3.1182 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=565 \[ \frac{\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{840 d^3 (d+e x)^5 (c d-b e)^3}-\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{1024 d^5 (d+e x)^2 (c d-b e)^5}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{384 d^4 (d+e x)^4 (c d-b e)^4}+\frac{b^4 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2048 d^{11/2} (c d-b e)^{11/2}}-\frac{\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{84 d^2 (d+e x)^6 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)} \]

[Out]

-(b^2*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b
^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(1024*d^5
*(c*d - b*e)^5*(d + e*x)^2) + ((48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*
e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*(b*
x + c*x^2)^(3/2))/(384*d^4*(c*d - b*e)^4*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^
2)^(5/2))/(7*d*(c*d - b*e)*(d + e*x)^7) - ((9*A*e*(2*c*d - b*e) - B*d*(4*c*d + 5
*b*e))*(b*x + c*x^2)^(5/2))/(84*d^2*(c*d - b*e)^2*(d + e*x)^6) + ((B*d*(8*c^2*d^
2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*(b*
x + c*x^2)^(5/2))/(840*d^3*(c*d - b*e)^3*(d + e*x)^5) + (b^4*(48*A*c^3*d^3 - 24*
b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e
))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])
])/(2048*d^(11/2)*(c*d - b*e)^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 2.21598, antiderivative size = 565, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{840 d^3 (d+e x)^5 (c d-b e)^3}-\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{1024 d^5 (d+e x)^2 (c d-b e)^5}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{384 d^4 (d+e x)^4 (c d-b e)^4}+\frac{b^4 \left (b^3 \left (-e^2\right ) (9 A e+5 B d)+2 b^2 c d e (21 A e+10 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2048 d^{11/2} (c d-b e)^{11/2}}-\frac{\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{84 d^2 (d+e x)^6 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

-(b^2*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b
^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(1024*d^5
*(c*d - b*e)^5*(d + e*x)^2) + ((48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*
e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*(b*
x + c*x^2)^(3/2))/(384*d^4*(c*d - b*e)^4*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^
2)^(5/2))/(7*d*(c*d - b*e)*(d + e*x)^7) - ((9*A*e*(2*c*d - b*e) - B*d*(4*c*d + 5
*b*e))*(b*x + c*x^2)^(5/2))/(84*d^2*(c*d - b*e)^2*(d + e*x)^6) + ((B*d*(8*c^2*d^
2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*(b*
x + c*x^2)^(5/2))/(840*d^3*(c*d - b*e)^3*(d + e*x)^5) + (b^4*(48*A*c^3*d^3 - 24*
b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e
))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])
])/(2048*d^(11/2)*(c*d - b*e)^(11/2))

