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3.1198 \(\int \frac{A+B x}{(d+e x)^4 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=331 \[ \frac{\sqrt{b x+c x^2} \left (B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )-A e \left (15 b^2 e^2-44 b c d e+44 c^2 d^2\right )\right )}{24 d^3 (d+e x) (c d-b e)^3}+\frac{\left (b^3 \left (-e^2\right ) (5 A e+B d)+2 b^2 c d e (9 A e+2 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{7/2} (c d-b e)^{7/2}}-\frac{\sqrt{b x+c x^2} (5 A e (2 c d-b e)-B d (b e+4 c d))}{12 d^2 (d+e x)^2 (c d-b e)^2}+\frac{\sqrt{b x+c x^2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \]

[Out]

((B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^3) - ((5*A*e*(2*c*d -
 b*e) - B*d*(4*c*d + b*e))*Sqrt[b*x + c*x^2])/(12*d^2*(c*d - b*e)^2*(d + e*x)^2)
 + ((B*d*(8*c^2*d^2 + 10*b*c*d*e - 3*b^2*e^2) - A*e*(44*c^2*d^2 - 44*b*c*d*e + 1
5*b^2*e^2))*Sqrt[b*x + c*x^2])/(24*d^3*(c*d - b*e)^3*(d + e*x)) + ((16*A*c^3*d^3
 - 8*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(B*d + 5*A*e) + 2*b^2*c*d*e*(2*B*d + 9*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]
)])/(16*d^(7/2)*(c*d - b*e)^(7/2))

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Rubi [A]  time = 1.24931, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{b x+c x^2} \left (B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )-A e \left (15 b^2 e^2-44 b c d e+44 c^2 d^2\right )\right )}{24 d^3 (d+e x) (c d-b e)^3}+\frac{\left (b^3 \left (-e^2\right ) (5 A e+B d)+2 b^2 c d e (9 A e+2 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{7/2} (c d-b e)^{7/2}}-\frac{\sqrt{b x+c x^2} (5 A e (2 c d-b e)-B d (b e+4 c d))}{12 d^2 (d+e x)^2 (c d-b e)^2}+\frac{\sqrt{b x+c x^2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^4*Sqrt[b*x + c*x^2]),x]

[Out]

((B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^3) - ((5*A*e*(2*c*d -
 b*e) - B*d*(4*c*d + b*e))*Sqrt[b*x + c*x^2])/(12*d^2*(c*d - b*e)^2*(d + e*x)^2)
 + ((B*d*(8*c^2*d^2 + 10*b*c*d*e - 3*b^2*e^2) - A*e*(44*c^2*d^2 - 44*b*c*d*e + 1
5*b^2*e^2))*Sqrt[b*x + c*x^2])/(24*d^3*(c*d - b*e)^3*(d + e*x)) + ((16*A*c^3*d^3
 - 8*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(B*d + 5*A*e) + 2*b^2*c*d*e*(2*B*d + 9*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]
)])/(16*d^(7/2)*(c*d - b*e)^(7/2))

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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((B*x+A)/(e*x+d)**4/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.42102, size = 299, normalized size = 0.9 \[ \frac{\sqrt{x} \left (\frac{\sqrt{d} \sqrt{x} (b+c x) \left ((d+e x)^2 \left (A e \left (-15 b^2 e^2+44 b c d e-44 c^2 d^2\right )+B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )\right )+8 d^2 (B d-A e) (c d-b e)^2+2 d (d+e x) (c d-b e) (5 A e (b e-2 c d)+B d (b e+4 c d))\right )}{(d+e x)^3}-\frac{3 \sqrt{b+c x} \left (b^3 e^2 (5 A e+B d)-2 b^2 c d e (9 A e+2 B d)+8 b c^2 d^2 (3 A e+B d)-16 A c^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b e-c d}}\right )}{24 d^{7/2} \sqrt{x (b+c x)} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^4*Sqrt[b*x + c*x^2]),x]

