3.1557 \(\int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx\)
Optimal. Leaf size=379 \[ \frac{3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}-\frac{3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \]
[Out]
(-3*(b^2 - 4*a*c)^2*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)
*Sqrt[a + b*x + c*x^2])/(8192*c^5) + ((b^2 - 4*a*c)*e*(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))/(1024*c^4) + (3*(2*c*d
- b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(56*c) + ((d + e*x)^3*(a + b*x + c*
x^2)^(5/2))/4 + ((96*c^3*d^3 - 63*b^3*e^3 + 4*b*c*e^2*(56*b*d + 61*a*e) - 8*c^2*
d*e*(13*b*d + 96*a*e) + 10*c*e*(8*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(2*b*d + 7*a*e))*x
)*(a + b*x + c*x^2)^(5/2))/(2240*c^3) + (3*(b^2 - 4*a*c)^3*e*(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]
)])/(16384*c^(11/2))
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Rubi [A] time = 1.14508, antiderivative size = 379, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179
\[ \frac{3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}-\frac{3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(3/2),x]
[Out]
(-3*(b^2 - 4*a*c)^2*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)
*Sqrt[a + b*x + c*x^2])/(8192*c^5) + ((b^2 - 4*a*c)*e*(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))/(1024*c^4) + (3*(2*c*d
- b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(56*c) + ((d + e*x)^3*(a + b*x + c*
x^2)^(5/2))/4 + ((96*c^3*d^3 - 63*b^3*e^3 + 4*b*c*e^2*(56*b*d + 61*a*e) - 8*c^2*
d*e*(13*b*d + 96*a*e) + 10*c*e*(8*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(2*b*d + 7*a*e))*x
)*(a + b*x + c*x^2)^(5/2))/(2240*c^3) + (3*(b^2 - 4*a*c)^3*e*(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]
)])/(16384*c^(11/2))
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Rubi in Sympy [A] time = 158.909, size = 406, normalized size = 1.07 \[ \frac{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{4} - \frac{3 \left (d + e x\right )^{2} \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{56 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 183 a b c e^{3} + 576 a c^{2} d e^{2} + \frac{189 b^{3} e^{3}}{4} - 168 b^{2} c d e^{2} + 78 b c^{2} d^{2} e - 72 c^{3} d^{3} - \frac{15 c e x \left (- 28 a c e^{2} + 9 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{2}\right )}{1680 c^{3}} + \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{1024 c^{4}} - \frac{3 e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{8192 c^{5}} + \frac{3 e \left (- 4 a c + b^{2}\right )^{3} \left (- 4 a c e^{2} + 9 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16384 c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(3/2),x)
[Out]
(d + e*x)**3*(a + b*x + c*x**2)**(5/2)/4 - 3*(d + e*x)**2*(b*e - 2*c*d)*(a + b*x
+ c*x**2)**(5/2)/(56*c) - (a + b*x + c*x**2)**(5/2)*(-183*a*b*c*e**3 + 576*a*c*
*2*d*e**2 + 189*b**3*e**3/4 - 168*b**2*c*d*e**2 + 78*b*c**2*d**2*e - 72*c**3*d**
