3.1565 \(\int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=446 \[ -\frac{3 e \left (b^2-4 a c\right )^4 \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{131072 c^{13/2}}+\frac{3 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{65536 c^6}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{2560 c^4}+\frac{\left (a+b x+c x^2\right )^{7/2} \left (14 c e x \left (-4 c e (9 a e+2 b d)+11 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (160 a e+17 b d)+4 b c e^2 (97 a e+90 b d)-99 b^3 e^3+128 c^3 d^3\right )}{6720 c^3}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{30 c} \]

[Out]

(3*(b^2 - 4*a*c)^3*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x
)*Sqrt[a + b*x + c*x^2])/(65536*c^6) - ((b^2 - 4*a*c)^2*e*(40*c^2*d^2 + 11*b^2*e
^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8192*c^5) + ((b
^2 - 4*a*c)*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a +
b*x + c*x^2)^(5/2))/(2560*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(7
/2))/(30*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(7/2))/5 + ((128*c^3*d^3 - 99*b^3*e
^3 + 4*b*c*e^2*(90*b*d + 97*a*e) - 8*c^2*d*e*(17*b*d + 160*a*e) + 14*c*e*(8*c^2*
d^2 + 11*b^2*e^2 - 4*c*e*(2*b*d + 9*a*e))*x)*(a + b*x + c*x^2)^(7/2))/(6720*c^3)
- (3*(b^2 - 4*a*c)^4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*ArcTanh
[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(131072*c^(13/2))

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Rubi [A]  time = 1.36157, antiderivative size = 446, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{3 e \left (b^2-4 a c\right )^4 \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{131072 c^{13/2}}+\frac{3 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{65536 c^6}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{2560 c^4}+\frac{\left (a+b x+c x^2\right )^{7/2} \left (14 c e x \left (-4 c e (9 a e+2 b d)+11 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (160 a e+17 b d)+4 b c e^2 (97 a e+90 b d)-99 b^3 e^3+128 c^3 d^3\right )}{6720 c^3}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{30 c} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(3*(b^2 - 4*a*c)^3*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x
)*Sqrt[a + b*x + c*x^2])/(65536*c^6) - ((b^2 - 4*a*c)^2*e*(40*c^2*d^2 + 11*b^2*e
^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8192*c^5) + ((b
^2 - 4*a*c)*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a +
b*x + c*x^2)^(5/2))/(2560*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(7
/2))/(30*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(7/2))/5 + ((128*c^3*d^3 - 99*b^3*e
^3 + 4*b*c*e^2*(90*b*d + 97*a*e) - 8*c^2*d*e*(17*b*d + 160*a*e) + 14*c*e*(8*c^2*
