/7*d^2*e*(c*x^2+b*x+a)^(7/2)/c-55/32768*e^3*b^8/c^
6*(c*x^2+b*x+a)^(1/2)+11/224*e^3*b^2/c^3*(c*x^2+b*x+a)^(7/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.587189, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^3,x, algorithm="fricas")
[Out]
[1/8257536*(4*(229376*c^8*e^3*x^8 + 14336*(54*c^8*d*e^2 + 37*b*c^7*e^3)*x^7 + 10
24*(864*c^8*d^2*e + 1782*b*c^7*d*e^2 + (309*b^2*c^6 + 608*a*c^7)*e^3)*x^6 + 256*
(1344*c^8*d^3 + 8352*b*c^7*d^2*e + 18*(243*b^2*c^6 + 476*a*c^7)*d*e^2 + (5*b^3*c
^5 + 3012*a*b*c^6)*e^3)*x^5 + 128*(6720*b*c^7*d^3 + 288*(37*b^2*c^6 + 72*a*c^7)*
d^2*e + 18*(3*b^3*c^5 + 1228*a*b*c^6)*d*e^2 - (11*b^4*c^4 - 84*a*b^2*c^5 - 3840*
a^2*c^6)*e^3)*x^4 + 1344*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^3 - 288*
(105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d^2*e + 18*(945
*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4)*d*e^2 - (3465*b^
8 - 40740*a*b^6*c + 162288*a^2*b^4*c^2 - 234432*a^3*b^2*c^3 + 65536*a^4*c^4)*e^3
+ 16*(1344*(27*b^2*c^6 + 52*a*c^7)*d^3 + 288*(3*b^3*c^5 + 788*a*b*c^6)*d^2*e -
18*(27*b^4*c^4 - 216*a*b^2*c^5 - 6608*a^2*c^6)*d*e^2 + (99*b^5*c^3 - 856*a*b^3*c
^4 + 1968*a^2*b*c^5)*e^3)*x^3 + 8*(1344*(b^3*c^5 + 156*a*b*c^6)*d^3 - 288*(7*b^4
*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d^2*e + 18*(63*b^5*c^3 - 568*a*b^3*c^4 + 139
2*a^2*b*c^5)*d*e^2 - (231*b^6*c^2 - 2232*a*b^4*c^3 + 6384*a^2*b^2*c^4 - 4096*a^3
*c^5)*e^3)*x^2 - 2*(1344*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^3 - 288*(35*
b^5*c^3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d^2*e + 18*(315*b^6*c^2 - 3164*a*b^4*c^
3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*d*e^2 - (1155*b^7*c - 12348*a*b^5*c^2 + 421
92*a^2*b^3*c^3 - 44096*a^3*b*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 315*(6
4*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3 - 96*(b^7*c^2 - 12*
a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e + 6*(9*b^8*c - 112*a*b^6*c^2 +
480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a^4*c^5)*d*e^2 - (11*b^9 - 144*a*b^7*c +
672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 768*a^4*b*c^4)*e^3)*log(-4*(2*c^2*x + b*c)
*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/c^(13/2),
1/4128768*(2*(229376*c^8*e^3*x^8 + 14336*(54*c^8*d*e^2 + 37*b*c^7*e^3)*x^7 + 10
24*(864*c^8*d^2*e + 1782*b*c^7*d*e^2 + (309*b^2*c^6 + 608*a*c^7)*e^3)*x^6 + 256*
(1344*c^8*d^3 + 8352*b*c^7*d^2*e + 18*(243*b^2*c^6 + 476*a*c^7)*d*e^2 + (5*b^3*c
^5 + 3012*a*b*c^6)*e^3)*x^5 + 128*(6720*b*c^7*d^3 + 288*(37*b^2*c^6 + 72*a*c^7)*
d^2*e + 18*(3*b^3*c^5 + 1228*a*b*c^6)*d*e^2 - (11*b^4*c^4 - 84*a*b^2*c^5 - 3840*
a^2*c^6)*e^3)*x^4 + 1344*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^3 - 288*
(105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d^2*e + 18*(945
*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4)*d*e^2 - (3465*b^
8 - 40740*a*b^6*c + 162288*a^2*b^4*c^2 - 234432*a^3*b^2*c^3 + 65536*a^4*c^4)*e^3
+ 16*(1344*(27*b^2*c^6 + 52*a*c^7)*d^3 + 288*(3*b^3*c^5 + 788*a*b*c^6)*d^2*e -
18*(27*b^4*c^4 - 216*a*b^2*c^5 - 6608*a^2*c^6)*d*e^2 + (99*b^5*c^3 - 856*a*b^3*c
^4 + 1968*a^2*b*c^5)*e^3)*x^3 + 8*(1344*(b^3*c^5 + 156*a*b*c^6)*d^3 - 288*(7*b^4
*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d^2*e + 18*(63*b^5*c^3 - 568*a*b^3*c^4 + 139
2*a^2*b*c^5)*d*e^2 - (231*b^6*c^2 - 2232*a*b^4*c^3 + 6384*a^2*b^2*c^4 - 4096*a^3
*c^5)*e^3)*x^2 - 2*(1344*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^3 - 288*(35*
b^5*c^3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d^2*e + 18*(315*b^6*c^2 - 3164*a*b^4*c^
3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*d*e^2 - (1155*b^7*c - 12348*a*b^5*c^2 + 421
92*a^2*b^3*c^3 - 44096*a^3*b*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 315*(
64*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3 - 96*(b^7*c^2 - 12
*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e + 6*(9*b^8*c - 112*a*b^6*c^2 +
480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a^4*c^5)*d*e^2 - (11*b^9 - 144*a*b^7*c
+ 672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 768*a^4*b*c^4)*e^3)*arctan(1/2*(2*c*x + b
)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^6)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)
[Out]
Integral((d + e*x)**3*(a + b*x + c*x**2)**(5/2), x)
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GIAC/XCAS [A] time = 0.231434, 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)^(5/2)*(e*x + d)^3,x, algorithm="giac")
[Out]
Done