3.770 \(\int \frac{A+B x}{\sqrt{x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=130 \[ \frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{7/2} b^{3/2}}+\frac{\sqrt{x} (a B+5 A b)}{8 a^3 b (a+b x)}+\frac{\sqrt{x} (a B+5 A b)}{12 a^2 b (a+b x)^2}+\frac{\sqrt{x} (A b-a B)}{3 a b (a+b x)^3} \]

[Out]

((A*b - a*B)*Sqrt[x])/(3*a*b*(a + b*x)^3) + ((5*A*b + a*B)*Sqrt[x])/(12*a^2*b*(a
+ b*x)^2) + ((5*A*b + a*B)*Sqrt[x])/(8*a^3*b*(a + b*x)) + ((5*A*b + a*B)*ArcTan
[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(7/2)*b^(3/2))

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Rubi [A]  time = 0.141448, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{7/2} b^{3/2}}+\frac{\sqrt{x} (a B+5 A b)}{8 a^3 b (a+b x)}+\frac{\sqrt{x} (a B+5 A b)}{12 a^2 b (a+b x)^2}+\frac{\sqrt{x} (A b-a B)}{3 a b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*Sqrt[x])/(3*a*b*(a + b*x)^3) + ((5*A*b + a*B)*Sqrt[x])/(12*a^2*b*(a
+ b*x)^2) + ((5*A*b + a*B)*Sqrt[x])/(8*a^3*b*(a + b*x)) + ((5*A*b + a*B)*ArcTan
[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(7/2)*b^(3/2))

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Rubi in Sympy [A]  time = 37.6979, size = 112, normalized size = 0.86 \[ \frac{\sqrt{x} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{3}} + \frac{\sqrt{x} \left (5 A b + B a\right )}{12 a^{2} b \left (a + b x\right )^{2}} + \frac{\sqrt{x} \left (5 A b + B a\right )}{8 a^{3} b \left (a + b x\right )} + \frac{\left (5 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}} b^{\frac{3}{2}}} \]

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)**2/x**(1/2),x)

[Out]

sqrt(x)*(A*b - B*a)/(3*a*b*(a + b*x)**3) + sqrt(x)*(5*A*b + B*a)/(12*a**2*b*(a +
b*x)**2) + sqrt(x)*(5*A*b + B*a)/(8*a**3*b*(a + b*x)) + (5*A*b + B*a)*atan(sqrt
(b)*sqrt(x)/sqrt(a))/(8*a**(7/2)*b**(3/2))

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Mathematica [A]  time = 0.182444, size = 105, normalized size = 0.81 \[ \frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{7/2} b^{3/2}}+\frac{\sqrt{x} \left (-3 a^3 B+a^2 b (33 A+8 B x)+a b^2 x (40 A+3 B x)+15 A b^3 x^2\right )}{24 a^3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-3*a^3*B + 15*A*b^3*x^2 + a*b^2*x*(40*A + 3*B*x) + a^2*b*(33*A + 8*B*x
)))/(24*a^3*b*(a + b*x)^3) + ((5*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(
8*a^(7/2)*b^(3/2))

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Maple [A]  time = 0.022, size = 112, normalized size = 0.9 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{3}} \left ( 1/16\,{\frac{ \left ( 5\,Ab+Ba \right ) b{x}^{5/2}}{{a}^{3}}}+1/6\,{\frac{ \left ( 5\,Ab+Ba \right ){x}^{3/2}}{{a}^{2}}}+1/16\,{\frac{ \left ( 11\,Ab-Ba \right ) \sqrt{x}}{ab}} \right ) }+{\frac{5\,A}{8\,{a}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{8\,{a}^{2}b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)^2/x^(1/2),x)

[Out]

2*(1/16*(5*A*b+B*a)/a^3*b*x^(5/2)+1/6/a^2*(5*A*b+B*a)*x^(3/2)+1/16*(11*A*b-B*a)/
a/b*x^(1/2))/(b*x+a)^3+5/8/a^3/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+1/8/a
^2/b/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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)^2*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310041, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, B a^{3} - 33 \, A a^{2} b - 3 \,{\left (B a b^{2} + 5 \, A b^{3}\right )} x^{2} - 8 \,{\left (B a^{2} b + 5 \, A a b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 3 \,{\left (B a^{4} + 5 \, A a^{3} b +{\left (B a b^{3} + 5 \, A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{2} + 3 \,{\left (B a^{3} b + 5 \, A a^{2} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{48 \,{\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )} \sqrt{-a b}}, -\frac{{\left (3 \, B a^{3} - 33 \, A a^{2} b - 3 \,{\left (B a b^{2} + 5 \, A b^{3}\right )} x^{2} - 8 \,{\left (B a^{2} b + 5 \, A a b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 3 \,{\left (B a^{4} + 5 \, A a^{3} b +{\left (B a b^{3} + 5 \, A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{2} + 3 \,{\left (B a^{3} b + 5 \, A a^{2} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{24 \,{\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )} \sqrt{a b}}\right ] \]

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)^2*sqrt(x)),x, algorithm="fricas")

[Out]

[-1/48*(2*(3*B*a^3 - 33*A*a^2*b - 3*(B*a*b^2 + 5*A*b^3)*x^2 - 8*(B*a^2*b + 5*A*a
*b^2)*x)*sqrt(-a*b)*sqrt(x) - 3*(B*a^4 + 5*A*a^3*b + (B*a*b^3 + 5*A*b^4)*x^3 + 3
*(B*a^2*b^2 + 5*A*a*b^3)*x^2 + 3*(B*a^3*b + 5*A*a^2*b^2)*x)*log((2*a*b*sqrt(x) +
sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a^3*b^4*x^3 + 3*a^4*b^3*x^2 + 3*a^5*b^2*x +
a^6*b)*sqrt(-a*b)), -1/24*((3*B*a^3 - 33*A*a^2*b - 3*(B*a*b^2 + 5*A*b^3)*x^2 -
8*(B*a^2*b + 5*A*a*b^2)*x)*sqrt(a*b)*sqrt(x) + 3*(B*a^4 + 5*A*a^3*b + (B*a*b^3 +
5*A*b^4)*x^3 + 3*(B*a^2*b^2 + 5*A*a*b^3)*x^2 + 3*(B*a^3*b + 5*A*a^2*b^2)*x)*arc
tan(a/(sqrt(a*b)*sqrt(x))))/((a^3*b^4*x^3 + 3*a^4*b^3*x^2 + 3*a^5*b^2*x + a^6*b)
*sqrt(a*b))]

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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)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277519, size = 144, normalized size = 1.11 \[ \frac{{\left (B a + 5 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3} b} + \frac{3 \, B a b^{2} x^{\frac{5}{2}} + 15 \, A b^{3} x^{\frac{5}{2}} + 8 \, B a^{2} b x^{\frac{3}{2}} + 40 \, A a b^{2} x^{\frac{3}{2}} - 3 \, B a^{3} \sqrt{x} + 33 \, A a^{2} b \sqrt{x}}{24 \,{\left (b x + a\right )}^{3} a^{3} b} \]

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)^2*sqrt(x)),x, algorithm="giac")

[Out]

1/8*(B*a + 5*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3*b) + 1/24*(3*B*a*b^
2*x^(5/2) + 15*A*b^3*x^(5/2) + 8*B*a^2*b*x^(3/2) + 40*A*a*b^2*x^(3/2) - 3*B*a^3*
sqrt(x) + 33*A*a^2*b*sqrt(x))/((b*x + a)^3*a^3*b)