3 - 15*c*e*x*(-28*a*c*e**2 + 9*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/2)/(1680*c**
3) + e*(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)/(1024*c**4) - 3*e*(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)/(8192*c**5) + 3*e*(-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)))/(16384*c
**(11/2))
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Mathematica [A] time = 1.4842, size = 613, normalized size = 1.62 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-192 a^3 c^3 e^2 (-221 b e+512 c d+70 c e x)+16 a^2 c^2 \left (-2359 b^3 e^3+6 b^2 c e^2 (1232 d+199 e x)-24 b c^2 e \left (308 d^2+152 d e x+29 e^2 x^2\right )+16 c^3 \left (448 d^3+420 d^2 e x+192 d e^2 x^2+35 e^3 x^3\right )\right )+4 a c \left (2625 b^5 e^3-14 b^4 c e^2 (640 d+113 e x)+16 b^3 c^2 e \left (560 d^2+336 d e x+71 e^2 x^2\right )-96 b^2 c^3 e x \left (56 d^2+40 d e x+9 e^2 x^2\right )+128 b c^4 x \left (448 d^3+798 d^2 e x+556 d e^2 x^2+141 e^3 x^3\right )+256 c^5 x^2 \left (224 d^3+490 d^2 e x+384 d e^2 x^2+105 e^3 x^3\right )\right )-945 b^7 e^3+210 b^6 c e^2 (16 d+3 e x)-56 b^5 c^2 e \left (60 d^2+40 d e x+9 e^2 x^2\right )+16 b^4 c^3 e x \left (140 d^2+112 d e x+27 e^2 x^2\right )-128 b^3 c^4 e x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )+256 b^2 c^5 x^2 \left (448 d^3+966 d^2 e x+752 d e^2 x^2+205 e^3 x^3\right )+1024 b c^6 x^3 \left (224 d^3+532 d^2 e x+440 d e^2 x^2+125 e^3 x^3\right )+2048 c^7 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )+105 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{573440 c^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(3/2),x]
[Out]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-945*b^7*e^3 + 210*b^6*c*e^2*(16*d + 3*e*x) -
192*a^3*c^3*e^2*(512*c*d - 221*b*e + 70*c*e*x) - 128*b^3*c^4*e*x^2*(14*d^2 + 12*
d*e*x + 3*e^2*x^2) - 56*b^5*c^2*e*(60*d^2 + 40*d*e*x + 9*e^2*x^2) + 16*b^4*c^3*e
*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2) + 2048*c^7*x^4*(56*d^3 + 140*d^2*e*x + 120
*d*e^2*x^2 + 35*e^3*x^3) + 1024*b*c^6*x^3*(224*d^3 + 532*d^2*e*x + 440*d*e^2*x^2
+ 125*e^3*x^3) + 256*b^2*c^5*x^2*(448*d^3 + 966*d^2*e*x + 752*d*e^2*x^2 + 205*e
^3*x^3) + 16*a^2*c^2*(-2359*b^3*e^3 + 6*b^2*c*e^2*(1232*d + 199*e*x) - 24*b*c^2*
e*(308*d^2 + 152*d*e*x + 29*e^2*x^2) + 16*c^3*(448*d^3 + 420*d^2*e*x + 192*d*e^2
*x^2 + 35*e^3*x^3)) + 4*a*c*(2625*b^5*e^3 - 14*b^4*c*e^2*(640*d + 113*e*x) - 96*
b^2*c^3*e*x*(56*d^2 + 40*d*e*x + 9*e^2*x^2) + 16*b^3*c^2*e*(560*d^2 + 336*d*e*x
+ 71*e^2*x^2) + 256*c^5*x^2*(224*d^3 + 490*d^2*e*x + 384*d*e^2*x^2 + 105*e^3*x^3
) + 128*b*c^4*x*(448*d^3 + 798*d^2*e*x + 556*d*e^2*x^2 + 141*e^3*x^3))) + 105*(b
^2 - 4*a*c)^3*e*(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)]])/(573440*c^(11/2))
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Maple [B] time = 0.023, size = 1607, normalized size = 4.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(3/2),x)
[Out]
3/8*b/c*a^2*(c*x^2+b*x+a)^(1/2)*x*d*e^2+3/16*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a*d^2*e
