3.1866 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=407 \[ -\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d
+ e*x)^(3/2))/(12*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(4*b*B*d + 5*A*b*e -
9*a*B*e)*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(9/2))/(2*b*(b*d - a*e)*(a
+ b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*
e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(1
1/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.922391, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^5*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d
+ e*x)^(3/2))/(12*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(4*b*B*d + 5*A*b*e -
9*a*B*e)*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(9/2))/(2*b*(b*d - a*e)*(a
+ b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*
e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(1
1/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.16713, size = 243, normalized size = 0.6 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (8 e \left (90 a^2 B e^2-15 a b e (3 A e+10 B d)+2 b^2 d (25 A e+29 B d)\right )+8 b e^2 x (-15 a B e+5 A b e+16 b B d)-\frac{15 (b d-a e)^2 (-17 a B e+13 A b e+4 b B d)}{a+b x}-\frac{30 (A b-a B) (b d-a e)^3}{(a+b x)^2}+24 b^2 B e^3 x^2\right )}{15 b^5}-\frac{7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((a + b*x)^3*((Sqrt[d + e*x]*(8*e*(90*a^2*B*e^2 - 15*a*b*e*(10*B*d + 3*A*e) + 2*
b^2*d*(29*B*d + 25*A*e)) + 8*b*e^2*(16*b*B*d + 5*A*b*e - 15*a*B*e)*x + 24*b^2*B*
e^3*x^2 - (30*(A*b - a*B)*(b*d - a*e)^3)/(a + b*x)^2 - (15*(b*d - a*e)^2*(4*b*B*
d + 13*A*b*e - 17*a*B*e))/(a + b*x)))/(15*b^5) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d
+ 5*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)
))/(4*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.039, size = 1873, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/60*(-945*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^5-1080*B*
(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a*b^3*d*e^3+720*A*(b*(a*e-b*d))^(1/2)*(e*x
+d)^(1/2)*x*a*b^3*d*e^3-2160*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^2*b^2*d*e^3
-155*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3-195*A*(b*(a*e-b*d))^(1/2)*(
e*x+d)^(3/2)*b^4*d^2*e+420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^3
*d^3*e^2-525*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+165*A*(b*(a*e-b*d))^(
1/2)*(e*x+d)^(1/2)*b^4*d^3*e+525*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x
^2*b^5*d^2*e^3+525*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*e^5
+525*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^3*d^2*e^3+135*B*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3+2310*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*a^4*b*d*e^4-1785*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b^2*d
^2*e^3+375*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e+1440*B*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(1/2)*x*a^3*b*e^4+360*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^4*
d*e^3-2100*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b^3*d*e^4+1050*A*
arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b^4*d^2*e^3-240*B*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(3/2)*x*a^2*b^2*e^3+720*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^
2*b^2*e^4+360*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^4*d^2*e^2+4620*B*arctan(
(e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^4-3570*B*arctan((e*x+d)^(1/2)
*b/(b*(a*e-b*d))^(1/2))*x*a^2*b^3*d^2*e^3+840*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b
*d))^(1/2))*x*a*b^4*d^3*e^2+390*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^3*d*e^2+
80*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a*b^3*e^3-360*A*(b*(a*e-b*d))^(1/2)*(e*
x+d)^(1/2)*x^2*a*b^3*e^4+855*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3-4
95*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2-1815*B*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(1/2)*a^3*b*d*e^3+1215*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e
^2-405*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e-720*A*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(1/2)*x*a^2*b^2*e^4-490*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*d*e
^2-1050*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^4*d*e^4+48*B*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(5/2)*x*a*b^3*e^2-120*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)
*x^2*a*b^3*e^3+80*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*b^4*d*e^2+2310*B*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*d*e^4-1785*B*arctan((e*x+d)^(
1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^4*d^2*e^3+720*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(1/2)*x*a*b^3*d^2*e^2+160*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a*b^3*d*e^2+525*
A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b*e^5-60*B*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(3/2)*b^4*d^3+945*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*e^4+60*B*(b*(
a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d^4-945*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*a^5*e^5+24*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x^2*b^4*e^2+40*A*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(3/2)*x^2*b^4*e^3-1050*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*a^3*b^2*d*e^4+420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^5
*d^3*e^2+1050*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*e^5+24*B*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2*b^2*e^2-1890*B*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*x*a^4*b*e^5)/e*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^5/((b*x+a)^2)^(3/2)

