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

Optimal. Leaf size=195 \[ \frac{(3 a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{9/2} b^{5/2}}+\frac{\sqrt{x} (3 a B+7 A b)}{128 a^4 b^2 (a+b x)}+\frac{\sqrt{x} (3 a B+7 A b)}{192 a^3 b^2 (a+b x)^2}+\frac{\sqrt{x} (3 a B+7 A b)}{240 a^2 b^2 (a+b x)^3}-\frac{\sqrt{x} (3 a B+7 A b)}{40 a b^2 (a+b x)^4}+\frac{x^{3/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

((A*b - a*B)*x^(3/2))/(5*a*b*(a + b*x)^5) - ((7*A*b + 3*a*B)*Sqrt[x])/(40*a*b^2*
(a + b*x)^4) + ((7*A*b + 3*a*B)*Sqrt[x])/(240*a^2*b^2*(a + b*x)^3) + ((7*A*b + 3
*a*B)*Sqrt[x])/(192*a^3*b^2*(a + b*x)^2) + ((7*A*b + 3*a*B)*Sqrt[x])/(128*a^4*b^
2*(a + b*x)) + ((7*A*b + 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(9/2)*
b^(5/2))

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Rubi [A]  time = 0.221699, antiderivative size = 195, 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 \[ \frac{(3 a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{9/2} b^{5/2}}+\frac{\sqrt{x} (3 a B+7 A b)}{128 a^4 b^2 (a+b x)}+\frac{\sqrt{x} (3 a B+7 A b)}{192 a^3 b^2 (a+b x)^2}+\frac{\sqrt{x} (3 a B+7 A b)}{240 a^2 b^2 (a+b x)^3}-\frac{\sqrt{x} (3 a B+7 A b)}{40 a b^2 (a+b x)^4}+\frac{x^{3/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^(3/2))/(5*a*b*(a + b*x)^5) - ((7*A*b + 3*a*B)*Sqrt[x])/(40*a*b^2*
(a + b*x)^4) + ((7*A*b + 3*a*B)*Sqrt[x])/(240*a^2*b^2*(a + b*x)^3) + ((7*A*b + 3
*a*B)*Sqrt[x])/(192*a^3*b^2*(a + b*x)^2) + ((7*A*b + 3*a*B)*Sqrt[x])/(128*a^4*b^
2*(a + b*x)) + ((7*A*b + 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(9/2)*
b^(5/2))

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Rubi in Sympy [A]  time = 57.7421, size = 180, normalized size = 0.92 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} - \frac{\sqrt{x} \left (7 A b + 3 B a\right )}{40 a b^{2} \left (a + b x\right )^{4}} + \frac{\sqrt{x} \left (7 A b + 3 B a\right )}{240 a^{2} b^{2} \left (a + b x\right )^{3}} + \frac{\sqrt{x} \left (7 A b + 3 B a\right )}{192 a^{3} b^{2} \left (a + b x\right )^{2}} + \frac{\sqrt{x} \left (7 A b + 3 B a\right )}{128 a^{4} b^{2} \left (a + b x\right )} + \frac{\left (7 A b + 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 a^{\frac{9}{2}} b^{\frac{5}{2}}} \]

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,x)

[Out]

x**(3/2)*(A*b - B*a)/(5*a*b*(a + b*x)**5) - sqrt(x)*(7*A*b + 3*B*a)/(40*a*b**2*(
a + b*x)**4) + sqrt(x)*(7*A*b + 3*B*a)/(240*a**2*b**2*(a + b*x)**3) + sqrt(x)*(7
*A*b + 3*B*a)/(192*a**3*b**2*(a + b*x)**2) + sqrt(x)*(7*A*b + 3*B*a)/(128*a**4*b
**2*(a + b*x)) + (7*A*b + 3*B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(128*a**(9/2)*b**
(5/2))

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Mathematica [A]  time = 0.322798, size = 145, normalized size = 0.74 \[ \frac{\frac{15 (3 a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{\sqrt{b} \sqrt{x} \left (-45 a^5 B-105 a^4 b (A+2 B x)+2 a^3 b^2 x (395 A+192 B x)+14 a^2 b^3 x^2 (64 A+15 B x)+5 a b^4 x^3 (98 A+9 B x)+105 A b^5 x^4\right )}{a^4 (a+b x)^5}}{1920 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*Sqrt[x]*(-45*a^5*B + 105*A*b^5*x^4 - 105*a^4*b*(A + 2*B*x) + 5*a*b^4*x
^3*(98*A + 9*B*x) + 14*a^2*b^3*x^2*(64*A + 15*B*x) + 2*a^3*b^2*x*(395*A + 192*B*
x)))/(a^4*(a + b*x)^5) + (15*(7*A*b + 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/
a^(9/2))/(1920*b^(5/2))

