3.2345 \(\int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx\)
Optimal. Leaf size=323 \[ -\frac{5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}+\frac{9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c} \]
[Out]
(5*(b^2 - 4*a*c)^2*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sq
rt[a + b*x + c*x^2])/(16384*c^5) - (5*(b^2 - 4*a*c)*(32*c^2*d^2 + 9*b^2*e^2 - 4*
c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*c^2*d^
2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c
^3) + (9*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*(a +
b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(
8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(
11/2))
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Rubi [A] time = 0.924884, antiderivative size = 323, 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
\[ -\frac{5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}+\frac{9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]
[Out]
(5*(b^2 - 4*a*c)^2*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sq
rt[a + b*x + c*x^2])/(16384*c^5) - (5*(b^2 - 4*a*c)*(32*c^2*d^2 + 9*b^2*e^2 - 4*
c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*c^2*d^
2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c
^3) + (9*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*(a +
b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(
8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(
11/2))
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Rubi in Sympy [A] time = 75.125, size = 335, normalized size = 1.04 \[ \frac{e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{8 c} - \frac{9 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{112 c^{2}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{384 c^{3}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{6144 c^{4}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{16384 c^{5}} - \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{32768 c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+b*x+a)**(5/2),x)
[Out]
e*(d + e*x)*(a + b*x + c*x**2)**(7/2)/(8*c) - 9*e*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(7/2)/(112*c**2) + (b + 2*c*x)*(a + b*x + c*x**2)**(5/2)*(-4*a*c*e**2 + 9*b*
*2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/(384*c**3) - 5*(b + 2*c*x)*(-4*a*c + b**2)*
(a + b*x + c*x**2)**(3/2)*(-4*a*c*e**2 + 9*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2
)/(6144*c**4) + 5*(b + 2*c*x)*(-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2)*(-4*a*c*
e**2 + 9*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/(16384*c**5) - 5*(-4*a*c + b**2)
**3*(-4*a*c*e**2 + 9*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)))/(32768*c**(11/2))
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Mathematica [A] time = 1.43209, size = 538, normalized size = 1.67 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^3 c^2 \left (2359 a^2 e^2-4 a c \left (560 d^2+336 d e x+71 e^2 x^2\right )+8 c^2 x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )\right )+32 b^2 c^3 \left (-3 a^2 e (1232 d+199 e x)+12 a c x \left (56 d^2+40 d e x+9 e^2 x^2\right )+8 c^2 x^3 \left (378 d^2+592 d e x+243 e^2 x^2\right )\right )+64 b c^3 \left (-663 a^3 e^2+6 a^2 c \left (308 d^2+152 d e x+29 e^2 x^2\right )+8 a c^2 x^2 \left (546 d^2+788 d e x+307 e^2 x^2\right )+16 c^3 x^4 \left (140 d^2+232 d e x+99 e^2 x^2\right )\right )+128 c^4 \left (3 a^3 e (256 d+35 e x)+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+28 b^5 c \left (2 c \left (60 d^2+40 d e x+9 e^2 x^2\right )-375 a e^2\right )+8 b^4 c^2 \left (7 a e (640 d+113 e x)-2 c x \left (140 d^2+112 d e x+27 e^2 x^2\right )\right )+945 b^7 e^2-210 b^6 c e (16 d+3 e x)\right )-105 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{688128 c^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]
[Out]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^7*e^2 - 210*b^6*c*e*(16*d + 3*e*x) + 28*
b^5*c*(-375*a*e^2 + 2*c*(60*d^2 + 40*d*e*x + 9*e^2*x^2)) + 8*b^4*c^2*(7*a*e*(640
*d + 113*e*x) - 2*c*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2)) + 16*b^3*c^2*(2359*a^2