d^2 + 11*b^2*e^2 - 4*c*e*(2*b*d + 9*a*e))*x)*(a + b*x + c*x^2)^(7/2))/(6720*c^3)
- (3*(b^2 - 4*a*c)^4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*ArcTanh
[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(131072*c^(13/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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Mathematica [B]  time = 2.62503, size = 927, normalized size = 2.08 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (3465 e^3 b^9-210 c e^2 (60 d+11 e x) b^8+168 c^2 e \left (75 d^2+50 e x d+11 e^2 x^2\right ) b^7-48 c^3 e x \left (175 d^2+140 e x d+33 e^2 x^2\right ) b^6+64 c^4 e x^2 \left (105 d^2+90 e x d+22 e^2 x^2\right ) b^5-640 c^5 e x^3 \left (9 d^2+8 e x d+2 e^2 x^2\right ) b^4+5120 c^6 x^3 \left (384 d^3+897 e x d^2+734 e^2 x^2 d+207 e^3 x^3\right ) b^3+2048 c^7 x^4 \left (2880 d^3+7125 e x d^2+6060 e^2 x^2 d+1757 e^3 x^3\right ) b^2+8192 c^8 x^5 \left (720 d^3+1845 e x d^2+1610 e^2 x^2 d+476 e^3 x^3\right ) b+16384 c^9 x^6 \left (120 d^3+315 e x d^2+280 e^2 x^2 d+84 e^3 x^3\right )-1280 a^4 c^4 e^2 (2 c (512 d+63 e x)-449 b e)+1280 a^3 c^3 \left (4 \left (384 d^3+315 e x d^2+128 e^2 x^2 d+21 e^3 x^3\right ) c^3-2 b e \left (837 d^2+374 e x d+65 e^2 x^2\right ) c^2+62 b^2 e^2 (27 d+4 e x) c-537 b^3 e^3\right )+96 a^2 c^2 \left (3003 e^3 b^5-10 c e^2 (1022 d+167 e x) b^4+20 c^2 e \left (511 d^2+282 e x d+55 e^2 x^2\right ) b^3-40 c^3 e x \left (141 d^2+92 e x d+19 e^2 x^2\right ) b^2+160 c^4 x \left (384 d^3+663 e x d^2+454 e^2 x^2 d+114 e^3 x^3\right ) b+64 c^5 x^2 \left (960 d^3+2065 e x d^2+1600 e^2 x^2 d+434 e^3 x^3\right )\right )+16 a c \left (-3255 e^3 b^7+42 c e^2 (275 d+48 e x) b^6-6 c^2 e \left (1925 d^2+1190 e x d+249 e^2 x^2\right ) b^5+20 c^3 e x \left (357 d^2+264 e x d+59 e^2 x^2\right ) b^4-160 c^4 e x^2 \left (33 d^2+26 e x d+6 e^2 x^2\right ) b^3+960 c^5 x^2 \left (384 d^3+815 e x d^2+628 e^2 x^2 d+170 e^3 x^3\right ) b^2+256 c^6 x^3 \left (2880 d^3+6765 e x d^2+5550 e^2 x^2 d+1567 e^3 x^3\right ) b+512 c^7 x^4 \left (720 d^3+1785 e x d^2+1520 e^2 x^2 d+441 e^3 x^3\right )\right )\right )-315 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) \log \left (b+2 c x+2 \sqrt{c} \sqrt{a+x (b+c x)}\right )}{13762560 c^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x)*(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^9*e^3 - 210*b^8*c*e^2*(60*d + 11*e*x) -
640*b^4*c^5*e*x^3*(9*d^2 + 8*d*e*x + 2*e^2*x^2) + 168*b^7*c^2*e*(75*d^2 + 50*d*
e*x + 11*e^2*x^2) + 64*b^5*c^4*e*x^2*(105*d^2 + 90*d*e*x + 22*e^2*x^2) - 48*b^6*
c^3*e*x*(175*d^2 + 140*d*e*x + 33*e^2*x^2) + 16384*c^9*x^6*(120*d^3 + 315*d^2*e*