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Maple [A]  time = 0.026, size = 154, normalized size = 0.8 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{5}} \left ({\frac{ \left ( 7\,Ab+3\,Ba \right ){b}^{2}{x}^{9/2}}{256\,{a}^{4}}}+{\frac{7\,b \left ( 7\,Ab+3\,Ba \right ){x}^{7/2}}{384\,{a}^{3}}}+1/30\,{\frac{ \left ( 7\,Ab+3\,Ba \right ){x}^{5/2}}{{a}^{2}}}+{\frac{ \left ( 79\,Ab-21\,Ba \right ){x}^{3/2}}{384\,ab}}-{\frac{ \left ( 7\,Ab+3\,Ba \right ) \sqrt{x}}{256\,{b}^{2}}} \right ) }+{\frac{7\,A}{128\,{a}^{4}b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{128\,{a}^{3}{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(1/256*(7*A*b+3*B*a)*b^2/a^4*x^(9/2)+7/384/a^3*b*(7*A*b+3*B*a)*x^(7/2)+1/30/a^
2*(7*A*b+3*B*a)*x^(5/2)+1/384*(79*A*b-21*B*a)/a/b*x^(3/2)-1/256*(7*A*b+3*B*a)/b^
2*x^(1/2))/(b*x+a)^5+7/128/b/a^4/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+3/1
28/b^2/a^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)*sqrt(x)/(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 = 0.31557, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (45 \, B a^{5} + 105 \, A a^{4} b - 15 \,{\left (3 \, B a b^{4} + 7 \, A b^{5}\right )} x^{4} - 70 \,{\left (3 \, B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} - 128 \,{\left (3 \, B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 10 \,{\left (21 \, B a^{4} b - 79 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 15 \,{\left (3 \, B a^{6} + 7 \, A a^{5} b +{\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{5} + 5 \,{\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{4} + 10 \,{\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{3840 \,{\left (a^{4} b^{7} x^{5} + 5 \, a^{5} b^{6} x^{4} + 10 \, a^{6} b^{5} x^{3} + 10 \, a^{7} b^{4} x^{2} + 5 \, a^{8} b^{3} x + a^{9} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (45 \, B a^{5} + 105 \, A a^{4} b - 15 \,{\left (3 \, B a b^{4} + 7 \, A b^{5}\right )} x^{4} - 70 \,{\left (3 \, B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} - 128 \,{\left (3 \, B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 10 \,{\left (21 \, B a^{4} b - 79 \, A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 15 \,{\left (3 \, B a^{6} + 7 \, A a^{5} b +{\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{5} + 5 \,{\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{4} + 10 \,{\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{1920 \,{\left (a^{4} b^{7} x^{5} + 5 \, a^{5} b^{6} x^{4} + 10 \, a^{6} b^{5} x^{3} + 10 \, a^{7} b^{4} x^{2} + 5 \, a^{8} b^{3} x + a^{9} b^{2}\right )} \sqrt{a b}}\right ] \]

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,x, algorithm="fricas")

[Out]

[-1/3840*(2*(45*B*a^5 + 105*A*a^4*b - 15*(3*B*a*b^4 + 7*A*b^5)*x^4 - 70*(3*B*a^2
*b^3 + 7*A*a*b^4)*x^3 - 128*(3*B*a^3*b^2 + 7*A*a^2*b^3)*x^2 + 10*(21*B*a^4*b - 7
9*A*a^3*b^2)*x)*sqrt(-a*b)*sqrt(x) - 15*(3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*
b^6)*x^5 + 5*(3*B*a^2*b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*b^3 + 7*A*a^2*b^4)*x^3
+ 10*(3*B*a^4*b^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*x)*log((2*a*b
*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a^4*b^7*x^5 + 5*a^5*b^6*x^4 + 10*
a^6*b^5*x^3 + 10*a^7*b^4*x^2 + 5*a^8*b^3*x + a^9*b^2)*sqrt(-a*b)), -1/1920*((45*
B*a^5 + 105*A*a^4*b - 15*(3*B*a*b^4 + 7*A*b^5)*x^4 - 70*(3*B*a^2*b^3 + 7*A*a*b^4
)*x^3 - 128*(3*B*a^3*b^2 + 7*A*a^2*b^3)*x^2 + 10*(21*B*a^4*b - 79*A*a^3*b^2)*x)*
sqrt(a*b)*sqrt(x) + 15*(3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*b^6)*x^5 + 5*(3*B
*a^2*b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*b^3 + 7*A*a^2*b^4)*x^3 + 10*(3*B*a^4*b^2
+ 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x)
)))/((a^4*b^7*x^5 + 5*a^5*b^6*x^4 + 10*a^6*b^5*x^3 + 10*a^7*b^4*x^2 + 5*a^8*b^3*
x + a^9*b^2)*sqrt(a*b))]