_______________________________________________________________________________________

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((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 6.77445, size = 1017, normalized size = 1.8 \[ \frac{(x (b+c x))^{3/2} \left (\frac{d (B d-A e) \sqrt{x} (c d-b e)}{7 e^4 (d+e x)^7}-\frac{\left (44 B c d^2-29 b B e d-30 A c e d+15 A b e^2\right ) \sqrt{x}}{84 e^4 (d+e x)^6}+\frac{\left (568 B c^2 d^3-204 A c^2 e d^2-750 b B c e d^2+185 b^2 B e^2 d+204 A b c e^2 d-3 A b^2 e^3\right ) \sqrt{x}}{840 d e^4 (d+e x)^5 (c d-b e)}-\frac{\left (2176 B c^3 d^4-48 A c^3 e d^3-4328 b B c^2 e d^3+72 A b c^2 e^2 d^2+2140 b^2 B c e^2 d^2-15 b^3 B e^3 d+30 A b^2 c e^3 d-27 A b^3 e^4\right ) \sqrt{x}}{6720 d^2 e^4 (d+e x)^4 (c d-b e)^2}+\frac{\left (128 B c^4 d^5+96 A c^4 e d^4-368 b B c^3 e d^4-192 A b c^3 e^2 d^3+288 b^2 B c^2 e^2 d^3-36 A b^2 c^2 e^3 d^2+50 b^3 B c e^3 d^2-35 b^4 B e^4 d+132 A b^3 c e^4 d-63 A b^4 e^5\right ) \sqrt{x}}{13440 d^3 e^4 (d+e x)^3 (c d-b e)^3}+\frac{\left (512 B c^5 d^6+384 A c^5 e d^5-1728 b B c^4 e d^5-960 A b c^4 e^2 d^4+1696 b^2 B c^3 e^2 d^4+96 A b^2 c^3 e^3 d^3+80 b^3 B c^2 e^3 d^3+816 A b^3 c^2 e^4 d^2-420 b^4 B c e^4 d^2+175 b^5 B e^5 d-966 A b^4 c e^5 d+315 A b^5 e^6\right ) \sqrt{x}}{53760 d^4 e^4 (d+e x)^2 (c d-b e)^4}+\frac{\left (1024 B c^6 d^7+768 A c^6 e d^6-3968 b B c^5 e d^6-2304 A b c^5 e^2 d^5+4864 b^2 B c^4 e^2 d^5+960 A b^2 c^4 e^3 d^4-800 b^3 B c^3 e^3 d^4+1920 A b^3 c^3 e^4 d^3-1400 b^4 B c^2 e^4 d^3-5124 A b^4 c^2 e^5 d^2+1750 b^5 B c e^5 d^2-525 b^6 B e^6 d+3780 A b^5 c e^6 d-945 A b^6 e^7\right ) \sqrt{x}}{107520 d^5 e^4 (d+e x) (c d-b e)^5}\right )}{x^{3/2} (b+c x)}-\frac{b^4 \left (9 A e^3 b^3+5 B d e^2 b^3-42 A c d e^2 b^2-20 B c d^2 e b^2+24 B c^2 d^3 b+72 A c^2 d^2 e b-48 A c^3 d^3\right ) (x (b+c x))^{3/2} \tan ^{-1}\left (\frac{\sqrt{b e-c d} \sqrt{x}}{\sqrt{d} \sqrt{b+c x}}\right )}{1024 d^{11/2} (c d-b e)^5 \sqrt{b e-c d} x^{3/2} (b+c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

((x*(b + c*x))^(3/2)*((d*(B*d - A*e)*(c*d - b*e)*Sqrt[x])/(7*e^4*(d + e*x)^7) -
((44*B*c*d^2 - 29*b*B*d*e - 30*A*c*d*e + 15*A*b*e^2)*Sqrt[x])/(84*e^4*(d + e*x)^
6) + ((568*B*c^2*d^3 - 750*b*B*c*d^2*e - 204*A*c^2*d^2*e + 185*b^2*B*d*e^2 + 204
*A*b*c*d*e^2 - 3*A*b^2*e^3)*Sqrt[x])/(840*d*e^4*(c*d - b*e)*(d + e*x)^5) - ((217
6*B*c^3*d^4 - 4328*b*B*c^2*d^3*e - 48*A*c^3*d^3*e + 2140*b^2*B*c*d^2*e^2 + 72*A*
b*c^2*d^2*e^2 - 15*b^3*B*d*e^3 + 30*A*b^2*c*d*e^3 - 27*A*b^3*e^4)*Sqrt[x])/(6720
*d^2*e^4*(c*d - b*e)^2*(d + e*x)^4) + ((128*B*c^4*d^5 - 368*b*B*c^3*d^4*e + 96*A
*c^4*d^4*e + 288*b^2*B*c^2*d^3*e^2 - 192*A*b*c^3*d^3*e^2 + 50*b^3*B*c*d^2*e^3 -
36*A*b^2*c^2*d^2*e^3 - 35*b^4*B*d*e^4 + 132*A*b^3*c*d*e^4 - 63*A*b^4*e^5)*Sqrt[x
])/(13440*d^3*e^4*(c*d - b*e)^3*(d + e*x)^3) + ((512*B*c^5*d^6 - 1728*b*B*c^4*d^
5*e + 384*A*c^5*d^5*e + 1696*b^2*B*c^3*d^4*e^2 - 960*A*b*c^4*d^4*e^2 + 80*b^3*B*
c^2*d^3*e^3 + 96*A*b^2*c^3*d^3*e^3 - 420*b^4*B*c*d^2*e^4 + 816*A*b^3*c^2*d^2*e^4
+ 175*b^5*B*d*e^5 - 966*A*b^4*c*d*e^5 + 315*A*b^5*e^6)*Sqrt[x])/(53760*d^4*e^4*
(c*d - b*e)^4*(d + e*x)^2) + ((1024*B*c^6*d^7 - 3968*b*B*c^5*d^6*e + 768*A*c^6*d
^6*e + 4864*b^2*B*c^4*d^5*e^2 - 2304*A*b*c^5*d^5*e^2 - 800*b^3*B*c^3*d^4*e^3 + 9
60*A*b^2*c^4*d^4*e^3 - 1400*b^4*B*c^2*d^3*e^4 + 1920*A*b^3*c^3*d^3*e^4 + 1750*b^
5*B*c*d^2*e^5 - 5124*A*b^4*c^2*d^2*e^5 - 525*b^6*B*d*e^6 + 3780*A*b^5*c*d*e^6 -
945*A*b^6*e^7)*Sqrt[x])/(107520*d^5*e^4*(c*d - b*e)^5*(d + e*x))))/(x^(3/2)*(b +
c*x)) - (b^4*(24*b*B*c^2*d^3 - 48*A*c^3*d^3 - 20*b^2*B*c*d^2*e + 72*A*b*c^2*d^2
*e + 5*b^3*B*d*e^2 - 42*A*b^2*c*d*e^2 + 9*A*b^3*e^3)*(x*(b + c*x))^(3/2)*ArcTan[
(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(1024*d^(11/2)*(c*d - b*e
)^5*Sqrt[-(c*d) + b*e]*x^(3/2)*(b + c*x)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.1, size = 37630, normalized size = 66.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^8,x)

