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

Optimal. Leaf size=158 \[ \frac{\sqrt{x} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (a B+3 A b)}{4 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.25202, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{\sqrt{x} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (a B+3 A b)}{4 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((3*A*b + a*B)*Sqrt[x])/(4*a^2*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*S
qrt[x])/(2*a*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((3*A*b + a*B)*(a + b*
x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/2)*b^(3/2)*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)**(3/2)/x**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.111332, size = 106, normalized size = 0.67 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{x} \left (a^2 (-B)+a b (5 A+B x)+3 A b^2 x\right )+(a+b x)^2 (a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} b^{3/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 194, normalized size = 1.2 \[{\frac{bx+a}{4\,{a}^{2}b} \left ( 3\,A\sqrt{ab}{x}^{3/2}{b}^{2}+3\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{b}^{3}+B\sqrt{ab}{x}^{{\frac{3}{2}}}ab+B\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){x}^{2}a{b}^{2}+6\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) xa{b}^{2}+2\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{2}b+5\,A\sqrt{ab}\sqrt{x}ab+3\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{2}b-B\sqrt{ab}\sqrt{x}{a}^{2}+B\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^(1/2),x)

[Out]

1/4*(3*A*(a*b)^(1/2)*x^(3/2)*b^2+3*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*b^3+B*(a*
b)^(1/2)*x^(3/2)*a*b+B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a*b^2+6*A*arctan(x^(1/2
)*b/(a*b)^(1/2))*x*a*b^2+2*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^2*b+5*A*(a*b)^(1/
2)*x^(1/2)*a*b+3*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^2*b-B*(a*b)^(1/2)*x^(1/2)*a^2
+B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^3)*(b*x+a)/(a*b)^(1/2)/b/a^2/((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)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31294, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (B a^{2} - 5 \, A a b -{\left (B a b + 3 \, A b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left (B a^{3} + 3 \, A a^{2} b +{\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )} \sqrt{-a b}}, -\frac{{\left (B a^{2} - 5 \, A a b -{\left (B a b + 3 \, A b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} +{\left (B a^{3} + 3 \, A a^{2} b +{\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )} \sqrt{a b}}\right ] \]

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

[Out]

[-1/8*(2*(B*a^2 - 5*A*a*b - (B*a*b + 3*A*b^2)*x)*sqrt(-a*b)*sqrt(x) - (B*a^3 + 3
*A*a^2*b + (B*a*b^2 + 3*A*b^3)*x^2 + 2*(B*a^2*b + 3*A*a*b^2)*x)*log((2*a*b*sqrt(
x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a^2*b^3*x^2 + 2*a^3*b^2*x + a^4*b)*sqrt
(-a*b)), -1/4*((B*a^2 - 5*A*a*b - (B*a*b + 3*A*b^2)*x)*sqrt(a*b)*sqrt(x) + (B*a^
3 + 3*A*a^2*b + (B*a*b^2 + 3*A*b^3)*x^2 + 2*(B*a^2*b + 3*A*a*b^2)*x)*arctan(a/(s
qrt(a*b)*sqrt(x))))/((a^2*b^3*x^2 + 2*a^3*b^2*x + a^4*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)**(3/2)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278963, size = 132, normalized size = 0.84 \[ \frac{{\left (B a + 3 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{2} b{\rm sign}\left (b x + a\right )} + \frac{B a b x^{\frac{3}{2}} + 3 \, A b^{2} x^{\frac{3}{2}} - B a^{2} \sqrt{x} + 5 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{2} b{\rm sign}\left (b x + a\right )} \]

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)*sqrt(x)),x, algorithm="giac")

[Out]

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