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Sympy [A]  time = 58.6128, size = 1838, normalized size = 9.43 \[ \text{result too large to display} \]

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,x)

[Out]

-1930*A*a**5*sqrt(x)/(1280*a**10*b + 6400*a**9*b**2*x + 12800*a**8*b**3*x**2 + 1
2800*a**7*b**4*x**3 + 6400*a**6*b**5*x**4 + 1280*a**5*b**6*x**5) - 4740*A*a**4*x
**(3/2)/(1280*a**10 + 6400*a**9*b*x + 12800*a**8*b**2*x**2 + 12800*a**7*b**3*x**
3 + 6400*a**6*b**4*x**4 + 1280*a**5*b**5*x**5) - 5376*A*a**3*b*x**(5/2)/(1280*a*
*10 + 6400*a**9*b*x + 12800*a**8*b**2*x**2 + 12800*a**7*b**3*x**3 + 6400*a**6*b*
*4*x**4 + 1280*a**5*b**5*x**5) + 558*A*a**3*sqrt(x)/(384*a**8*b + 1536*a**7*b**2
*x + 2304*a**6*b**3*x**2 + 1536*a**5*b**4*x**3 + 384*a**4*b**5*x**4) - 2940*A*a*
*2*b**2*x**(7/2)/(1280*a**10 + 6400*a**9*b*x + 12800*a**8*b**2*x**2 + 12800*a**7
*b**3*x**3 + 6400*a**6*b**4*x**4 + 1280*a**5*b**5*x**5) + 1022*A*a**2*x**(3/2)/(
384*a**8 + 1536*a**7*b*x + 2304*a**6*b**2*x**2 + 1536*a**5*b**3*x**3 + 384*a**4*
b**4*x**4) - 630*A*a*b**3*x**(9/2)/(1280*a**10 + 6400*a**9*b*x + 12800*a**8*b**2
*x**2 + 12800*a**7*b**3*x**3 + 6400*a**6*b**4*x**4 + 1280*a**5*b**5*x**5) + 770*
A*a*b*x**(5/2)/(384*a**8 + 1536*a**7*b*x + 2304*a**6*b**2*x**2 + 1536*a**5*b**3*
x**3 + 384*a**4*b**4*x**4) + 63*A*a*sqrt(-1/(a**11*b))*log(-a**6*sqrt(-1/(a**11*
b)) + sqrt(x))/(256*b) - 63*A*a*sqrt(-1/(a**11*b))*log(a**6*sqrt(-1/(a**11*b)) +
sqrt(x))/(256*b) + 210*A*b**2*x**(7/2)/(384*a**8 + 1536*a**7*b*x + 2304*a**6*b*
*2*x**2 + 1536*a**5*b**3*x**3 + 384*a**4*b**4*x**4) - 35*A*sqrt(-1/(a**9*b))*log
(-a**5*sqrt(-1/(a**9*b)) + sqrt(x))/(128*b) + 35*A*sqrt(-1/(a**9*b))*log(a**5*sq
rt(-1/(a**9*b)) + sqrt(x))/(128*b) + 1930*B*a**6*sqrt(x)/(1280*a**10*b**2 + 6400
*a**9*b**3*x + 12800*a**8*b**4*x**2 + 12800*a**7*b**5*x**3 + 6400*a**6*b**6*x**4
+ 1280*a**5*b**7*x**5) + 4740*B*a**5*x**(3/2)/(1280*a**10*b + 6400*a**9*b**2*x
+ 12800*a**8*b**3*x**2 + 12800*a**7*b**4*x**3 + 6400*a**6*b**5*x**4 + 1280*a**5*
b**6*x**5) + 5376*B*a**4*x**(5/2)/(1280*a**10 + 6400*a**9*b*x + 12800*a**8*b**2*
x**2 + 12800*a**7*b**3*x**3 + 6400*a**6*b**4*x**4 + 1280*a**5*b**5*x**5) - 1116*
B*a**4*sqrt(x)/(384*a**8*b**2 + 1536*a**7*b**3*x + 2304*a**6*b**4*x**2 + 1536*a*
*5*b**5*x**3 + 384*a**4*b**6*x**4) + 2940*B*a**3*b*x**(7/2)/(1280*a**10 + 6400*a
**9*b*x + 12800*a**8*b**2*x**2 + 12800*a**7*b**3*x**3 + 6400*a**6*b**4*x**4 + 12
80*a**5*b**5*x**5) - 2044*B*a**3*x**(3/2)/(384*a**8*b + 1536*a**7*b**2*x + 2304*
a**6*b**3*x**2 + 1536*a**5*b**4*x**3 + 384*a**4*b**5*x**4) + 630*B*a**2*b**2*x**
(9/2)/(1280*a**10 + 6400*a**9*b*x + 12800*a**8*b**2*x**2 + 12800*a**7*b**3*x**3
+ 6400*a**6*b**4*x**4 + 1280*a**5*b**5*x**5) - 1540*B*a**2*x**(5/2)/(384*a**8 +
1536*a**7*b*x + 2304*a**6*b**2*x**2 + 1536*a**5*b**3*x**3 + 384*a**4*b**4*x**4)
+ 66*B*a**2*sqrt(x)/(48*a**6*b**2 + 144*a**5*b**3*x + 144*a**4*b**4*x**2 + 48*a*
*3*b**5*x**3) - 63*B*a**2*sqrt(-1/(a**11*b))*log(-a**6*sqrt(-1/(a**11*b)) + sqrt
(x))/(256*b**2) + 63*B*a**2*sqrt(-1/(a**11*b))*log(a**6*sqrt(-1/(a**11*b)) + sqr
t(x))/(256*b**2) - 420*B*a*b*x**(7/2)/(384*a**8 + 1536*a**7*b*x + 2304*a**6*b**2
*x**2 + 1536*a**5*b**3*x**3 + 384*a**4*b**4*x**4) + 80*B*a*x**(3/2)/(48*a**6*b +
144*a**5*b**2*x + 144*a**4*b**3*x**2 + 48*a**3*b**4*x**3) + 35*B*a*sqrt(-1/(a**
9*b))*log(-a**5*sqrt(-1/(a**9*b)) + sqrt(x))/(64*b**2) - 35*B*a*sqrt(-1/(a**9*b)
)*log(a**5*sqrt(-1/(a**9*b)) + sqrt(x))/(64*b**2) + 30*B*x**(5/2)/(48*a**6 + 144
*a**5*b*x + 144*a**4*b**2*x**2 + 48*a**3*b**3*x**3) - 5*B*sqrt(-1/(a**7*b))*log(
-a**4*sqrt(-1/(a**7*b)) + sqrt(x))/(16*b**2) + 5*B*sqrt(-1/(a**7*b))*log(a**4*sq
rt(-1/(a**7*b)) + sqrt(x))/(16*b**2)