[Out]

result too large to display

_______________________________________________________________________________________

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)^(3/2)*(B*x + A)/(e*x + d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.351949, 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)^(3/2)*(B*x + A)/(e*x + d)^8,x, algorithm="fricas")

[Out]

[1/215040*(2*(945*A*b^6*d^6*e^3 + 2520*(B*b^4*c^2 - 2*A*b^3*c^3)*d^9 - 420*(5*B*
b^5*c - 18*A*b^4*c^2)*d^8*e + 105*(5*B*b^6 - 42*A*b^5*c)*d^7*e^2 + (1024*B*c^6*d
^7*e^2 - 945*A*b^6*e^9 - 128*(31*B*b*c^5 - 6*A*c^6)*d^6*e^3 + 256*(19*B*b^2*c^4
- 9*A*b*c^5)*d^5*e^4 - 160*(5*B*b^3*c^3 - 6*A*b^2*c^4)*d^4*e^5 - 40*(35*B*b^4*c^
2 - 48*A*b^3*c^3)*d^3*e^6 + 14*(125*B*b^5*c - 366*A*b^4*c^2)*d^2*e^7 - 105*(5*B*
b^6 - 36*A*b^5*c)*d*e^8)*x^6 + 2*(3584*B*c^6*d^8*e - 3150*A*b^6*d*e^8 - 64*(221*
B*b*c^5 - 42*A*c^6)*d^7*e^2 + 32*(563*B*b^2*c^4 - 258*A*b*c^5)*d^6*e^3 - 16*(251
*B*b^3*c^3 - 246*A*b^2*c^4)*d^5*e^4 - 20*(235*B*b^4*c^2 - 324*A*b^3*c^3)*d^4*e^5
+ (5845*B*b^5*c - 17154*A*b^4*c^2)*d^3*e^6 - 7*(250*B*b^6 - 1803*A*b^5*c)*d^2*e
^7)*x^5 + (21504*B*c^6*d^9 - 17829*A*b^6*d^2*e^7 - 896*(97*B*b*c^5 - 18*A*c^6)*d
^8*e + 64*(1819*B*b^2*c^4 - 798*A*b*c^5)*d^7*e^2 - 16*(2207*B*b^3*c^3 - 1782*A*b
^2*c^4)*d^6*e^3 - 8*(3097*B*b^4*c^2 - 4512*A*b^3*c^3)*d^5*e^4 + 8*(4145*B*b^5*c
- 12198*A*b^4*c^2)*d^4*e^5 - (9905*B*b^6 - 71574*A*b^5*c)*d^3*e^6)*x^4 - 4*(6912
*A*b^6*d^3*e^6 - 672*(11*B*b*c^5 + 10*A*c^6)*d^9 + 336*(107*B*b^2*c^4 + 66*A*b*c
^5)*d^8*e - 4*(16787*B*b^3*c^3 + 3738*A*b^2*c^4)*d^7*e^2 + 4*(12862*B*b^4*c^2 -
3177*A*b^3*c^3)*d^6*e^3 - 5*(4285*B*b^5*c - 7566*A*b^4*c^2)*d^5*e^4 + 3*(1280*B*
b^6 - 9271*A*b^5*c)*d^4*e^5)*x^3 - 7*(3597*A*b^6*d^4*e^5 - 192*(B*b^2*c^4 + 30*A
*b*c^5)*d^9 + 16*(109*B*b^3*c^3 + 1686*A*b^2*c^4)*d^8*e - 8*(1019*B*b^4*c^2 + 53
46*A*b^3*c^3)*d^7*e^2 + 10*(599*B*b^5*c + 3822*A*b^4*c^2)*d^6*e^3 - 5*(283*B*b^6
+ 3648*A*b^5*c)*d^5*e^4)*x^2 + 70*(90*A*b^6*d^5*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c
^4)*d^9 + 4*(65*B*b^4*c^2 - 138*A*b^3*c^3)*d^8*e - (205*B*b^5*c - 762*A*b^4*c^2)
*d^7*e^2 + (50*B*b^6 - 429*A*b^5*c)*d^6*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 +