x + 280*d*e^2*x^2 + 84*e^3*x^3) + 5120*b^3*c^6*x^3*(384*d^3 + 897*d^2*e*x + 734*
d*e^2*x^2 + 207*e^3*x^3) + 8192*b*c^8*x^5*(720*d^3 + 1845*d^2*e*x + 1610*d*e^2*x
^2 + 476*e^3*x^3) + 2048*b^2*c^7*x^4*(2880*d^3 + 7125*d^2*e*x + 6060*d*e^2*x^2 +
1757*e^3*x^3) - 1280*a^4*c^4*e^2*(-449*b*e + 2*c*(512*d + 63*e*x)) + 1280*a^3*c
^3*(-537*b^3*e^3 + 62*b^2*c*e^2*(27*d + 4*e*x) - 2*b*c^2*e*(837*d^2 + 374*d*e*x
+ 65*e^2*x^2) + 4*c^3*(384*d^3 + 315*d^2*e*x + 128*d*e^2*x^2 + 21*e^3*x^3)) + 96
*a^2*c^2*(3003*b^5*e^3 - 10*b^4*c*e^2*(1022*d + 167*e*x) - 40*b^2*c^3*e*x*(141*d
^2 + 92*d*e*x + 19*e^2*x^2) + 20*b^3*c^2*e*(511*d^2 + 282*d*e*x + 55*e^2*x^2) +
160*b*c^4*x*(384*d^3 + 663*d^2*e*x + 454*d*e^2*x^2 + 114*e^3*x^3) + 64*c^5*x^2*(
960*d^3 + 2065*d^2*e*x + 1600*d*e^2*x^2 + 434*e^3*x^3)) + 16*a*c*(-3255*b^7*e^3
+ 42*b^6*c*e^2*(275*d + 48*e*x) - 160*b^3*c^4*e*x^2*(33*d^2 + 26*d*e*x + 6*e^2*x
^2) + 20*b^4*c^3*e*x*(357*d^2 + 264*d*e*x + 59*e^2*x^2) - 6*b^5*c^2*e*(1925*d^2
+ 1190*d*e*x + 249*e^2*x^2) + 960*b^2*c^5*x^2*(384*d^3 + 815*d^2*e*x + 628*d*e^2
*x^2 + 170*e^3*x^3) + 512*c^7*x^4*(720*d^3 + 1785*d^2*e*x + 1520*d*e^2*x^2 + 441
*e^3*x^3) + 256*b*c^6*x^3*(2880*d^3 + 6765*d^2*e*x + 5550*d*e^2*x^2 + 1567*e^3*x
^3))) - 315*(b^2 - 4*a*c)^4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*L
og[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(13762560*c^(13/2))

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Maple [B]  time = 0.027, size = 2387, normalized size = 5.4 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-5/32*a^2*(c*x^2+b*x+a)^(3/2)*x*d^2*e+1/5*e^3*x^3*(c*x^2+b*x+a)^(7/2)-33/2240*b^
3/c^3*(c*x^2+b*x+a)^(7/2)*e^3+11/2560*b^5/c^4*(c*x^2+b*x+a)^(5/2)*e^3-11/8192*b^
7/c^5*(c*x^2+b*x+a)^(3/2)*e^3+33/65536*b^9/c^6*(c*x^2+b*x+a)^(1/2)*e^3-33/131072
*b^10/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^3+3/4*x*(c*x^2+b*x+
a)^(7/2)*d^2*e+2/3*x^2*(c*x^2+b*x+a)^(7/2)*d*e^2+3/128*e^3/c^(3/2)*a^5*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+15/64*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a^3*d^2*e-45/512*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*a^2*d^2*e+15/1024*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))*a*d^2*e-5/128*b^4/c^3*(c*x^2+b*x+a)^(3/2)*a*d*e^2-3/80*b^2/c^2*a*(c*x^2+b*x+
a)^(5/2)*x*e^3-1/32*b^3/c^2*(c*x^2+b*x+a)^(5/2)*x*d*e^2-15/4096*b^7/c^4*(c*x^2+b
*x+a)^(1/2)*x*d*e^2+15/128*b^2/c^2*a^3*(c*x^2+b*x+a)^(1/2)*d*e^2+45/2048*b^6/c^4
*(c*x^2+b*x+a)^(1/2)*a*d*e^2+23/1024*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x*a*e^3+5/64*b^
2/c^2*a^2*(c*x^2+b*x+a)^(3/2)*d*e^2+45/512*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a^2*d*e^2+15/64*b/c^(3/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*d*e^2-1/12*b/c*x*(c*x^2+b*x+a)^(7/2)*d*e^2+5/512*b^5/c^3*(c*x^2+b*x
