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3.1878 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ -\frac{(d+e x)^{5/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e-3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{e^3 (a+b x) (-5 a B e-3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac{e^2 \sqrt{d+e x} (-5 a B e-3 A b e+8 b B d)}{64 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e \sqrt{d+e x} (-5 a B e-3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

-(e^2*(8*b*B*d - 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) - (e*(8*b*B*d - 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(32*b^
3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 3*A*b*e - 5
*a*B*e)*(d + e*x)^(3/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((A*b - a*B)*(d + e*x)^(5/2))/(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 - 3*A*b*e - 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(7/2)*(b*d - a*e)^(5/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])

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Rubi [A]  time = 0.877339, 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 \[ -\frac{(d+e x)^{5/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e-3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{e^3 (a+b x) (-5 a B e-3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac{e^2 \sqrt{d+e x} (-5 a B e-3 A b e+8 b B d)}{64 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e \sqrt{d+e x} (-5 a B e-3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*(8*b*B*d - 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) - (e*(8*b*B*d - 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(32*b^
3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 3*A*b*e - 5
*a*B*e)*(d + e*x)^(3/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((A*b - a*B)*(d + e*x)^(5/2))/(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 - 3*A*b*e - 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(7/2)*(b*d - a*e)^(5/2)*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.03483, size = 235, normalized size = 0.65 \[ \frac{(a+b x) \left (\frac{e^3 (-5 a B e-3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} \left (-3 e^2 (a+b x)^3 (5 a B e+3 A b e-8 b B d)+8 (a+b x) (b d-a e)^2 (-17 a B e+9 A b e+8 b B d)+2 e (a+b x)^2 (b d-a e) (-59 a B e+3 A b e+56 b B d)+48 (A b-a B) (b d-a e)^3\right )}{3 b^3 (a+b x)^4 (b d-a e)^2}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(3/2))/(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 + 9*A*b*e - 17*a*B*e)*(a + b*x) + 2*e*(b*d - a*e)*(56*b*B*d + 3*A*b*e - 59*
a*B*e)*(a + b*x)^2 - 3*e^2*(-8*b*B*d + 3*A*b*e + 5*a*B*e)*(a + b*x)^3))/(3*b^3*(
b*d - a*e)^2*(a + b*x)^4) + (e^3*(8*b*B*d - 3*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(7/2)*(b*d - a*e)^(5/2))))/(64*Sqrt[(a + b*x
)^2])