b*x) - 105*(9*A*b^7*d^7*e^3 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^10 - 4*(5*B*b^6*c
- 18*A*b^5*c^2)*d^9*e + (5*B*b^7 - 42*A*b^6*c)*d^8*e^2 + (9*A*b^7*e^10 + 24*(B*b
^5*c^2 - 2*A*b^4*c^3)*d^3*e^7 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^2*e^8 + (5*B*b^7
- 42*A*b^6*c)*d*e^9)*x^7 + 7*(9*A*b^7*d*e^9 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^4*e
^6 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^3*e^7 + (5*B*b^7 - 42*A*b^6*c)*d^2*e^8)*x^6
+ 21*(9*A*b^7*d^2*e^8 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^5*e^5 - 4*(5*B*b^6*c - 18
*A*b^5*c^2)*d^4*e^6 + (5*B*b^7 - 42*A*b^6*c)*d^3*e^7)*x^5 + 35*(9*A*b^7*d^3*e^7
+ 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^6*e^4 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^5*e^5 +
(5*B*b^7 - 42*A*b^6*c)*d^4*e^6)*x^4 + 35*(9*A*b^7*d^4*e^6 + 24*(B*b^5*c^2 - 2*A*
b^4*c^3)*d^7*e^3 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^6*e^4 + (5*B*b^7 - 42*A*b^6*c)
*d^5*e^5)*x^3 + 21*(9*A*b^7*d^5*e^5 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^8*e^2 - 4*(
5*B*b^6*c - 18*A*b^5*c^2)*d^7*e^3 + (5*B*b^7 - 42*A*b^6*c)*d^6*e^4)*x^2 + 7*(9*A
*b^7*d^6*e^4 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^9*e - 4*(5*B*b^6*c - 18*A*b^5*c^2)
*d^8*e^2 + (5*B*b^7 - 42*A*b^6*c)*d^7*e^3)*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2
+ b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((c^5*d^17 - 5
*b*c^4*d^16*e + 10*b^2*c^3*d^15*e^2 - 10*b^3*c^2*d^14*e^3 + 5*b^4*c*d^13*e^4 - b
^5*d^12*e^5 + (c^5*d^10*e^7 - 5*b*c^4*d^9*e^8 + 10*b^2*c^3*d^8*e^9 - 10*b^3*c^2*
d^7*e^10 + 5*b^4*c*d^6*e^11 - b^5*d^5*e^12)*x^7 + 7*(c^5*d^11*e^6 - 5*b*c^4*d^10
*e^7 + 10*b^2*c^3*d^9*e^8 - 10*b^3*c^2*d^8*e^9 + 5*b^4*c*d^7*e^10 - b^5*d^6*e^11
)*x^6 + 21*(c^5*d^12*e^5 - 5*b*c^4*d^11*e^6 + 10*b^2*c^3*d^10*e^7 - 10*b^3*c^2*d
^9*e^8 + 5*b^4*c*d^8*e^9 - b^5*d^7*e^10)*x^5 + 35*(c^5*d^13*e^4 - 5*b*c^4*d^12*e
^5 + 10*b^2*c^3*d^11*e^6 - 10*b^3*c^2*d^10*e^7 + 5*b^4*c*d^9*e^8 - b^5*d^8*e^9)*
x^4 + 35*(c^5*d^14*e^3 - 5*b*c^4*d^13*e^4 + 10*b^2*c^3*d^12*e^5 - 10*b^3*c^2*d^1
1*e^6 + 5*b^4*c*d^10*e^7 - b^5*d^9*e^8)*x^3 + 21*(c^5*d^15*e^2 - 5*b*c^4*d^14*e^
3 + 10*b^2*c^3*d^13*e^4 - 10*b^3*c^2*d^12*e^5 + 5*b^4*c*d^11*e^6 - b^5*d^10*e^7)