+a)^(3/2)*x*d*e^2+27/512*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a^2*e^3-51/4096*b^6/c^4*(
c*x^2+b*x+a)^(1/2)*x*a*e^3+1/32*b^2/c*(c*x^2+b*x+a)^(5/2)*x*d^2*e-15/128*a^3/c*(
c*x^2+b*x+a)^(1/2)*b*d^2*e-45/2048*b^5/c^3*(c*x^2+b*x+a)^(1/2)*a*d^2*e-5/64*a^2/
c*(c*x^2+b*x+a)^(3/2)*b*d^2*e+5/128*b^3/c^2*(c*x^2+b*x+a)^(3/2)*a*d^2*e+15/4096*
b^6/c^3*(c*x^2+b*x+a)^(1/2)*x*d^2*e-5/512*b^4/c^2*(c*x^2+b*x+a)^(3/2)*x*d^2*e+45
/512*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a^2*d^2*e-1/16*a/c*(c*x^2+b*x+a)^(5/2)*b*d^2*e-
45/512*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a^2*d*e^2+1/16*b^2/c^2*a*(c*x^2+b*x+a)^(5/2)*
d*e^2-13/256*b^2/c^2*a^2*(c*x^2+b*x+a)^(3/2)*x*e^3-21/256*b^2/c^2*a^3*(c*x^2+b*x
+a)^(1/2)*x*e^3+2/7*(c*x^2+b*x+a)^(7/2)*d^3+1/8*b/c*a*(c*x^2+b*x+a)^(5/2)*x*d*e^
2+5/32*b/c*a^2*(c*x^2+b*x+a)^(3/2)*x*d*e^2+15/64*b/c*a^3*(c*x^2+b*x+a)^(1/2)*x*d
*e^2-45/256*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2*d*e^2+45/1024*b^5/c^3*(c*x^2+b*x+a
)^(1/2)*x*a*d*e^2-5/64*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x*a*d*e^2+5/64*b^2/c*(c*x^2+b
*x+a)^(3/2)*x*a*d^2*e+45/256*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a^2*d^2*e-45/1024*b^4/c
^2*(c*x^2+b*x+a)^(1/2)*x*a*d^2*e-15/16384*b^8/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*d^2*e-15/64*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))*a^3*d*e^2-15/1024*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*
a*d*e^2-3/56*b/c*(c*x^2+b*x+a)^(7/2)*d^2*e+1/64*b^3/c^2*(c*x^2+b*x+a)^(5/2)*d^2*
e-5/1024*b^5/c^3*(c*x^2+b*x+a)^(3/2)*d^2*e+15/8192*b^7/c^4*(c*x^2+b*x+a)^(1/2)*d
^2*e-15/64*a^3*(c*x^2+b*x+a)^(1/2)*x*d^2*e-1/8*a*(c*x^2+b*x+a)^(5/2)*x*d^2*e+15/
16384*b^9/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2-4/21*a/c*(c
*x^2+b*x+a)^(7/2)*d*e^2-1/30*x^2*(c*x^2+b*x+a)^(7/2)/c*b*e^3+11/480*b^2/c^2*x*(c
*x^2+b*x+a)^(7/2)*e^3+11/1280*b^4/c^3*(c*x^2+b*x+a)^(5/2)*x*e^3-15/8192*b^8/c^5*
(c*x^2+b*x+a)^(1/2)*d*e^2-11/4096*b^6/c^4*(c*x^2+b*x+a)^(3/2)*x*e^3+23/2048*b^5/
c^4*(c*x^2+b*x+a)^(3/2)*a*e^3+33/32768*b^8/c^5*(c*x^2+b*x+a)^(1/2)*x*e^3+27/1024
*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a^2*e^3-3/160*b^3/c^3*a*(c*x^2+b*x+a)^(5/2)*e^3-13/
512*b^3/c^3*a^2*(c*x^2+b*x+a)^(3/2)*e^3-21/512*b^3/c^3*a^3*(c*x^2+b*x+a)^(1/2)*e
^3-51/8192*b^7/c^5*(c*x^2+b*x+a)^(1/2)*a*e^3+135/32768*b^8/c^(11/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e^3+75/1024*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*a^3*e^3-105/4096*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))*a^2*e^3-45/512*b^2/c^(5/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*e^3+3/56*b^2/c^2*(c*x^2+b*x+a)^(7/2)*d*e^2-1/64*b^4/c^3*(c*x^2+b*x+a)^
