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

Optimal. Leaf size=255 \[ \frac{7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{b} (a+b x) (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (7 A b-3 a B)}{4 a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 (a+b x) (7 A b-3 a B)}{12 a^3 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(7*A*b - 3*a*B)/(4*a^2*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(2
*a*b*x^(3/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*(7*A*b - 3*a*B)*(a +
b*x))/(12*a^3*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b - 3*a*B)*(a +
b*x))/(4*a^4*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*Sqrt[b]*(7*A*b - 3*a*B
)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])

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Rubi [A]  time = 0.360238, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{2 a b x^{3/2} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{b} (a+b x) (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (7 A b-3 a B)}{4 a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 (a+b x) (7 A b-3 a B)}{12 a^3 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(7*A*b - 3*a*B)/(4*a^2*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(2
*a*b*x^(3/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*(7*A*b - 3*a*B)*(a +
b*x))/(12*a^3*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b - 3*a*B)*(a +
b*x))/(4*a^4*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*Sqrt[b]*(7*A*b - 3*a*B
)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(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)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.14652, size = 136, normalized size = 0.53 \[ \frac{\sqrt{a} \left (-8 a^3 (A+3 B x)+a^2 b x (56 A-75 B x)+5 a b^2 x^2 (35 A-9 B x)+105 A b^3 x^3\right )+15 \sqrt{b} x^{3/2} (a+b x)^2 (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{12 a^{9/2} x^{3/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*(105*A*b^3*x^3 + a^2*b*x*(56*A - 75*B*x) + 5*a*b^2*x^2*(35*A - 9*B*x) -
8*a^3*(A + 3*B*x)) + 15*Sqrt[b]*(7*A*b - 3*a*B)*x^(3/2)*(a + b*x)^2*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/(12*a^(9/2)*x^(3/2)*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.014, size = 253, normalized size = 1. \[{\frac{bx+a}{12\,{a}^{4}} \left ( 105\,A\sqrt{ab}{x}^{3}{b}^{3}+105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{b}^{4}-45\,B\sqrt{ab}{x}^{3}a{b}^{2}-45\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}a{b}^{3}+210\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}a{b}^{3}-90\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{2}{b}^{2}+175\,A\sqrt{ab}{x}^{2}a{b}^{2}+105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{2}{b}^{2}-75\,B\sqrt{ab}{x}^{2}{a}^{2}b-45\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{3}b+56\,A\sqrt{ab}x{a}^{2}b-24\,B\sqrt{ab}x{a}^{3}-8\,A{a}^{3}\sqrt{ab} \right ){\frac{1}{\sqrt{ab}}}{x}^{-{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(105*A*(a*b)^(1/2)*x^3*b^3+105*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*b^4-
45*B*(a*b)^(1/2)*x^3*a*b^2-45*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*a*b^3+210*
A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a*b^3-90*B*arctan(x^(1/2)*b/(a*b)^(1/2))
*x^(5/2)*a^2*b^2+175*A*(a*b)^(1/2)*x^2*a*b^2+105*A*arctan(x^(1/2)*b/(a*b)^(1/2))
*x^(3/2)*a^2*b^2-75*B*(a*b)^(1/2)*x^2*a^2*b-45*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x
^(3/2)*a^3*b+56*A*(a*b)^(1/2)*x*a^2*b-24*B*(a*b)^(1/2)*x*a^3-8*A*a^3*(a*b)^(1/2)
)*(b*x+a)/(a*b)^(1/2)/x^(3/2)/a^4/((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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320828, size = 1, normalized size = 0. \[ \left [-\frac{16 \, A a^{3} + 30 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 50 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 16 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x}{24 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )} \sqrt{x}}, -\frac{8 \, A a^{3} + 15 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} - 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 8 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x}{12 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )} \sqrt{x}}\right ] \]

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)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(16*A*a^3 + 30*(3*B*a*b^2 - 7*A*b^3)*x^3 + 50*(3*B*a^2*b - 7*A*a*b^2)*x^2
+ 15*((3*B*a*b^2 - 7*A*b^3)*x^3 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^2 + (3*B*a^3 - 7*
A*a^2*b)*x)*sqrt(x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a))
+ 16*(3*B*a^3 - 7*A*a^2*b)*x)/((a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)*sqrt(x)), -1
/12*(8*A*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^2 - 1
5*((3*B*a*b^2 - 7*A*b^3)*x^3 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^2 + (3*B*a^3 - 7*A*a^
2*b)*x)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 8*(3*B*a^3 - 7*A*a^2
*b)*x)/((a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)*sqrt(x))]

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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)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279372, size = 178, normalized size = 0.7 \[ -\frac{5 \,{\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{4}{\rm sign}\left (b x + a\right )} - \frac{2 \,{\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right )} - \frac{7 \, B a b^{2} x^{\frac{3}{2}} - 11 \, A b^{3} x^{\frac{3}{2}} + 9 \, B a^{2} b \sqrt{x} - 13 \, A a b^{2} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{4}{\rm sign}\left (b x + a\right )} \]

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)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

-5/4*(3*B*a*b - 7*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*sign(b*x + a
)) - 2/3*(3*B*a*x - 9*A*b*x + A*a)/(a^4*x^(3/2)*sign(b*x + a)) - 1/4*(7*B*a*b^2*
x^(3/2) - 11*A*b^3*x^(3/2) + 9*B*a^2*b*sqrt(x) - 13*A*a*b^2*sqrt(x))/((b*x + a)^
2*a^4*sign(b*x + a))