-3/16*a^2/c*(c*x^2+b*x+a)^(1/2)*b*d^2*e-9/32*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))*a^2*d*e^2+9/128*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a*d*e^2+3/32*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a*d^2*e+1/16*b^2/c*(c*x^2+
b*x+a)^(3/2)*x*d^2*e-3/128*b^4/c^2*(c*x^2+b*x+a)^(1/2)*x*d^2*e-1/8*a/c*(c*x^2+b*
x+a)^(3/2)*b*d^2*e-1/16*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x*d*e^2+3/128*b^5/c^3*(c*x^2
+b*x+a)^(1/2)*x*d*e^2-5/64*b^2/c^2*a*(c*x^2+b*x+a)^(3/2)*x*e^3-33/256*b^2/c^2*a^
2*(c*x^2+b*x+a)^(1/2)*x*e^3-1/7*b/c*x*(c*x^2+b*x+a)^(5/2)*d*e^2+1/8*b^2/c^2*a*(c
*x^2+b*x+a)^(3/2)*d*e^2+3/16*b^2/c^2*a^2*(c*x^2+b*x+a)^(1/2)*d*e^2+57/1024*b^4/c
^3*(c*x^2+b*x+a)^(1/2)*x*a*e^3-3/32*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a*d*e^2+3/8*b/c^
(3/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2+9/32*b^2/c^(3/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d^2*e-9/128*b^4/c^(5/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^2*e-12/35*a/c*(c*x^2+b*x+a)^(5/2)*d*e^2-21/
1024*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e^3-9/64*b^2/c^(5
/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^3+9/224*b^2/c^2*x*(c*x^2+b
*x+a)^(5/2)*e^3+9/512*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x*e^3-27/4096*b^6/c^4*(c*x^2+b
*x+a)^(1/2)*x*e^3-3/256*b^5/c^3*(c*x^2+b*x+a)^(1/2)*d^2*e-1/10*b/c*(c*x^2+b*x+a)
^(5/2)*d^2*e-1/8*e^3/c*a*x*(c*x^2+b*x+a)^(5/2)+1/32*e^3/c*a^2*(c*x^2+b*x+a)^(3/2
)*x+1/64*e^3/c^2*a^2*(c*x^2+b*x+a)^(3/2)*b+3/64*e^3/c*a^3*(c*x^2+b*x+a)^(1/2)*x+
3/128*e^3/c^2*a^3*(c*x^2+b*x+a)^(1/2)*b-3/512*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*d*e^2+3/512*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*d^2*e-3/8*a^2*(c*x^2+b*x+a)^(1/2)*x*d^2*e-1/4*a*(c*x^2+b*x+a)^(3/2)*x*
d^2*e+1/32*b^3/c^2*(c*x^2+b*x+a)^(3/2)*d^2*e-5/128*b^3/c^3*a*(c*x^2+b*x+a)^(3/2)
*e^3-33/512*b^3/c^3*a^2*(c*x^2+b*x+a)^(1/2)*e^3-3/56*x^2*(c*x^2+b*x+a)^(5/2)/c*b
*e^3+57/2048*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a*e^3+1/10*b^2/c^2*(c*x^2+b*x+a)^(5/2)*
d*e^2-1/32*b^4/c^3*(c*x^2+b*x+a)^(3/2)*d*e^2+3/256*b^6/c^4*(c*x^2+b*x+a)^(1/2)*d
*e^2+45/512*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*e^3-3/8*
a^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e+2/5*d^3*(c*x^2+b*x
+a)^(5/2)+61/560*a/c^2*(c*x^2+b*x+a)^(5/2)*b*e^3+1/4*e^3*x^3*(c*x^2+b*x+a)^(5/2)
-3/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a*d*e^2+1/4*b/c*a*(c*x^2+b*x+a)^(3/2)*x*d*e^
2+3/64*e^3/c^(3/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-9/320*b^3/c^3
*(c*x^2+b*x+a)^(5/2)*e^3+9/1024*b^5/c^4*(c*x^2+b*x+a)^(3/2)*e^3-27/8192*b^7/c^5*
(c*x^2+b*x+a)^(1/2)*e^3+27/16384*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*e^3+6/7*d*e^2*x^2*(c*x^2+b*x+a)^(5/2)+d^2*e*x*(c*x^2+b*x+a)^(5/2)
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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)^(3/2)*(2*c*x + b)*(e*x + d)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.510339, 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)^(3/2)*(2*c*x + b)*(e*x + d)^3,x, algorithm="fricas")
