3.1880 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ -\frac{5 e^2 \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{5 e \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{5 e^3 (a+b x) (-a B e-7 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

[Out]

(-5*e^2*(8*b*B*d - 7*A*b*e - a*B*e)*Sqrt[d + e*x])/(64*b*(b*d - a*e)^4*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*Sqrt[d + e*x])/(4*b*(b*d - a*e)*(a + b*x)^3
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 7*A*b*e - a*B*e)*Sqrt[d + e*x])/(2
4*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e*(8*b*B*d - 7
*A*b*e - a*B*e)*Sqrt[d + e*x])/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (5*e^3*(8*b*B*d - 7*A*b*e - a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt
[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])

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Rubi [A]  time = 0.927328, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{5 e^2 \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{5 e \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{5 e^3 (a+b x) (-a B e-7 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

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]

(-5*e^2*(8*b*B*d - 7*A*b*e - a*B*e)*Sqrt[d + e*x])/(64*b*(b*d - a*e)^4*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*Sqrt[d + e*x])/(4*b*(b*d - a*e)*(a + b*x)^3
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 7*A*b*e - a*B*e)*Sqrt[d + e*x])/(2
4*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e*(8*b*B*d - 7
*A*b*e - a*B*e)*Sqrt[d + e*x])/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (5*e^3*(8*b*B*d - 7*A*b*e - a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt
[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(9/2)*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)/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.2286, size = 233, normalized size = 0.65 \[ \frac{(a+b x) \left (-\frac{5 e^3 (a B e+7 A b e-8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} (b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (-15 e^2 (a+b x)^3 (a B e+7 A b e-8 b B d)+8 (a+b x) (b d-a e)^2 (-a B e-7 A b e+8 b B d)-10 e (a+b x)^2 (a e-b d) (a B e+7 A b e-8 b B d)+48 (A b-a B) (b d-a e)^3\right )}{3 b (a+b x)^4 (b d-a e)^4}\right )}{64 \sqrt{(a+b x)^2}} \]

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 - 7*A*b*e - a*B*e)*(a + b*x) - 10*e*(-(b*d) + a*e)*(-8*b*B*d + 7*A*b*e + a*
B*e)*(a + b*x)^2 - 15*e^2*(-8*b*B*d + 7*A*b*e + a*B*e)*(a + b*x)^3))/(3*b*(b*d -
a*e)^4*(a + b*x)^4) - (5*e^3*(-8*b*B*d + 7*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt
[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*(b*d - a*e)^(9/2))))/(64*Sqrt[(a + b*x)^2]
)

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Maple [B]  time = 0.042, 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)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x)

[Out]