*x^2 + 7*(c^5*d^16*e - 5*b*c^4*d^15*e^2 + 10*b^2*c^3*d^14*e^3 - 10*b^3*c^2*d^13*
e^4 + 5*b^4*c*d^12*e^5 - b^5*d^11*e^6)*x)*sqrt(c*d^2 - b*d*e)), 1/107520*((945*A
*b^6*d^6*e^3 + 2520*(B*b^4*c^2 - 2*A*b^3*c^3)*d^9 - 420*(5*B*b^5*c - 18*A*b^4*c^
2)*d^8*e + 105*(5*B*b^6 - 42*A*b^5*c)*d^7*e^2 + (1024*B*c^6*d^7*e^2 - 945*A*b^6*
e^9 - 128*(31*B*b*c^5 - 6*A*c^6)*d^6*e^3 + 256*(19*B*b^2*c^4 - 9*A*b*c^5)*d^5*e^
4 - 160*(5*B*b^3*c^3 - 6*A*b^2*c^4)*d^4*e^5 - 40*(35*B*b^4*c^2 - 48*A*b^3*c^3)*d
^3*e^6 + 14*(125*B*b^5*c - 366*A*b^4*c^2)*d^2*e^7 - 105*(5*B*b^6 - 36*A*b^5*c)*d
*e^8)*x^6 + 2*(3584*B*c^6*d^8*e - 3150*A*b^6*d*e^8 - 64*(221*B*b*c^5 - 42*A*c^6)
*d^7*e^2 + 32*(563*B*b^2*c^4 - 258*A*b*c^5)*d^6*e^3 - 16*(251*B*b^3*c^3 - 246*A*
b^2*c^4)*d^5*e^4 - 20*(235*B*b^4*c^2 - 324*A*b^3*c^3)*d^4*e^5 + (5845*B*b^5*c -
17154*A*b^4*c^2)*d^3*e^6 - 7*(250*B*b^6 - 1803*A*b^5*c)*d^2*e^7)*x^5 + (21504*B*
c^6*d^9 - 17829*A*b^6*d^2*e^7 - 896*(97*B*b*c^5 - 18*A*c^6)*d^8*e + 64*(1819*B*b
^2*c^4 - 798*A*b*c^5)*d^7*e^2 - 16*(2207*B*b^3*c^3 - 1782*A*b^2*c^4)*d^6*e^3 - 8
*(3097*B*b^4*c^2 - 4512*A*b^3*c^3)*d^5*e^4 + 8*(4145*B*b^5*c - 12198*A*b^4*c^2)*
d^4*e^5 - (9905*B*b^6 - 71574*A*b^5*c)*d^3*e^6)*x^4 - 4*(6912*A*b^6*d^3*e^6 - 67
2*(11*B*b*c^5 + 10*A*c^6)*d^9 + 336*(107*B*b^2*c^4 + 66*A*b*c^5)*d^8*e - 4*(1678
7*B*b^3*c^3 + 3738*A*b^2*c^4)*d^7*e^2 + 4*(12862*B*b^4*c^2 - 3177*A*b^3*c^3)*d^6
*e^3 - 5*(4285*B*b^5*c - 7566*A*b^4*c^2)*d^5*e^4 + 3*(1280*B*b^6 - 9271*A*b^5*c)
*d^4*e^5)*x^3 - 7*(3597*A*b^6*d^4*e^5 - 192*(B*b^2*c^4 + 30*A*b*c^5)*d^9 + 16*(1
09*B*b^3*c^3 + 1686*A*b^2*c^4)*d^8*e - 8*(1019*B*b^4*c^2 + 5346*A*b^3*c^3)*d^7*e
^2 + 10*(599*B*b^5*c + 3822*A*b^4*c^2)*d^6*e^3 - 5*(283*B*b^6 + 3648*A*b^5*c)*d^
5*e^4)*x^2 + 70*(90*A*b^6*d^5*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^9 + 4*(65*B*b
^4*c^2 - 138*A*b^3*c^3)*d^8*e - (205*B*b^5*c - 762*A*b^4*c^2)*d^7*e^2 + (50*B*b^
6 - 429*A*b^5*c)*d^6*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 105*(9*A*b
^7*d^7*e^3 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^10 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^
9*e + (5*B*b^7 - 42*A*b^6*c)*d^8*e^2 + (9*A*b^7*e^10 + 24*(B*b^5*c^2 - 2*A*b^4*c
^3)*d^3*e^7 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^2*e^8 + (5*B*b^7 - 42*A*b^6*c)*d*e^