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Maple [B]  time = 0.039, size = 1273, 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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(b*x+a)/e*(90*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^
5-33*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3-33*A*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(3/2)*b^4*d^2*e-9*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+9*A*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e+54*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^
(1/2))*x^2*a^2*b^3*e^5-55*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3-24*B*arc
tan((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+113*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*
b^3*d*e+15*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*b^4*e^5-24*B*arct
an((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*d*e^4+36*A*arctan((e*x+d)^(1/2)*
b/(b*(a*e-b*d))^(1/2))*x^3*a*b^4*e^5+60*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(
1/2))*x^3*a^2*b^3*e^5+15*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^3*e+33*A*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-33*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^4
*d*e-231*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+66*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)
*a*b^3*d*e^2+27*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3-27*A*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2+69*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^
3*b*d*e^3-117*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2+87*B*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+198*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+9
*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*e^5+9*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-40*B*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d^2+9*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2
))*a^4*b*e^5+88*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^4*d^3-15*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+15*B*a
rctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*e^5+36*A*arctan((e*x+d)^(1/2)*b/(
b*(a*e-b*d))^(1/2))*x*a^3*b^2*e^5-73*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2*b^2
*e^2+60*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)^2/b^3/((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)*(e*x + d)^(3/2)/(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 = 0.303387, 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)^(3/2)/(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*(B*a^2*b^2 + 9*A*a*b^3)*d^2*e - 2*(7*
B*a^3*b - 3*A*a^2*b^2)*d*e^2 + 3*(5*B*a^4 + 3*A*a^3*b)*e^3 + 3*(8*B*b^4*d*e^2 -
(5*B*a*b^3 + 3*A*b^4)*e^3)*x^3 + (112*B*b^4*d^2*e - 2*(79*B*a*b^3 - 3*A*b^4)*d*e
^2 + (73*B*a^2*b^2 - 33*A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 - 8*(5*B*a*b^3 - 9*A*b
^4)*d^2*e - 4*(13*B*a^2*b^2 + 33*A*a*b^3)*d*e^2 + 11*(5*B*a^3*b + 3*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 - (5*B*a^5 + 3*A*a
^4*b)*e^4 + (8*B*b^5*d*e^3 - (5*B*a*b^4 + 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3
 - (5*B*a^2*b^3 + 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (5*B*a^3*b^2 + 3*
A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (5*B*a^4*b + 3*A*a^3*b^2)*e^4)*x)*l
og((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^2 - 2*a^5*b^4*d*e + a^6*b^3*e^2 + (b^9*d^2 - 2*a*b^8*d*
e + a^2*b^7*e^2)*x^4 + 4*(a*b^8*d^2 - 2*a^2*b^7*d*e + a^3*b^6*e^2)*x^3 + 6*(a^2*
b^7*d^2 - 2*a^3*b^6*d*e + a^4*b^5*e^2)*x^2 + 4*(a^3*b^6*d^2 - 2*a^4*b^5*d*e + a^
5*b^4*e^2)*x)*sqrt(b^2*d - a*b*e)), -1/192*((16*(B*a*b^3 + 3*A*b^4)*d^3 - 8*(B*a
^2*b^2 + 9*A*a*b^3)*d^2*e - 2*(7*B*a^3*b - 3*A*a^2*b^2)*d*e^2 + 3*(5*B*a^4 + 3*A
*a^3*b)*e^3 + 3*(8*B*b^4*d*e^2 - (5*B*a*b^3 + 3*A*b^4)*e^3)*x^3 + (112*B*b^4*d^2
*e - 2*(79*B*a*b^3 - 3*A*b^4)*d*e^2 + (73*B*a^2*b^2 - 33*A*a*b^3)*e^3)*x^2 + (64
*B*b^4*d^3 - 8*(5*B*a*b^3 - 9*A*b^4)*d^2*e - 4*(13*B*a^2*b^2 + 33*A*a*b^3)*d*e^2
 + 11*(5*B*a^3*b + 3*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 - (5*B*a^5 + 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (5*B*a*b^4 + 3*A*
b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (5*B*a^2*b^3 + 3*A*a*b^4)*e^4)*x^3 + 6*(8*B
*a^2*b^3*d*e^3 - (5*B*a^3*b^2 + 3*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (
5*B*a^4*b + 3*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^2 - 2*a^5*b^4*d*e + a^6*b^3*e^2 + (b^9*d^2 - 2*a*b^8*d*e
 + a^2*b^7*e^2)*x^4 + 4*(a*b^8*d^2 - 2*a^2*b^7*d*e + a^3*b^6*e^2)*x^3 + 6*(a^2*b
^7*d^2 - 2*a^3*b^6*d*e + a^4*b^5*e^2)*x^2 + 4*(a^3*b^6*d^2 - 2*a^4*b^5*d*e + a^5
*b^4*e^2)*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.368715, size = 973, normalized size = 2.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

1/64*(8*B*b*d*e^3 - 5*B*a*e^4 - 3*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d +
a*b*e))/((b^5*d^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 2*a*b^4*d*e*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) + a^2*b^3*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sq
rt(-b^2*d + a*b*e)) + 1/192*(24*(x*e + d)^(7/2)*B*b^4*d*e^3 + 40*(x*e + d)^(5/2)
*B*b^4*d^2*e^3 - 88*(x*e + d)^(3/2)*B*b^4*d^3*e^3 + 24*sqrt(x*e + d)*B*b^4*d^4*e
^3 - 15*(x*e + d)^(7/2)*B*a*b^3*e^4 - 9*(x*e + d)^(7/2)*A*b^4*e^4 - 113*(x*e + d
)^(5/2)*B*a*b^3*d*e^4 + 33*(x*e + d)^(5/2)*A*b^4*d*e^4 + 231*(x*e + d)^(3/2)*B*a
*b^3*d^2*e^4 + 33*(x*e + d)^(3/2)*A*b^4*d^2*e^4 - 87*sqrt(x*e + d)*B*a*b^3*d^3*e
^4 - 9*sqrt(x*e + d)*A*b^4*d^3*e^4 + 73*(x*e + d)^(5/2)*B*a^2*b^2*e^5 - 33*(x*e
+ d)^(5/2)*A*a*b^3*e^5 - 198*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 - 66*(x*e + d)^(3/2
)*A*a*b^3*d*e^5 + 117*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 + 27*sqrt(x*e + d)*A*a*b^3
*d^2*e^5 + 55*(x*e + d)^(3/2)*B*a^3*b*e^6 + 33*(x*e + d)^(3/2)*A*a^2*b^2*e^6 - 6
9*sqrt(x*e + d)*B*a^3*b*d*e^6 - 27*sqrt(x*e + d)*A*a^2*b^2*d*e^6 + 15*sqrt(x*e +
 d)*B*a^4*e^7 + 9*sqrt(x*e + d)*A*a^3*b*e^7)/((b^5*d^2*sign(-(x*e + d)*b*e + b*d
*e - a*e^2) - 2*a*b^4*d*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + a^2*b^3*e^2*sig
n(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^4)

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