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

Optimal. Leaf size=121 \[ -\frac{A b-2 a B}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{B}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.215449, antiderivative size = 121, 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 \[ -\frac{A b-2 a B}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a (A b-a B)}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{B}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.0522, size = 110, normalized size = 0.91 \[ - \frac{B x^{2} \left (2 a + 2 b x\right )}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{a \left (2 a + 2 b x\right ) \left (A b + B a\right )}{8 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{A b + B a}{3 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

-B*x**2*(2*a + 2*b*x)/(4*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) + a*(2*a + 2*b*x
)*(A*b + B*a)/(8*b**3*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) - (A*b + B*a)/(3*b**3
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2))

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Mathematica [A]  time = 0.0474535, size = 56, normalized size = 0.46 \[ \frac{a^2 (-B)-a b (A+4 B x)-2 b^2 x (2 A+3 B x)}{12 b^3 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 52, normalized size = 0.4 \[ -{\frac{ \left ( bx+a \right ) \left ( 6\,{b}^{2}B{x}^{2}+4\,Ax{b}^{2}+4\,Bxab+abA+{a}^{2}B \right ) }{12\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

-1/12*(b*x+a)/b^3*(6*B*b^2*x^2+4*A*b^2*x+4*B*a*b*x+A*a*b+B*a^2)/((b*x+a)^2)^(5/2
)

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Maxima [A]  time = 0.691039, size = 142, normalized size = 1.17 \[ -\frac{A}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{B a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, B a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{B}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{A a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

-1/3*A/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 1/4*B*a^2*b^2/((b^2)^(9/2)*(x + a
/b)^4) + 2/3*B*a*b/((b^2)^(7/2)*(x + a/b)^3) - 1/2*B/((b^2)^(5/2)*(x + a/b)^2) +
1/4*A*a/((b^2)^(5/2)*b*(x + a/b)^4)

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Fricas [A]  time = 0.290388, size = 108, normalized size = 0.89 \[ -\frac{6 \, B b^{2} x^{2} + B a^{2} + A a b + 4 \,{\left (B a b + A b^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

-1/12*(6*B*b^2*x^2 + B*a^2 + A*a*b + 4*(B*a*b + A*b^2)*x)/(b^7*x^4 + 4*a*b^6*x^3
+ 6*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral(x*(A + B*x)/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.584095, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

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)^(5/2),x, algorithm="giac")

[Out]

sage0*x