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

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

[Out]

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

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Rubi [A]  time = 0.259302, 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 \[ -\frac{\sqrt{x} (3 a B+A b)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.116263, size = 105, normalized size = 0.66 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{x} \left (-3 a^2 B-a b (A+5 B x)+A b^2 x\right )+(a+b x)^2 (3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289061, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, B a^{2} + A a b +{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left (3 \, B a^{3} + A a^{2} b +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (3 \, B a^{2} + A a b +{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} +{\left (3 \, B a^{3} + A a^{2} b +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(x)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)*(A + B*x)/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.276463, size = 132, normalized size = 0.84 \[ \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b^{2}{\rm sign}\left (b x + a\right )} - \frac{5 \, B a b x^{\frac{3}{2}} - A b^{2} x^{\frac{3}{2}} + 3 \, B a^{2} \sqrt{x} + A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a b^{2}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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