9)*x^7 + 7*(9*A*b^7*d*e^9 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^4*e^6 - 4*(5*B*b^6*c
- 18*A*b^5*c^2)*d^3*e^7 + (5*B*b^7 - 42*A*b^6*c)*d^2*e^8)*x^6 + 21*(9*A*b^7*d^2*
e^8 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^5*e^5 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^4*e^
6 + (5*B*b^7 - 42*A*b^6*c)*d^3*e^7)*x^5 + 35*(9*A*b^7*d^3*e^7 + 24*(B*b^5*c^2 -
2*A*b^4*c^3)*d^6*e^4 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^5*e^5 + (5*B*b^7 - 42*A*b^
6*c)*d^4*e^6)*x^4 + 35*(9*A*b^7*d^4*e^6 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^7*e^3 -
4*(5*B*b^6*c - 18*A*b^5*c^2)*d^6*e^4 + (5*B*b^7 - 42*A*b^6*c)*d^5*e^5)*x^3 + 21
*(9*A*b^7*d^5*e^5 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^8*e^2 - 4*(5*B*b^6*c - 18*A*b
^5*c^2)*d^7*e^3 + (5*B*b^7 - 42*A*b^6*c)*d^6*e^4)*x^2 + 7*(9*A*b^7*d^6*e^4 + 24*
(B*b^5*c^2 - 2*A*b^4*c^3)*d^9*e - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^8*e^2 + (5*B*b^
7 - 42*A*b^6*c)*d^7*e^3)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d
- b*e)*x)))/((c^5*d^17 - 5*b*c^4*d^16*e + 10*b^2*c^3*d^15*e^2 - 10*b^3*c^2*d^14
*e^3 + 5*b^4*c*d^13*e^4 - b^5*d^12*e^5 + (c^5*d^10*e^7 - 5*b*c^4*d^9*e^8 + 10*b^
2*c^3*d^8*e^9 - 10*b^3*c^2*d^7*e^10 + 5*b^4*c*d^6*e^11 - b^5*d^5*e^12)*x^7 + 7*(
c^5*d^11*e^6 - 5*b*c^4*d^10*e^7 + 10*b^2*c^3*d^9*e^8 - 10*b^3*c^2*d^8*e^9 + 5*b^
4*c*d^7*e^10 - b^5*d^6*e^11)*x^6 + 21*(c^5*d^12*e^5 - 5*b*c^4*d^11*e^6 + 10*b^2*
c^3*d^10*e^7 - 10*b^3*c^2*d^9*e^8 + 5*b^4*c*d^8*e^9 - b^5*d^7*e^10)*x^5 + 35*(c^
5*d^13*e^4 - 5*b*c^4*d^12*e^5 + 10*b^2*c^3*d^11*e^6 - 10*b^3*c^2*d^10*e^7 + 5*b^
4*c*d^9*e^8 - b^5*d^8*e^9)*x^4 + 35*(c^5*d^14*e^3 - 5*b*c^4*d^13*e^4 + 10*b^2*c^
3*d^12*e^5 - 10*b^3*c^2*d^11*e^6 + 5*b^4*c*d^10*e^7 - b^5*d^9*e^8)*x^3 + 21*(c^5
*d^15*e^2 - 5*b*c^4*d^14*e^3 + 10*b^2*c^3*d^13*e^4 - 10*b^3*c^2*d^12*e^5 + 5*b^4
*c*d^11*e^6 - b^5*d^10*e^7)*x^2 + 7*(c^5*d^16*e - 5*b*c^4*d^15*e^2 + 10*b^2*c^3*
d^14*e^3 - 10*b^3*c^2*d^13*e^4 + 5*b^4*c*d^12*e^5 - b^5*d^11*e^6)*x)*sqrt(-c*d^2
+ b*d*e))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.659586, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/(e*x + d)^8,x, algorithm="giac")

[Out]

sage0*x