3.1879 \(\int \frac{(A+B x) \sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ \frac{e^2 \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{96 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(d+e x)^{3/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e^3 (a+b x) (-3 a B e-5 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

[Out]

(e^2*(8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(24*b^2*(
b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(8*b*B*d - 5*A*b*e -
3*a*B*e)*Sqrt[d + e*x])/(96*b^2*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2]) - ((A*b - a*B)*(d + e*x)^(3/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (e^3*(8*b*B*d - 5*A*b*e - 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(7/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])

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Rubi [A]  time = 0.884827, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ \frac{e^2 \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{64 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e \sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{96 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{(d+e x)^{3/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x} (-3 a B e-5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e^3 (a+b x) (-3 a B e-5 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(e^2*(8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(24*b^2*(
b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(8*b*B*d - 5*A*b*e -
3*a*B*e)*Sqrt[d + e*x])/(96*b^2*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2]) - ((A*b - a*B)*(d + e*x)^(3/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (e^3*(8*b*B*d - 5*A*b*e - 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(7/2)*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)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.13918, size = 234, normalized size = 0.65 \[ \frac{(a+b x) \left (\frac{e^3 (3 a B e+5 A b e-8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (3 e^2 (a+b x)^3 (3 a B e+5 A b e-8 b B d)+8 (a+b x) (b d-a e)^2 (-9 a B e+A b e+8 b B d)+2 e (a+b x)^2 (a e-b d) (3 a B e+5 A b e-8 b B d)+48 (A b-a B) (b d-a e)^3\right )}{3 b^2 (a+b x)^4 (b d-a e)^3}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)*(-(Sqrt[d + e*x]*(48*(A*b - a*B)*(b*d - a*e)^3 + 8*(b*d - a*e)^2*(8*b
*B*d + A*b*e - 9*a*B*e)*(a + b*x) + 2*e*(-(b*d) + a*e)*(-8*b*B*d + 5*A*b*e + 3*a
*B*e)*(a + b*x)^2 + 3*e^2*(-8*b*B*d + 5*A*b*e + 3*a*B*e)*(a + b*x)^3))/(3*b^2*(b
*d - a*e)^3*(a + b*x)^4) + (e^3*(-8*b*B*d + 5*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*(b*d - a*e)^(7/2))))/(64*Sqrt[(a + b*x
)^2])