[Out]
[1/1146880*(4*(71680*c^7*e^3*x^7 + 114688*a^2*c^5*d^3 + 5120*(48*c^7*d*e^2 + 25*
b*c^6*e^3)*x^6 + 1280*(224*c^7*d^2*e + 352*b*c^6*d*e^2 + (41*b^2*c^5 + 84*a*c^6)
*e^3)*x^5 + 128*(896*c^7*d^3 + 4256*b*c^6*d^2*e + 32*(47*b^2*c^5 + 96*a*c^6)*d*e
^2 - 3*(b^3*c^4 - 188*a*b*c^5)*e^3)*x^4 - 224*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*
a^2*b*c^4)*d^2*e + 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^2 - (945*b^7 - 10500*a*b^5*c + 37744*a^2*b^3*c^2 - 42432*a^3*b*c^3)*e^3
+ 16*(14336*b*c^6*d^3 + 224*(69*b^2*c^5 + 140*a*c^6)*d^2*e - 32*(3*b^3*c^4 - 55
6*a*b*c^5)*d*e^2 + (27*b^4*c^3 - 216*a*b^2*c^4 + 560*a^2*c^5)*e^3)*x^3 + 8*(1433
6*(b^2*c^5 + 2*a*c^6)*d^3 - 224*(b^3*c^4 - 228*a*b*c^5)*d^2*e + 32*(7*b^4*c^3 -
60*a*b^2*c^4 + 192*a^2*c^5)*d*e^2 - (63*b^5*c^2 - 568*a*b^3*c^3 + 1392*a^2*b*c^4
)*e^3)*x^2 + 2*(114688*a*b*c^5*d^3 + 224*(5*b^4*c^3 - 48*a*b^2*c^4 + 240*a^2*c^5
)*d^2*e - 32*(35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*d*e^2 + (315*b^6*c - 3
164*a*b^4*c^2 + 9552*a^2*b^2*c^3 - 6720*a^3*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a)*s
qrt(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*e -
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^2 + (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^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^(1
1/2), 1/573440*(2*(71680*c^7*e^3*x^7 + 114688*a^2*c^5*d^3 + 5120*(48*c^7*d*e^2 +
25*b*c^6*e^3)*x^6 + 1280*(224*c^7*d^2*e + 352*b*c^6*d*e^2 + (41*b^2*c^5 + 84*a*
c^6)*e^3)*x^5 + 128*(896*c^7*d^3 + 4256*b*c^6*d^2*e + 32*(47*b^2*c^5 + 96*a*c^6)
*d*e^2 - 3*(b^3*c^4 - 188*a*b*c^5)*e^3)*x^4 - 224*(15*b^5*c^2 - 160*a*b^3*c^3 +
528*a^2*b*c^4)*d^2*e + 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^2 - (945*b^7 - 10500*a*b^5*c + 37744*a^2*b^3*c^2 - 42432*a^3*b*c^3)
*e^3 + 16*(14336*b*c^6*d^3 + 224*(69*b^2*c^5 + 140*a*c^6)*d^2*e - 32*(3*b^3*c^4
- 556*a*b*c^5)*d*e^2 + (27*b^4*c^3 - 216*a*b^2*c^4 + 560*a^2*c^5)*e^3)*x^3 + 8*(
14336*(b^2*c^5 + 2*a*c^6)*d^3 - 224*(b^3*c^4 - 228*a*b*c^5)*d^2*e + 32*(7*b^4*c^
3 - 60*a*b^2*c^4 + 192*a^2*c^5)*d*e^2 - (63*b^5*c^2 - 568*a*b^3*c^3 + 1392*a^2*b
*c^4)*e^3)*x^2 + 2*(114688*a*b*c^5*d^3 + 224*(5*b^4*c^3 - 48*a*b^2*c^4 + 240*a^2
*c^5)*d^2*e - 32*(35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*d*e^2 + (315*b^6*c
- 3164*a*b^4*c^2 + 9552*a^2*b^2*c^3 - 6720*a^3*c^4)*e^3)*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
*e - 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^2 + (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^3)*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. \[ \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(3/2),x)
[Out]
Integral((b + 2*c*x)*(d + e*x)**3*(a + b*x + c*x**2)**(3/2), x)
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GIAC/XCAS [A] time = 0.288535, 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)^(3/2)*(2*c*x + b)*(e*x + d)^3,x, algorithm="giac")
[Out]
Done