*e^2 + 8*c^2*x^2*(14*d^2 + 12*d*e*x + 3*e^2*x^2) - 4*a*c*(560*d^2 + 336*d*e*x +
71*e^2*x^2)) + 32*b^2*c^3*(-3*a^2*e*(1232*d + 199*e*x) + 12*a*c*x*(56*d^2 + 40*d
*e*x + 9*e^2*x^2) + 8*c^2*x^3*(378*d^2 + 592*d*e*x + 243*e^2*x^2)) + 64*b*c^3*(-
663*a^3*e^2 + 6*a^2*c*(308*d^2 + 152*d*e*x + 29*e^2*x^2) + 16*c^3*x^4*(140*d^2 +
232*d*e*x + 99*e^2*x^2) + 8*a*c^2*x^2*(546*d^2 + 788*d*e*x + 307*e^2*x^2)) + 12
8*c^4*(3*a^3*e*(256*d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) +
8*a*c^2*x^3*(182*d^2 + 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(924*d^2 + 1152*d*e*
x + 413*e^2*x^2))) - 105*(b^2 - 4*a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d
+ a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(688128*c^(11/2))
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 1517, normalized size = 4.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x)
[Out]
5/48*d^2/c*(c*x^2+b*x+a)^(3/2)*b*a-75/1024*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+5/192*d*e*b^4/c^3*(c*x^2+b*x+a)^(3/2)-5/512*d*e*b^6/c
^4*(c*x^2+b*x+a)^(1/2)+45/8192*e^2*b^6/c^4*(c*x^2+b*x+a)^(1/2)*x-1/12*d*e*b^2/c^
2*(c*x^2+b*x+a)^(5/2)-5/96*d^2/c*(c*x^2+b*x+a)^(3/2)*x*b^2+5/32*d^2/c*(c*x^2+b*x
+a)^(1/2)*b*a^2-5/64*d^2/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a+55/1024*e^2*b^3/c^3*(c*x^
2+b*x+a)^(1/2)*a^2-95/4096*e^2*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a-1/48*e^2*a/c*(c*x^2
+b*x+a)^(5/2)*x-1/96*e^2*a/c^2*(c*x^2+b*x+a)^(5/2)*b+5/32*d*e*b^3/c^2*(c*x^2+b*x
+a)^(1/2)*x*a-5/24*d*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a-5/16*d*e*b/c*(c*x^2+b*x+a)^(1
/2)*x*a^2+1/6*d^2*(c*x^2+b*x+a)^(5/2)*x-15/64*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*b^2*a^2-5/32*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a^2+5/64*d*e*
b^4/c^3*(c*x^2+b*x+a)^(1/2)*a-5/16*d*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a^3+15/64*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*a^2-15/256*d*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-95/2
048*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a-1/6*d*e*b/c*(c*x^2+b*x+a)^(5/2)*x-5/128*
e^2*a^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/1024*d*e*b^7/c^(9/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/1024*e^2*b^4/c^3*(c*x^2+b*x+a)
^(3/2)*x+5/256*d^2/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+3/64*e^2*b^2/c^2*(c*x^2+b*x+a)^
(5/2)*x-5/128*e^2*a^3/c*(c*x^2+b*x+a)^(1/2)*x-5/256*e^2*a^3/c^2*(c*x^2+b*x+a)^(1
/2)*b+15/128*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+15/
256*d^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-5/192*e^2*a^2/
c*(c*x^2+b*x+a)^(3/2)*x+25/768*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a+35/2048*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-5/384*e^2*a^2/c^2*(c*x^2+
b*x+a)^(3/2)*b-5/256*d*e*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x+55/512*e^2*b^2/c^2*(c*x^2
+b*x+a)^(1/2)*x*a^2-5/32*d^2/c*(c*x^2+b*x+a)^(1/2)*x*a*b^2-5/48*d*e*b^2/c^2*(c*x
^2+b*x+a)^(3/2)*a+25/384*e^2*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x*a+5/96*d*e*b^3/c^2*(c
*x^2+b*x+a)^(3/2)*x-9/112*e^2*b/c^2*(c*x^2+b*x+a)^(7/2)+3/128*e^2*b^3/c^3*(c*x^2
+b*x+a)^(5/2)-15/2048*e^2*b^5/c^4*(c*x^2+b*x+a)^(3/2)+45/16384*e^2*b^7/c^5*(c*x^
2+b*x+a)^(1/2)+1/8*e^2*x*(c*x^2+b*x+a)^(7/2)/c+1/12*d^2/c*(c*x^2+b*x+a)^(5/2)*b+
5/24*d^2*(c*x^2+b*x+a)^(3/2)*x*a-5/192*d^2/c^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*d^2*
(c*x^2+b*x+a)^(1/2)*x*a^2+2/7*d*e*(c*x^2+b*x+a)^(7/2)/c+5/512*d^2/c^3*(c*x^2+b*x
+a)^(1/2)*b^5+5/16*d^2/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^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^6-45/32768*e^2*b
^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
_______________________________________________________________________________________
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)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.452347, 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)^2,x, algorithm="fricas")
[Out]
[1/1376256*(4*(43008*c^7*e^2*x^7 + 3072*(32*c^7*d*e + 33*b*c^6*e^2)*x^6 + 256*(2