(5/2)*d*e^2+5/1024*b^6/c^4*(c*x^2+b*x+a)^(3/2)*d*e^2+97/1680*a/c^2*(c*x^2+b*x+a)
^(7/2)*b*e^3+3/256*e^3/c^2*a^4*(c*x^2+b*x+a)^(1/2)*b-3/40*e^3/c*a*x*(c*x^2+b*x+a
)^(7/2)-15/64*a^4/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e+1/80
*e^3/c*a^2*(c*x^2+b*x+a)^(5/2)*x+1/160*e^3/c^2*a^2*(c*x^2+b*x+a)^(5/2)*b+1/64*e^
3/c*a^3*(c*x^2+b*x+a)^(3/2)*x+1/128*e^3/c^2*a^3*(c*x^2+b*x+a)^(3/2)*b+3/128*e^3/
c*a^4*(c*x^2+b*x+a)^(1/2)*x

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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)*(2*c*x + b)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.800521, 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)*(2*c*x + b)*(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/27525120*(4*(1376256*c^9*e^3*x^9 + 1966080*a^3*c^6*d^3 + 229376*(20*c^9*d*e^2
+ 17*b*c^8*e^3)*x^8 + 14336*(360*c^9*d^2*e + 920*b*c^8*d*e^2 + (251*b^2*c^7 + 2
52*a*c^8)*e^3)*x^7 + 1024*(1920*c^9*d^3 + 14760*b*c^8*d^2*e + 40*(303*b^2*c^7 +
304*a*c^8)*d*e^2 + (1035*b^3*c^6 + 6268*a*b*c^7)*e^3)*x^6 + 256*(23040*b*c^8*d^3
+ 120*(475*b^2*c^7 + 476*a*c^8)*d^2*e + 40*(367*b^3*c^6 + 2220*a*b*c^7)*d*e^2 -
(5*b^4*c^5 - 10200*a*b^2*c^6 - 10416*a^2*c^7)*e^3)*x^5 + 128*(46080*(b^2*c^7 +
a*c^8)*d^3 + 120*(299*b^3*c^6 + 1804*a*b*c^7)*d^2*e - 40*(b^4*c^5 - 1884*a*b^2*c
^6 - 1920*a^2*c^7)*d*e^2 + (11*b^5*c^4 - 120*a*b^3*c^5 + 13680*a^2*b*c^6)*e^3)*x
^4 + 120*(105*b^7*c^2 - 1540*a*b^5*c^3 + 8176*a^2*b^3*c^4 - 17856*a^3*b*c^5)*d^2
*e - 40*(315*b^8*c - 4620*a*b^6*c^2 + 24528*a^2*b^4*c^3 - 53568*a^3*b^2*c^4 + 32
768*a^4*c^5)*d*e^2 + (3465*b^9 - 52080*a*b^7*c + 288288*a^2*b^5*c^2 - 687360*a^3
*b^3*c^3 + 574720*a^4*b*c^4)*e^3 + 16*(122880*(b^3*c^6 + 6*a*b*c^7)*d^3 - 120*(3
*b^4*c^5 - 6520*a*b^2*c^6 - 6608*a^2*c^7)*d^2*e + 40*(9*b^5*c^4 - 104*a*b^3*c^5
+ 10896*a^2*b*c^6)*d*e^2 - (99*b^6*c^3 - 1180*a*b^4*c^4 + 4560*a^2*b^2*c^5 - 672
0*a^3*c^6)*e^3)*x^3 + 8*(737280*(a*b^2*c^6 + a^2*c^7)*d^3 + 120*(7*b^5*c^4 - 88*
a*b^3*c^5 + 10608*a^2*b*c^6)*d^2*e - 40*(21*b^6*c^3 - 264*a*b^4*c^4 + 1104*a^2*b
^2*c^5 - 2048*a^3*c^6)*d*e^2 + (231*b^7*c^2 - 2988*a*b^5*c^3 + 13200*a^2*b^3*c^4
- 20800*a^3*b*c^5)*e^3)*x^2 + 2*(2949120*a^2*b*c^6*d^3 - 120*(35*b^6*c^3 - 476*
a*b^4*c^4 + 2256*a^2*b^2*c^5 - 6720*a^3*c^6)*d^2*e + 40*(105*b^7*c^2 - 1428*a*b^
5*c^3 + 6768*a^2*b^3*c^4 - 11968*a^3*b*c^5)*d*e^2 - (1155*b^8*c - 16128*a*b^6*c^
2 + 80160*a^2*b^4*c^3 - 158720*a^3*b^2*c^4 + 80640*a^4*c^5)*e^3)*x)*sqrt(c*x^2 +
b*x + a)*sqrt(c) - 315*(40*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^3*b