[Out]

(Sqrt[x]*((Sqrt[d]*Sqrt[x]*(b + c*x)*(8*d^2*(B*d - A*e)*(c*d - b*e)^2 + 2*d*(c*d
 - b*e)*(5*A*e*(-2*c*d + b*e) + B*d*(4*c*d + b*e))*(d + e*x) + (A*e*(-44*c^2*d^2
 + 44*b*c*d*e - 15*b^2*e^2) + B*d*(8*c^2*d^2 + 10*b*c*d*e - 3*b^2*e^2))*(d + e*x
)^2))/(d + e*x)^3 - (3*(-16*A*c^3*d^3 + 8*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(B*d
 + 5*A*e) - 2*b^2*c*d*e*(2*B*d + 9*A*e))*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e
]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[-(c*d) + b*e]))/(24*d^(7/2)*(c*d - b*e
)^3*Sqrt[x*(b + c*x)])

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Maple [B]  time = 0.024, size = 3242, normalized size = 9.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^4/(c*x^2+b*x)^(1/2),x)

[Out]

-5/2*e/d^2/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2)*b*c*A+15/8*e/d^2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d
)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c*A-5/2/e/(b*e-c*d)^3/(d/e+x)*(c*
(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B+1/2*B/e^2/d/(b*e-c*
d)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+1/3/e^2/d
/(b*e-c*d)/(d/e+x)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A
+5/2/e/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e
*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(1/2))/(d/e+x))*c^3*A+5/6/e^2/(b*e-c*d)^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c*B+5/2/d/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2
+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A+2/3*c/d^2/(b*e-c*d)^2/(d/e+x
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A+5/12/d^2/(b*e-c*d)
^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*A+3/4*B
/d^2/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2)*b-3/8*B/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-
2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2-3*B/e^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)
*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^2-5/8*e/d^2/(b*e-
c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B+5
/2/d/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2)*b*c*B-5/16*e^2/d^3/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^
2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*A+5/16*e/d^2/(b*e-c*d)^3/(-d*(b*e-c*d
)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(
1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*B-5
/2/e^2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e
*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(1/2))/(d/e+x))*c^3*B*d+3/2/e/d/(b*e-c*d)^2*c^2/(-d*(b*e-c*d)/e^2)^(1/2)
*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*A+15/4/e/(b*e-c*d)^
3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b
*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d
/e+x))*b*c^2*B-3/4/d^2/(b*e-c*d)^2*c/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)
/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e
*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*A-15/4/d/(b*e-c*d)^3/(-d*(b*e-c*d)/e
^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c^2*A-15
/8/d/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(
d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)
/e^2)^(1/2))/(d/e+x))*b^2*c*B-13/6*B/e/d/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c-1/2*B/e^2*c/d/(b*e-c*d)/(-d*(b*e-c*d)/e
^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))-5/12/e/d/(
b*e-c*d)^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b
*B-5/6/e/d/(b*e-c*d)^2/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/
e^2)^(1/2)*c*A+5/8*e^2/d^3/(b*e-c*d)^3/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A-1/3/e^3/(b*e-c*d)/(d/e+x)^3*(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*B+9/4*B/e/d/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^
(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c
*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c

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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((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310333, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^4),x, algorithm="fricas")

[Out]