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Maple [B]  time = 0.036, size = 1296, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(b*x+a)/e*(54*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^
5+73*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3+73*A*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(3/2)*b^4*d^2*e-15*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+15*A*(b*(
a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e+90*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*x^2*a^2*b^3*e^5-33*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3-24*B*a
rctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b*d*e^4-96*B*arctan((e*x+d)^(1/2)
*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^4*d*e^4-121*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*
a*b^3*d*e+9*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*b^4*e^5-24*B*arc
tan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*d*e^4+60*A*arctan((e*x+d)^(1/2)
*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^4*e^5+36*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*x^3*a^2*b^3*e^5+9*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^3*e+55*A*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-55*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^4
*d*e+47*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e-96*B*arctan((e*x+d)^(1/2
)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^4-146*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)
*a*b^3*d*e^2+45*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3-45*A*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2+51*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^
3*b*d*e^3-99*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2+81*B*(b*(a*e-b*
d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+26*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b
^2*d*e^2-144*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*d*e^4+15*
A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*e^5+15*A*(b*(a*e-b*d))^(1/
2)*(e*x+d)^(7/2)*b^4*e-24*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^4*d+88*B*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d^2+15*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/
2))*a^4*b*e^5-40*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^4*d^3-9*B*(b*(a*e-b*d))^(
1/2)*(e*x+d)^(1/2)*a^4*e^4-24*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d^4+9*B*ar
ctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*e^5+60*A*arctan((e*x+d)^(1/2)*b/(b
*(a*e-b*d))^(1/2))*x*a^3*b^2*e^5+33*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2*b^2*
e^2+36*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b*e^5)/(b*(a*e-b*d))^
(1/2)/(a*e-b*d)/b^2/(a^2*e^2-2*a*b*d*e+b^2*d^2)/((b*x+a)^2)^(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((B*x + A)*sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306851, 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(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(2*(16*(B*a*b^3 + 3*A*b^4)*d^3 - 8*(5*B*a^2*b^2 + 17*A*a*b^3)*d^2*e + 2*
(9*B*a^3*b + 59*A*a^2*b^2)*d*e^2 - 3*(3*B*a^4 + 5*A*a^3*b)*e^3 - 3*(8*B*b^4*d*e^
2 - (3*B*a*b^3 + 5*A*b^4)*e^3)*x^3 + (16*B*b^4*d^2*e - 2*(47*B*a*b^3 + 5*A*b^4)*
d*e^2 + 11*(3*B*a^2*b^2 + 5*A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 - 8*(21*B*a*b^3 -
A*b^4)*d^2*e + 4*(23*B*a^2*b^2 - 9*A*a*b^3)*d*e^2 - (33*B*a^3*b - 73*A*a^2*b^2)*
e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) - 3*(8*B*a^4*b*d*e^3 - (3*B*a^5 + 5*A*
a^4*b)*e^4 + (8*B*b^5*d*e^3 - (3*B*a*b^4 + 5*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^
3 - (3*B*a^2*b^3 + 5*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (3*B*a^3*b^2 + 5
*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (3*B*a^4*b + 5*A*a^3*b^2)*e^4)*x)*
log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d)
)/(b*x + a)))/((a^4*b^5*d^3 - 3*a^5*b^4*d^2*e + 3*a^6*b^3*d*e^2 - a^7*b^2*e^3 +
(b^9*d^3 - 3*a*b^8*d^2*e + 3*a^2*b^7*d*e^2 - a^3*b^6*e^3)*x^4 + 4*(a*b^8*d^3 - 3
*a^2*b^7*d^2*e + 3*a^3*b^6*d*e^2 - a^4*b^5*e^3)*x^3 + 6*(a^2*b^7*d^3 - 3*a^3*b^6
*d^2*e + 3*a^4*b^5*d*e^2 - a^5*b^4*e^3)*x^2 + 4*(a^3*b^6*d^3 - 3*a^4*b^5*d^2*e +
3*a^5*b^4*d*e^2 - a^6*b^3*e^3)*x)*sqrt(b^2*d - a*b*e)), -1/192*((16*(B*a*b^3 +
3*A*b^4)*d^3 - 8*(5*B*a^2*b^2 + 17*A*a*b^3)*d^2*e + 2*(9*B*a^3*b + 59*A*a^2*b^2)
*d*e^2 - 3*(3*B*a^4 + 5*A*a^3*b)*e^3 - 3*(8*B*b^4*d*e^2 - (3*B*a*b^3 + 5*A*b^4)*
e^3)*x^3 + (16*B*b^4*d^2*e - 2*(47*B*a*b^3 + 5*A*b^4)*d*e^2 + 11*(3*B*a^2*b^2 +
5*A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 - 8*(21*B*a*b^3 - A*b^4)*d^2*e + 4*(23*B*a^2
*b^2 - 9*A*a*b^3)*d*e^2 - (33*B*a^3*b - 73*A*a^2*b^2)*e^3)*x)*sqrt(-b^2*d + a*b*
e)*sqrt(e*x + d) + 3*(8*B*a^4*b*d*e^3 - (3*B*a^5 + 5*A*a^4*b)*e^4 + (8*B*b^5*d*e
^3 - (3*B*a*b^4 + 5*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (3*B*a^2*b^3 + 5*A*a*
b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (3*B*a^3*b^2 + 5*A*a^2*b^3)*e^4)*x^2 + 4*
(8*B*a^3*b^2*d*e^3 - (3*B*a^4*b + 5*A*a^3*b^2)*e^4)*x)*arctan(-(b*d - a*e)/(sqrt
(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^4*b^5*d^3 - 3*a^5*b^4*d^2*e + 3*a^6*b^3*d*
e^2 - a^7*b^2*e^3 + (b^9*d^3 - 3*a*b^8*d^2*e + 3*a^2*b^7*d*e^2 - a^3*b^6*e^3)*x^
4 + 4*(a*b^8*d^3 - 3*a^2*b^7*d^2*e + 3*a^3*b^6*d*e^2 - a^4*b^5*e^3)*x^3 + 6*(a^2
*b^7*d^3 - 3*a^3*b^6*d^2*e + 3*a^4*b^5*d*e^2 - a^5*b^4*e^3)*x^2 + 4*(a^3*b^6*d^3
- 3*a^4*b^5*d^2*e + 3*a^5*b^4*d*e^2 - a^6*b^3*e^3)*x)*sqrt(-b^2*d + a*b*e))]

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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)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.339263, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

Done