^2*c^5 + 256*a^4*c^6)*d^2*e - 40*(b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 256*a^
3*b^3*c^4 + 256*a^4*b*c^5)*d*e^2 + (11*b^10 - 180*a*b^8*c + 1120*a^2*b^6*c^2 - 3
200*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e^3)*log(-4*(2*c^2*x + b*c)*s
qrt(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
/13762560*(2*(1376256*c^9*e^3*x^9 + 1966080*a^3*c^6*d^3 + 229376*(20*c^9*d*e^2 +
17*b*c^8*e^3)*x^8 + 14336*(360*c^9*d^2*e + 920*b*c^8*d*e^2 + (251*b^2*c^7 + 252
*a*c^8)*e^3)*x^7 + 1024*(1920*c^9*d^3 + 14760*b*c^8*d^2*e + 40*(303*b^2*c^7 + 30
4*a*c^8)*d*e^2 + (1035*b^3*c^6 + 6268*a*b*c^7)*e^3)*x^6 + 256*(23040*b*c^8*d^3 +
120*(475*b^2*c^7 + 476*a*c^8)*d^2*e + 40*(367*b^3*c^6 + 2220*a*b*c^7)*d*e^2 - (
5*b^4*c^5 - 10200*a*b^2*c^6 - 10416*a^2*c^7)*e^3)*x^5 + 128*(46080*(b^2*c^7 + a*
c^8)*d^3 + 120*(299*b^3*c^6 + 1804*a*b*c^7)*d^2*e - 40*(b^4*c^5 - 1884*a*b^2*c^6
- 1920*a^2*c^7)*d*e^2 + (11*b^5*c^4 - 120*a*b^3*c^5 + 13680*a^2*b*c^6)*e^3)*x^4
+ 120*(105*b^7*c^2 - 1540*a*b^5*c^3 + 8176*a^2*b^3*c^4 - 17856*a^3*b*c^5)*d^2*e
- 40*(315*b^8*c - 4620*a*b^6*c^2 + 24528*a^2*b^4*c^3 - 53568*a^3*b^2*c^4 + 3276
8*a^4*c^5)*d*e^2 + (3465*b^9 - 52080*a*b^7*c + 288288*a^2*b^5*c^2 - 687360*a^3*b
^3*c^3 + 574720*a^4*b*c^4)*e^3 + 16*(122880*(b^3*c^6 + 6*a*b*c^7)*d^3 - 120*(3*b
^4*c^5 - 6520*a*b^2*c^6 - 6608*a^2*c^7)*d^2*e + 40*(9*b^5*c^4 - 104*a*b^3*c^5 +
10896*a^2*b*c^6)*d*e^2 - (99*b^6*c^3 - 1180*a*b^4*c^4 + 4560*a^2*b^2*c^5 - 6720*
a^3*c^6)*e^3)*x^3 + 8*(737280*(a*b^2*c^6 + a^2*c^7)*d^3 + 120*(7*b^5*c^4 - 88*a*
b^3*c^5 + 10608*a^2*b*c^6)*d^2*e - 40*(21*b^6*c^3 - 264*a*b^4*c^4 + 1104*a^2*b^2
*c^5 - 2048*a^3*c^6)*d*e^2 + (231*b^7*c^2 - 2988*a*b^5*c^3 + 13200*a^2*b^3*c^4 -
20800*a^3*b*c^5)*e^3)*x^2 + 2*(2949120*a^2*b*c^6*d^3 - 120*(35*b^6*c^3 - 476*a*
b^4*c^4 + 2256*a^2*b^2*c^5 - 6720*a^3*c^6)*d^2*e + 40*(105*b^7*c^2 - 1428*a*b^5*
c^3 + 6768*a^2*b^3*c^4 - 11968*a^3*b*c^5)*d*e^2 - (1155*b^8*c - 16128*a*b^6*c^2
+ 80160*a^2*b^4*c^3 - 158720*a^3*b^2*c^4 + 80640*a^4*c^5)*e^3)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(-c) - 315*(40*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^3*b^
2*c^5 + 256*a^4*c^6)*d^2*e - 40*(b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 256*a^3
*b^3*c^4 + 256*a^4*b*c^5)*d*e^2 + (11*b^10 - 180*a*b^8*c + 1120*a^2*b^6*c^2 - 32
00*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e^3)*arctan(1/2*(2*c*x + b)*sq
rt(-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. \[ \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**3*(a + b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.305059, 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)*(2*c*x + b)*(e*x + d)^3,x, algorithm="giac")

[Out]

Done