[1/48*(2*(24*B*c^2*d^5 - 33*A*b^2*d^2*e^3 - 12*(B*b*c + 6*A*c^2)*d^4*e + 3*(B*b^
2 + 30*A*b*c)*d^3*e^2 + (8*B*c^2*d^3*e^2 - 15*A*b^2*e^5 + 2*(5*B*b*c - 22*A*c^2)
*d^2*e^3 - (3*B*b^2 - 44*A*b*c)*d*e^4)*x^2 + 2*(12*B*c^2*d^4*e - 20*A*b^2*d*e^4
+ (7*B*b*c - 54*A*c^2)*d^3*e^2 - (4*B*b^2 - 59*A*b*c)*d^2*e^3)*x)*sqrt(c*d^2 - b
*d*e)*sqrt(c*x^2 + b*x) - 3*(5*A*b^3*d^3*e^3 + 8*(B*b*c^2 - 2*A*c^3)*d^6 - 4*(B*
b^2*c - 6*A*b*c^2)*d^5*e + (B*b^3 - 18*A*b^2*c)*d^4*e^2 + (5*A*b^3*e^6 + 8*(B*b*
c^2 - 2*A*c^3)*d^3*e^3 - 4*(B*b^2*c - 6*A*b*c^2)*d^2*e^4 + (B*b^3 - 18*A*b^2*c)*
d*e^5)*x^3 + 3*(5*A*b^3*d*e^5 + 8*(B*b*c^2 - 2*A*c^3)*d^4*e^2 - 4*(B*b^2*c - 6*A
*b*c^2)*d^3*e^3 + (B*b^3 - 18*A*b^2*c)*d^2*e^4)*x^2 + 3*(5*A*b^3*d^2*e^4 + 8*(B*
b*c^2 - 2*A*c^3)*d^5*e - 4*(B*b^2*c - 6*A*b*c^2)*d^4*e^2 + (B*b^3 - 18*A*b^2*c)*
d^3*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^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^
3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3
+ 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3
*d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c*d^6*e^3 - b^3*d^5*e^4)*x)*sqrt(c*d^2 - b*d*e)
), 1/24*((24*B*c^2*d^5 - 33*A*b^2*d^2*e^3 - 12*(B*b*c + 6*A*c^2)*d^4*e + 3*(B*b^
2 + 30*A*b*c)*d^3*e^2 + (8*B*c^2*d^3*e^2 - 15*A*b^2*e^5 + 2*(5*B*b*c - 22*A*c^2)
*d^2*e^3 - (3*B*b^2 - 44*A*b*c)*d*e^4)*x^2 + 2*(12*B*c^2*d^4*e - 20*A*b^2*d*e^4
+ (7*B*b*c - 54*A*c^2)*d^3*e^2 - (4*B*b^2 - 59*A*b*c)*d^2*e^3)*x)*sqrt(-c*d^2 +
b*d*e)*sqrt(c*x^2 + b*x) + 3*(5*A*b^3*d^3*e^3 + 8*(B*b*c^2 - 2*A*c^3)*d^6 - 4*(B
*b^2*c - 6*A*b*c^2)*d^5*e + (B*b^3 - 18*A*b^2*c)*d^4*e^2 + (5*A*b^3*e^6 + 8*(B*b
*c^2 - 2*A*c^3)*d^3*e^3 - 4*(B*b^2*c - 6*A*b*c^2)*d^2*e^4 + (B*b^3 - 18*A*b^2*c)
*d*e^5)*x^3 + 3*(5*A*b^3*d*e^5 + 8*(B*b*c^2 - 2*A*c^3)*d^4*e^2 - 4*(B*b^2*c - 6*
A*b*c^2)*d^3*e^3 + (B*b^3 - 18*A*b^2*c)*d^2*e^4)*x^2 + 3*(5*A*b^3*d^2*e^4 + 8*(B
*b*c^2 - 2*A*c^3)*d^5*e - 4*(B*b^2*c - 6*A*b*c^2)*d^4*e^2 + (B*b^3 - 18*A*b^2*c)
*d^3*e^3)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((
c^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2
*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3
 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c
*d^6*e^3 - b^3*d^5*e^4)*x)*sqrt(-c*d^2 + b*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x+d)**4/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((A + B*x)/(sqrt(x*(b + c*x))*(d + e*x)**4), x)

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GIAC/XCAS [A]  time = 0.671548, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^4),x, algorithm="giac")

[Out]

sage0*x

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