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GIAC/XCAS [A]  time = 0.275245, size = 211, normalized size = 1.08 \[ \frac{{\left (3 \, B a + 7 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{4} b^{2}} + \frac{45 \, B a b^{4} x^{\frac{9}{2}} + 105 \, A b^{5} x^{\frac{9}{2}} + 210 \, B a^{2} b^{3} x^{\frac{7}{2}} + 490 \, A a b^{4} x^{\frac{7}{2}} + 384 \, B a^{3} b^{2} x^{\frac{5}{2}} + 896 \, A a^{2} b^{3} x^{\frac{5}{2}} - 210 \, B a^{4} b x^{\frac{3}{2}} + 790 \, A a^{3} b^{2} x^{\frac{3}{2}} - 45 \, B a^{5} \sqrt{x} - 105 \, A a^{4} b \sqrt{x}}{1920 \,{\left (b x + a\right )}^{5} a^{4} b^{2}} \]

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,x, algorithm="giac")

[Out]

1/128*(3*B*a + 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*b^2) + 1/1920*(
45*B*a*b^4*x^(9/2) + 105*A*b^5*x^(9/2) + 210*B*a^2*b^3*x^(7/2) + 490*A*a*b^4*x^(
7/2) + 384*B*a^3*b^2*x^(5/2) + 896*A*a^2*b^3*x^(5/2) - 210*B*a^4*b*x^(3/2) + 790
*A*a^3*b^2*x^(3/2) - 45*B*a^5*sqrt(x) - 105*A*a^4*b*sqrt(x))/((b*x + a)^5*a^4*b^
2)