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

Optimal. Leaf size=113 \[ -\frac{A b-2 a B}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.203442, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{A b-2 a B}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.5287, size = 138, normalized size = 1.22 \[ \frac{A x^{2} \left (2 a + 2 b x\right )}{4 a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{B x^{2} \left (2 a + 2 b x\right )}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{B x}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{B \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*x**2*(2*a + 2*b*x)/(4*a*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - B*x**2*(2*a + 2
*b*x)/(4*b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - B*x/(b**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2)) + B*(a + b*x)*log(a + b*x)/(b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.048367, size = 65, normalized size = 0.58 \[ \frac{3 a^2 B-a b (A-4 B x)+2 B (a+b x)^2 \log (a+b x)-2 A b^2 x}{2 b^3 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 83, normalized size = 0.7 \[ -{\frac{ \left ( -2\,B\ln \left ( bx+a \right ){x}^{2}{b}^{2}-4\,B\ln \left ( bx+a \right ) xab+2\,Ax{b}^{2}-2\,B\ln \left ( bx+a \right ){a}^{2}-4\,Bxab+abA-3\,{a}^{2}B \right ) \left ( bx+a \right ) }{2\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-2*B*ln(b*x+a)*x^2*b^2-4*B*ln(b*x+a)*x*a*b+2*A*x*b^2-2*B*ln(b*x+a)*a^2-4*B
*x*a*b+a*b*A-3*a^2*B)*(b*x+a)/b^3/((b*x+a)^2)^(3/2)

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Maxima [A]  time = 0.715938, size = 140, normalized size = 1.24 \[ \frac{B \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, B a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, B a b x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{A}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{A a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*log(x + a/b)/(b^2)^(3/2) + 3/2*B*a^2*b^2/((b^2)^(7/2)*(x + a/b)^2) + 2*B*a*b*x
/((b^2)^(5/2)*(x + a/b)^2) - A/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) + 1/2*A*a/((b
^2)^(3/2)*b*(x + a/b)^2)

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Fricas [A]  time = 0.284248, size = 109, normalized size = 0.96 \[ \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x + 2 \,{\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.579041, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x