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

Optimal. Leaf size=185 \[ \frac{3 (a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{7/2} b^{7/2}}+\frac{3 \sqrt{x} (a B+A b)}{128 a^3 b^3 (a+b x)}+\frac{\sqrt{x} (a B+A b)}{64 a^2 b^3 (a+b x)^2}-\frac{\sqrt{x} (a B+A b)}{16 a b^3 (a+b x)^3}-\frac{x^{3/2} (a B+A b)}{8 a b^2 (a+b x)^4}+\frac{x^{5/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

((A*b - a*B)*x^(5/2))/(5*a*b*(a + b*x)^5) - ((A*b + a*B)*x^(3/2))/(8*a*b^2*(a +
b*x)^4) - ((A*b + a*B)*Sqrt[x])/(16*a*b^3*(a + b*x)^3) + ((A*b + a*B)*Sqrt[x])/(
64*a^2*b^3*(a + b*x)^2) + (3*(A*b + a*B)*Sqrt[x])/(128*a^3*b^3*(a + b*x)) + (3*(
A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(7/2)*b^(7/2))

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Rubi [A]  time = 0.218277, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{3 (a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{7/2} b^{7/2}}+\frac{3 \sqrt{x} (a B+A b)}{128 a^3 b^3 (a+b x)}+\frac{\sqrt{x} (a B+A b)}{64 a^2 b^3 (a+b x)^2}-\frac{\sqrt{x} (a B+A b)}{16 a b^3 (a+b x)^3}-\frac{x^{3/2} (a B+A b)}{8 a b^2 (a+b x)^4}+\frac{x^{5/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^(5/2))/(5*a*b*(a + b*x)^5) - ((A*b + a*B)*x^(3/2))/(8*a*b^2*(a +
b*x)^4) - ((A*b + a*B)*Sqrt[x])/(16*a*b^3*(a + b*x)^3) + ((A*b + a*B)*Sqrt[x])/(
64*a^2*b^3*(a + b*x)^2) + (3*(A*b + a*B)*Sqrt[x])/(128*a^3*b^3*(a + b*x)) + (3*(
A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(7/2)*b^(7/2))

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Rubi in Sympy [A]  time = 55.7531, size = 165, normalized size = 0.89 \[ \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} - \frac{x^{\frac{3}{2}} \left (A b + B a\right )}{8 a b^{2} \left (a + b x\right )^{4}} - \frac{\sqrt{x} \left (A b + B a\right )}{16 a b^{3} \left (a + b x\right )^{3}} + \frac{\sqrt{x} \left (A b + B a\right )}{64 a^{2} b^{3} \left (a + b x\right )^{2}} + \frac{3 \sqrt{x} \left (A b + B a\right )}{128 a^{3} b^{3} \left (a + b x\right )} + \frac{3 \left (A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 a^{\frac{7}{2}} b^{\frac{7}{2}}} \]

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)**3,x)

[Out]

x**(5/2)*(A*b - B*a)/(5*a*b*(a + b*x)**5) - x**(3/2)*(A*b + B*a)/(8*a*b**2*(a +
b*x)**4) - sqrt(x)*(A*b + B*a)/(16*a*b**3*(a + b*x)**3) + sqrt(x)*(A*b + B*a)/(6
4*a**2*b**3*(a + b*x)**2) + 3*sqrt(x)*(A*b + B*a)/(128*a**3*b**3*(a + b*x)) + 3*
(A*b + B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(128*a**(7/2)*b**(7/2))

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Mathematica [A]  time = 0.329201, size = 145, normalized size = 0.78 \[ \frac{\frac{15 (a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{\sqrt{b} \sqrt{x} \left (-15 a^5 B-5 a^4 b (3 A+14 B x)-2 a^3 b^2 x (35 A+64 B x)+2 a^2 b^3 x^2 (64 A+35 B x)+5 a b^4 x^3 (14 A+3 B x)+15 A b^5 x^4\right )}{a^3 (a+b x)^5}}{640 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*Sqrt[x]*(-15*a^5*B + 15*A*b^5*x^4 + 5*a*b^4*x^3*(14*A + 3*B*x) - 5*a^4
*b*(3*A + 14*B*x) + 2*a^2*b^3*x^2*(64*A + 35*B*x) - 2*a^3*b^2*x*(35*A + 64*B*x))
)/(a^3*(a + b*x)^5) + (15*(A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)
)/(640*b^(7/2))

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Maple [A]  time = 0.027, size = 143, normalized size = 0.8 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{5}} \left ({\frac{ \left ( 3\,Ab+3\,Ba \right ) b{x}^{9/2}}{256\,{a}^{3}}}+{\frac{ \left ( 7\,Ab+7\,Ba \right ){x}^{7/2}}{128\,{a}^{2}}}+1/10\,{\frac{ \left ( Ab-Ba \right ){x}^{5/2}}{ab}}-{\frac{ \left ( 7\,Ab+7\,Ba \right ){x}^{3/2}}{128\,{b}^{2}}}-{\frac{ \left ( 3\,Ab+3\,Ba \right ) a\sqrt{x}}{256\,{b}^{3}}} \right ) }+{\frac{3\,A}{128\,{a}^{3}{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{128\,{a}^{2}{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)^3,x)

[Out]