24*c^7*d^2 + 928*b*c^6*d*e + (243*b^2*c^5 + 476*a*c^6)*e^2)*x^5 + 128*(1120*b*c^
6*d^2 + 32*(37*b^2*c^5 + 72*a*c^6)*d*e + (3*b^3*c^4 + 1228*a*b*c^5)*e^2)*x^4 + 1
6*(224*(27*b^2*c^5 + 52*a*c^6)*d^2 + 32*(3*b^3*c^4 + 788*a*b*c^5)*d*e - (27*b^4*
c^3 - 216*a*b^2*c^4 - 6608*a^2*c^5)*e^2)*x^3 + 224*(15*b^5*c^2 - 160*a*b^3*c^3 +
528*a^2*b*c^4)*d^2 - 32*(105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a
^3*c^4)*d*e + (945*b^7 - 10500*a*b^5*c + 37744*a^2*b^3*c^2 - 42432*a^3*b*c^3)*e^
2 + 8*(224*(b^3*c^4 + 156*a*b*c^5)*d^2 - 32*(7*b^4*c^3 - 60*a*b^2*c^4 - 1152*a^2
*c^5)*d*e + (63*b^5*c^2 - 568*a*b^3*c^3 + 1392*a^2*b*c^4)*e^2)*x^2 - 2*(224*(5*b
^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d^2 - 32*(35*b^5*c^2 - 336*a*b^3*c^3 + 912*
a^2*b*c^4)*d*e + (315*b^6*c - 3164*a*b^4*c^2 + 9552*a^2*b^2*c^3 - 6720*a^3*c^4)*
e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2
*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*
b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*
c^4)*e^2)*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^(11/2), 1/688128*(2*(43008*c^7*e^2*x^7 + 3072*(32*c^7*d*
e + 33*b*c^6*e^2)*x^6 + 256*(224*c^7*d^2 + 928*b*c^6*d*e + (243*b^2*c^5 + 476*a*
c^6)*e^2)*x^5 + 128*(1120*b*c^6*d^2 + 32*(37*b^2*c^5 + 72*a*c^6)*d*e + (3*b^3*c^
4 + 1228*a*b*c^5)*e^2)*x^4 + 16*(224*(27*b^2*c^5 + 52*a*c^6)*d^2 + 32*(3*b^3*c^4
+ 788*a*b*c^5)*d*e - (27*b^4*c^3 - 216*a*b^2*c^4 - 6608*a^2*c^5)*e^2)*x^3 + 224
*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d^2 - 32*(105*b^6*c - 1120*a*b^4*c
^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*d*e + (945*b^7 - 10500*a*b^5*c + 37744*a^2
*b^3*c^2 - 42432*a^3*b*c^3)*e^2 + 8*(224*(b^3*c^4 + 156*a*b*c^5)*d^2 - 32*(7*b^4
*c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*d*e + (63*b^5*c^2 - 568*a*b^3*c^3 + 1392*a^2
*b*c^4)*e^2)*x^2 - 2*(224*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d^2 - 32*(35*
b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*d*e + (315*b^6*c - 3164*a*b^4*c^2 + 955
2*a^2*b^2*c^3 - 6720*a^3*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 105*(32*(
b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12*a*b^5
*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c
^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*e^2)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c
*x^2 + b*x + a)*c)))/(sqrt(-c)*c^5)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+b*x+a)**(5/2),x)
[Out]
Integral((d + e*x)**2*(a + b*x + c*x**2)**(5/2), x)
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GIAC/XCAS [A] time = 0.248361, size = 1035, normalized size = 3.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^2,x, algorithm="giac")
[Out]
1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e +
33*b*c^8*e^2)/c^7)*x + (224*c^9*d^2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2 + 476*a*c^
8*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 2304*a*c^8*d*e + 3*b^3*c^6*
e^2 + 1228*a*b*c^7*e^2)/c^7)*x + (6048*b^2*c^7*d^2 + 11648*a*c^8*d^2 + 96*b^3*c^
6*d*e + 25216*a*b*c^7*d*e - 27*b^4*c^5*e^2 + 216*a*b^2*c^6*e^2 + 6608*a^2*c^7*e^
2)/c^7)*x + (224*b^3*c^6*d^2 + 34944*a*b*c^7*d^2 - 224*b^4*c^5*d*e + 1920*a*b^2*
c^6*d*e + 36864*a^2*c^7*d*e + 63*b^5*c^4*e^2 - 568*a*b^3*c^5*e^2 + 1392*a^2*b*c^
6*e^2)/c^7)*x - (1120*b^4*c^5*d^2 - 10752*a*b^2*c^6*d^2 - 118272*a^2*c^7*d^2 - 1
120*b^5*c^4*d*e + 10752*a*b^3*c^5*d*e - 29184*a^2*b*c^6*d*e + 315*b^6*c^3*e^2 -
3164*a*b^4*c^4*e^2 + 9552*a^2*b^2*c^5*e^2 - 6720*a^3*c^6*e^2)/c^7)*x + (3360*b^5
*c^4*d^2 - 35840*a*b^3*c^5*d^2 + 118272*a^2*b*c^6*d^2 - 3360*b^6*c^3*d*e + 35840
*a*b^4*c^4*d*e - 118272*a^2*b^2*c^5*d*e + 98304*a^3*c^6*d*e + 945*b^7*c^2*e^2 -
10500*a*b^5*c^3*e^2 + 37744*a^2*b^3*c^4*e^2 - 42432*a^3*b*c^5*e^2)/c^7) + 5/3276
8*(32*b^6*c^2*d^2 - 384*a*b^4*c^3*d^2 + 1536*a^2*b^2*c^4*d^2 - 2048*a^3*c^5*d^2
- 32*b^7*c*d*e + 384*a*b^5*c^2*d*e - 1536*a^2*b^3*c^3*d*e + 2048*a^3*b*c^4*d*e +
9*b^8*e^2 - 112*a*b^6*c*e^2 + 480*a^2*b^4*c^2*e^2 - 768*a^3*b^2*c^3*e^2 + 256*a
^4*c^4*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)