1/192*(90*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^5+511*A*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3+511*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3
/2)*b^4*d^2*e+279*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4-279*A*(b*(a*e-b*
d))^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e+630*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/
2))*x^2*a^2*b^3*e^5+73*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3-120*B*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b*d*e^4-480*B*arctan((e*x+d)^(1/2)*b/
(b*(a*e-b*d))^(1/2))*x^3*a*b^4*d*e^4-495*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^5-120*B*arct
an((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*d*e^4+420*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+385*A*(b*(
a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-385*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*
b^4*d*e+1241*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e-480*B*arctan((e*x+d
)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^4-1022*A*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(3/2)*a*b^3*d*e^2-837*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3+837*A*
(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2-219*B*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(1/2)*a^3*b*d*e^3+747*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2-777*
B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e-730*B*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(3/2)*a^2*b^2*d*e^2-720*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*
b^3*d*e^4+105*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*e^5+105*A*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^4*e-120*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^
4*d+440*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d^2+105*A*arctan((e*x+d)^(1/2)*b
/(b*(a*e-b*d))^(1/2))*a^4*b*e^5-584*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+264*B*(b*(a*e-b*d))^(1/2)*(e*x+d)
^(1/2)*b^4*d^4+15*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*e^5+420*A*ar
ctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*e^5+55*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)/e*(b*x+a)/(b*(a*e-b*d))^(1/2)/(a*e-b*d)^2/b/(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)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.318082, 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)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[-1/384*(2*(16*(B*a*b^3 + 3*A*b^4)*d^3 - 8*(9*B*a^2*b^2 + 25*A*a*b^3)*d^2*e + 2*
(73*B*a^3*b + 163*A*a^2*b^2)*d*e^2 + 3*(5*B*a^4 - 93*A*a^3*b)*e^3 + 15*(8*B*b^4*
d*e^2 - (B*a*b^3 + 7*A*b^4)*e^3)*x^3 - 5*(16*B*b^4*d^2*e - 2*(45*B*a*b^3 + 7*A*b
^4)*d*e^2 + 11*(B*a^2*b^2 + 7*A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 - 8*(37*B*a*b^3
+ 7*A*b^4)*d^2*e + 4*(155*B*a^2*b^2 + 63*A*a*b^3)*d*e^2 - 73*(B*a^3*b + 7*A*a^2*
b^2)*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15*(8*B*a^4*b*d*e^3 - (B*a^5 +
7*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (B*a*b^4 + 7*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*
e^3 - (B*a^2*b^3 + 7*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (B*a^3*b^2 + 7*A
*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (B*a^4*b + 7*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^4 - 4*a^5*b^4*d^3*e + 6*a^6*b^3*d^2*e^2 - 4*a^7*b^2*d*e^3
+ a^8*b*e^4 + (b^9*d^4 - 4*a*b^8*d^3*e + 6*a^2*b^7*d^2*e^2 - 4*a^3*b^6*d*e^3 + a
^4*b^5*e^4)*x^4 + 4*(a*b^8*d^4 - 4*a^2*b^7*d^3*e + 6*a^3*b^6*d^2*e^2 - 4*a^4*b^5
*d*e^3 + a^5*b^4*e^4)*x^3 + 6*(a^2*b^7*d^4 - 4*a^3*b^6*d^3*e + 6*a^4*b^5*d^2*e^2
- 4*a^5*b^4*d*e^3 + a^6*b^3*e^4)*x^2 + 4*(a^3*b^6*d^4 - 4*a^4*b^5*d^3*e + 6*a^5
*b^4*d^2*e^2 - 4*a^6*b^3*d*e^3 + a^7*b^2*e^4)*x)*sqrt(b^2*d - a*b*e)), -1/192*((
16*(B*a*b^3 + 3*A*b^4)*d^3 - 8*(9*B*a^2*b^2 + 25*A*a*b^3)*d^2*e + 2*(73*B*a^3*b
+ 163*A*a^2*b^2)*d*e^2 + 3*(5*B*a^4 - 93*A*a^3*b)*e^3 + 15*(8*B*b^4*d*e^2 - (B*a
*b^3 + 7*A*b^4)*e^3)*x^3 - 5*(16*B*b^4*d^2*e - 2*(45*B*a*b^3 + 7*A*b^4)*d*e^2 +
11*(B*a^2*b^2 + 7*A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 - 8*(37*B*a*b^3 + 7*A*b^4)*d
^2*e + 4*(155*B*a^2*b^2 + 63*A*a*b^3)*d*e^2 - 73*(B*a^3*b + 7*A*a^2*b^2)*e^3)*x)
*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 15*(8*B*a^4*b*d*e^3 - (B*a^5 + 7*A*a^4*b)*
e^4 + (8*B*b^5*d*e^3 - (B*a*b^4 + 7*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (B*a^
2*b^3 + 7*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (B*a^3*b^2 + 7*A*a^2*b^3)*e
^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (B*a^4*b + 7*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^4 - 4*a^5*b^4*d^3*e + 6*
a^6*b^3*d^2*e^2 - 4*a^7*b^2*d*e^3 + a^8*b*e^4 + (b^9*d^4 - 4*a*b^8*d^3*e + 6*a^2
*b^7*d^2*e^2 - 4*a^3*b^6*d*e^3 + a^4*b^5*e^4)*x^4 + 4*(a*b^8*d^4 - 4*a^2*b^7*d^3
*e + 6*a^3*b^6*d^2*e^2 - 4*a^4*b^5*d*e^3 + a^5*b^4*e^4)*x^3 + 6*(a^2*b^7*d^4 - 4
*a^3*b^6*d^3*e + 6*a^4*b^5*d^2*e^2 - 4*a^5*b^4*d*e^3 + a^6*b^3*e^4)*x^2 + 4*(a^3
*b^6*d^4 - 4*a^4*b^5*d^3*e + 6*a^5*b^4*d^2*e^2 - 4*a^6*b^3*d*e^3 + a^7*b^2*e^4)*
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)/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.337184, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done