2*(3/256*(A*b+B*a)/a^3*b*x^(9/2)+7/128*(A*b+B*a)/a^2*x^(7/2)+1/10*(A*b-B*a)/a/b*
x^(5/2)-7/128*(A*b+B*a)/b^2*x^(3/2)-3/256*(A*b+B*a)*a/b^3*x^(1/2))/(b*x+a)^5+3/1
28/a^3/b^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+3/128/a^2/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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.329091, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \, B a^{5} + 15 \, A a^{4} b - 15 \,{\left (B a b^{4} + A b^{5}\right )} x^{4} - 70 \,{\left (B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 128 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 70 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 15 \,{\left (B a^{6} + A a^{5} b +{\left (B a b^{5} + A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b + A a^{4} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{1280 \,{\left (a^{3} b^{8} x^{5} + 5 \, a^{4} b^{7} x^{4} + 10 \, a^{5} b^{6} x^{3} + 10 \, a^{6} b^{5} x^{2} + 5 \, a^{7} b^{4} x + a^{8} b^{3}\right )} \sqrt{-a b}}, -\frac{{\left (15 \, B a^{5} + 15 \, A a^{4} b - 15 \,{\left (B a b^{4} + A b^{5}\right )} x^{4} - 70 \,{\left (B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 128 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 70 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 15 \,{\left (B a^{6} + A a^{5} b +{\left (B a b^{5} + A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b + A a^{4} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{640 \,{\left (a^{3} b^{8} x^{5} + 5 \, a^{4} b^{7} x^{4} + 10 \, a^{5} b^{6} x^{3} + 10 \, a^{6} b^{5} x^{2} + 5 \, a^{7} b^{4} x + a^{8} b^{3}\right )} \sqrt{a b}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[-1/1280*(2*(15*B*a^5 + 15*A*a^4*b - 15*(B*a*b^4 + A*b^5)*x^4 - 70*(B*a^2*b^3 +
A*a*b^4)*x^3 + 128*(B*a^3*b^2 - A*a^2*b^3)*x^2 + 70*(B*a^4*b + A*a^3*b^2)*x)*sqr
t(-a*b)*sqrt(x) - 15*(B*a^6 + A*a^5*b + (B*a*b^5 + A*b^6)*x^5 + 5*(B*a^2*b^4 + A
*a*b^5)*x^4 + 10*(B*a^3*b^3 + A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 + A*a^3*b^3)*x^2 +
5*(B*a^5*b + A*a^4*b^2)*x)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a))
)/((a^3*b^8*x^5 + 5*a^4*b^7*x^4 + 10*a^5*b^6*x^3 + 10*a^6*b^5*x^2 + 5*a^7*b^4*x
+ a^8*b^3)*sqrt(-a*b)), -1/640*((15*B*a^5 + 15*A*a^4*b - 15*(B*a*b^4 + A*b^5)*x^
4 - 70*(B*a^2*b^3 + A*a*b^4)*x^3 + 128*(B*a^3*b^2 - A*a^2*b^3)*x^2 + 70*(B*a^4*b
+ A*a^3*b^2)*x)*sqrt(a*b)*sqrt(x) + 15*(B*a^6 + A*a^5*b + (B*a*b^5 + A*b^6)*x^5
+ 5*(B*a^2*b^4 + A*a*b^5)*x^4 + 10*(B*a^3*b^3 + A*a^2*b^4)*x^3 + 10*(B*a^4*b^2
+ A*a^3*b^3)*x^2 + 5*(B*a^5*b + A*a^4*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((a
^3*b^8*x^5 + 5*a^4*b^7*x^4 + 10*a^5*b^6*x^3 + 10*a^6*b^5*x^2 + 5*a^7*b^4*x + a^8
*b^3)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\left (a + b x\right )^{6}}\, dx \]

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)**3,x)

[Out]

Integral(x**(3/2)*(A + B*x)/(a + b*x)**6, x)

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GIAC/XCAS [A]  time = 0.275249, size = 208, normalized size = 1.12 \[ \frac{3 \,{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{3} b^{3}} + \frac{15 \, B a b^{4} x^{\frac{9}{2}} + 15 \, A b^{5} x^{\frac{9}{2}} + 70 \, B a^{2} b^{3} x^{\frac{7}{2}} + 70 \, A a b^{4} x^{\frac{7}{2}} - 128 \, B a^{3} b^{2} x^{\frac{5}{2}} + 128 \, A a^{2} b^{3} x^{\frac{5}{2}} - 70 \, B a^{4} b x^{\frac{3}{2}} - 70 \, A a^{3} b^{2} x^{\frac{3}{2}} - 15 \, B a^{5} \sqrt{x} - 15 \, A a^{4} b \sqrt{x}}{640 \,{\left (b x + a\right )}^{5} a^{3} b^{3}} \]

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)^3,x, algorithm="giac")

[Out]

3/128*(B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3*b^3) + 1/640*(15*B*
a*b^4*x^(9/2) + 15*A*b^5*x^(9/2) + 70*B*a^2*b^3*x^(7/2) + 70*A*a*b^4*x^(7/2) - 1
28*B*a^3*b^2*x^(5/2) + 128*A*a^2*b^3*x^(5/2) - 70*B*a^4*b*x^(3/2) - 70*A*a^3*b^2
*x^(3/2) - 15*B*a^5*sqrt(x) - 15*A*a^4*b*sqrt(x))/((b*x + a)^5*a^3*b^3)