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

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

[Out]

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

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Rubi [A]  time = 0.145889, 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 \[ \frac{(5 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{3/2} b^{7/2}}-\frac{\sqrt{x} (5 a B+A b)}{8 a b^3 (a+b x)}-\frac{x^{3/2} (5 a B+A b)}{12 a b^2 (a+b x)^2}+\frac{x^{5/2} (A b-a B)}{3 a b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.8654, size = 112, normalized size = 0.86 \[ \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{3}} - \frac{x^{\frac{3}{2}} \left (A b + 5 B a\right )}{12 a b^{2} \left (a + b x\right )^{2}} - \frac{\sqrt{x} \left (A b + 5 B a\right )}{8 a b^{3} \left (a + b x\right )} + \frac{\left (A b + 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.277858, size = 105, normalized size = 0.81 \[ \frac{(5 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{3/2} b^{7/2}}-\frac{\sqrt{x} \left (15 a^3 B+a^2 b (3 A+40 B x)+a b^2 x (8 A+33 B x)-3 A b^3 x^2\right )}{24 a b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 111, normalized size = 0.9 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{3}} \left ( 1/16\,{\frac{ \left ( Ab-11\,Ba \right ){x}^{5/2}}{ab}}-1/6\,{\frac{ \left ( Ab+5\,Ba \right ){x}^{3/2}}{{b}^{2}}}-1/16\,{\frac{ \left ( Ab+5\,Ba \right ) a\sqrt{x}}{{b}^{3}}} \right ) }+{\frac{A}{8\,a{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,B}{8\,{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.323948, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, B a^{3} + 3 \, A a^{2} b + 3 \,{\left (11 \, B a b^{2} - A b^{3}\right )} x^{2} + 8 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 3 \,{\left (5 \, B a^{4} + A a^{3} b +{\left (5 \, B a b^{3} + A b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \,{\left (5 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{48 \,{\left (a b^{6} x^{3} + 3 \, a^{2} b^{5} x^{2} + 3 \, a^{3} b^{4} x + a^{4} b^{3}\right )} \sqrt{-a b}}, -\frac{{\left (15 \, B a^{3} + 3 \, A a^{2} b + 3 \,{\left (11 \, B a b^{2} - A b^{3}\right )} x^{2} + 8 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 3 \,{\left (5 \, B a^{4} + A a^{3} b +{\left (5 \, B a b^{3} + A b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 3 \,{\left (5 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{24 \,{\left (a b^{6} x^{3} + 3 \, a^{2} b^{5} x^{2} + 3 \, a^{3} b^{4} x + a^{4} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(2*(15*B*a^3 + 3*A*a^2*b + 3*(11*B*a*b^2 - A*b^3)*x^2 + 8*(5*B*a^2*b + A*
a*b^2)*x)*sqrt(-a*b)*sqrt(x) - 3*(5*B*a^4 + A*a^3*b + (5*B*a*b^3 + A*b^4)*x^3 +
3*(5*B*a^2*b^2 + A*a*b^3)*x^2 + 3*(5*B*a^3*b + A*a^2*b^2)*x)*log((2*a*b*sqrt(x)
+ sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a*b^6*x^3 + 3*a^2*b^5*x^2 + 3*a^3*b^4*x +
a^4*b^3)*sqrt(-a*b)), -1/24*((15*B*a^3 + 3*A*a^2*b + 3*(11*B*a*b^2 - A*b^3)*x^2
+ 8*(5*B*a^2*b + A*a*b^2)*x)*sqrt(a*b)*sqrt(x) + 3*(5*B*a^4 + A*a^3*b + (5*B*a*b
^3 + A*b^4)*x^3 + 3*(5*B*a^2*b^2 + A*a*b^3)*x^2 + 3*(5*B*a^3*b + A*a^2*b^2)*x)*a
rctan(a/(sqrt(a*b)*sqrt(x))))/((a*b^6*x^3 + 3*a^2*b^5*x^2 + 3*a^3*b^4*x + a^4*b^
3)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\left (a + b x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**(3/2)*(A + B*x)/(a + b*x)**4, x)

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GIAC/XCAS [A]  time = 0.271376, size = 144, normalized size = 1.11 \[ \frac{{\left (5 \, B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{3}} - \frac{33 \, B a b^{2} x^{\frac{5}{2}} - 3 \, A b^{3} x^{\frac{5}{2}} + 40 \, B a^{2} b x^{\frac{3}{2}} + 8 \, A a b^{2} x^{\frac{3}{2}} + 15 \, B a^{3} \sqrt{x} + 3 \, A a^{2} b \sqrt{x}}{24 \,{\left (b x + a\right )}^{3} a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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