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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((B*x + A)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296581, 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)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/120*(105*(4*B*a^2*b^2*d^2*e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5
*A*a^3*b)*e^3 + (4*B*b^4*d^2*e - (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5
*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*
B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*
sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(24*B*b^4*e^3*x^4 - 30*(B*a*
b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*
a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (9*B*a*b^3
- 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9
*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 195*A*b^4)*d^
2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5*A*a^2*b^2)*e^3)
*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*(4*B*a^2*b^2*d^2*
e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*
e - (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a
*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x
)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (24*B*b^4*e^
3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(1
2*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e
^2 - (9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b
^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3
- 195*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5
*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]

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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)*(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.351958, size = 929, normalized size = 2.28 \[ -\frac{7 \,{\left (4 \, B b^{3} d^{3} e^{2} - 17 \, B a b^{2} d^{2} e^{3} + 5 \, A b^{3} d^{2} e^{3} + 22 \, B a^{2} b d e^{4} - 10 \, A a b^{2} d e^{4} - 9 \, B a^{3} e^{5} + 5 \, A a^{2} b e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt{-b^{2} d + a b e} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{2} - 4 \, \sqrt{x e + d} B b^{4} d^{4} e^{2} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{3} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{3} + 27 \, \sqrt{x e + d} B a b^{3} d^{3} e^{3} - 11 \, \sqrt{x e + d} A b^{4} d^{3} e^{3} + 38 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{4} - 26 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{4} - 57 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{4} + 33 \, \sqrt{x e + d} A a b^{3} d^{2} e^{4} - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{5} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{5} + 49 \, \sqrt{x e + d} B a^{3} b d e^{5} - 33 \, \sqrt{x e + d} A a^{2} b^{2} d e^{5} - 15 \, \sqrt{x e + d} B a^{4} e^{6} + 11 \, \sqrt{x e + d} A a^{3} b e^{6}\right )} e^{\left (-1\right )}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} e^{6} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d e^{6} + 45 \, \sqrt{x e + d} B b^{12} d^{2} e^{6} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} e^{7} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} e^{7} - 135 \, \sqrt{x e + d} B a b^{11} d e^{7} + 45 \, \sqrt{x e + d} A b^{12} d e^{7} + 90 \, \sqrt{x e + d} B a^{2} b^{10} e^{8} - 45 \, \sqrt{x e + d} A a b^{11} e^{8}\right )} e^{\left (-5\right )}}{15 \, b^{15}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

-7/4*(4*B*b^3*d^3*e^2 - 17*B*a*b^2*d^2*e^3 + 5*A*b^3*d^2*e^3 + 22*B*a^2*b*d*e^4
- 10*A*a*b^2*d*e^4 - 9*B*a^3*e^5 + 5*A*a^2*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b
^2*d + a*b*e))*e^(-1)/(sqrt(-b^2*d + a*b*e)*b^5*sign(-(x*e + d)*b*e + b*d*e - a*
e^2)) + 1/4*(4*(x*e + d)^(3/2)*B*b^4*d^3*e^2 - 4*sqrt(x*e + d)*B*b^4*d^4*e^2 - 2
5*(x*e + d)^(3/2)*B*a*b^3*d^2*e^3 + 13*(x*e + d)^(3/2)*A*b^4*d^2*e^3 + 27*sqrt(x
*e + d)*B*a*b^3*d^3*e^3 - 11*sqrt(x*e + d)*A*b^4*d^3*e^3 + 38*(x*e + d)^(3/2)*B*
a^2*b^2*d*e^4 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^4 - 57*sqrt(x*e + d)*B*a^2*b^2*d^
2*e^4 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^4 - 17*(x*e + d)^(3/2)*B*a^3*b*e^5 + 13*(
x*e + d)^(3/2)*A*a^2*b^2*e^5 + 49*sqrt(x*e + d)*B*a^3*b*d*e^5 - 33*sqrt(x*e + d)
*A*a^2*b^2*d*e^5 - 15*sqrt(x*e + d)*B*a^4*e^6 + 11*sqrt(x*e + d)*A*a^3*b*e^6)*e^
(-1)/(((x*e + d)*b - b*d + a*e)^2*b^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/
15*(3*(x*e + d)^(5/2)*B*b^12*e^6 + 10*(x*e + d)^(3/2)*B*b^12*d*e^6 + 45*sqrt(x*e
+ d)*B*b^12*d^2*e^6 - 15*(x*e + d)^(3/2)*B*a*b^11*e^7 + 5*(x*e + d)^(3/2)*A*b^1
2*e^7 - 135*sqrt(x*e + d)*B*a*b^11*d*e^7 + 45*sqrt(x*e + d)*A*b^12*d*e^7 + 90*sq
rt(x*e + d)*B*a^2*b^10*e^8 - 45*sqrt(x*e + d)*A*a*b^11*e^8)*e^(-5)/(b^15*sign(-(
x*e + d)*b*e + b*d*e - a*e^2))