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

Optimal. Leaf size=108 \[ -\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{\sqrt{x} (3 A b-5 a B)}{b^3}-\frac{x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac{x^{5/2} (A b-a B)}{a b (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.134549, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{a} (3 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{\sqrt{x} (3 A b-5 a B)}{b^3}-\frac{x^{3/2} (3 A b-5 a B)}{3 a b^2}+\frac{x^{5/2} (A b-a B)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.9273, size = 97, normalized size = 0.9 \[ - \frac{\sqrt{a} \left (3 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{\sqrt{x} \left (3 A b - 5 B a\right )}{b^{3}} + \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{a b \left (a + b x\right )} - \frac{x^{\frac{3}{2}} \left (3 A b - 5 B a\right )}{3 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.163777, size = 88, normalized size = 0.81 \[ \frac{\sqrt{x} \left (-15 a^2 B+a b (9 A-10 B x)+2 b^2 x (3 A+B x)\right )}{3 b^3 (a+b x)}+\frac{\sqrt{a} (5 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-15*a^2*B + a*b*(9*A - 10*B*x) + 2*b^2*x*(3*A + B*x)))/(3*b^3*(a + b*x
)) + (Sqrt[a]*(-3*A*b + 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(7/2)

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 113, normalized size = 1.1 \[{\frac{2\,B}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{2}}}-4\,{\frac{Ba\sqrt{x}}{{b}^{3}}}+{\frac{Aa}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}-{\frac{B{a}^{2}}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{Aa}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+5\,{\frac{B{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.228538, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, \frac{3 \,{\left (5 \, B a^{2} - 3 \, A a b +{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (2 \, B b^{2} x^{2} - 15 \, B a^{2} + 9 \, A a b - 2 \,{\left (5 \, B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*(5*B*a^2 - 3*A*a*b + (5*B*a*b - 3*A*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*s
qrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(2*B*b^2*x^2 - 15*B*a^2 + 9*A*a*b - 2*(5*B
*a*b - 3*A*b^2)*x)*sqrt(x))/(b^4*x + a*b^3), 1/3*(3*(5*B*a^2 - 3*A*a*b + (5*B*a*
b - 3*A*b^2)*x)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) + (2*B*b^2*x^2 - 15*B*a^2 +
9*A*a*b - 2*(5*B*a*b - 3*A*b^2)*x)*sqrt(x))/(b^4*x + a*b^3)]

_______________________________________________________________________________________

Sympy [A]  time = 45.8553, size = 461, normalized size = 4.27 \[ A \left (- \frac{3 a^{\frac{17}{2}} b^{4} x^{\frac{13}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} - \frac{3 a^{\frac{15}{2}} b^{5} x^{\frac{15}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} + \frac{3 a^{8} b^{\frac{9}{2}} x^{7}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}} + \frac{2 a^{7} b^{\frac{11}{2}} x^{8}}{a^{8} b^{\frac{13}{2}} x^{\frac{13}{2}} + a^{7} b^{\frac{15}{2}} x^{\frac{15}{2}}}\right ) + B \left (\frac{15 a^{\frac{61}{2}} b^{17} x^{\frac{41}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} + \frac{15 a^{\frac{59}{2}} b^{18} x^{\frac{43}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} - \frac{15 a^{30} b^{\frac{35}{2}} x^{21}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} - \frac{10 a^{29} b^{\frac{37}{2}} x^{22}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} + \frac{2 a^{28} b^{\frac{39}{2}} x^{23}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-3*a**(17/2)*b**4*x**(13/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**8*b**(13/2)*x**
(13/2) + a**7*b**(15/2)*x**(15/2)) - 3*a**(15/2)*b**5*x**(15/2)*atan(sqrt(b)*sqr
t(x)/sqrt(a))/(a**8*b**(13/2)*x**(13/2) + a**7*b**(15/2)*x**(15/2)) + 3*a**8*b**
(9/2)*x**7/(a**8*b**(13/2)*x**(13/2) + a**7*b**(15/2)*x**(15/2)) + 2*a**7*b**(11
/2)*x**8/(a**8*b**(13/2)*x**(13/2) + a**7*b**(15/2)*x**(15/2))) + B*(15*a**(61/2
)*b**17*x**(41/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**29*b**(41/2)*x**(41/2) + 3
*a**28*b**(43/2)*x**(43/2)) + 15*a**(59/2)*b**18*x**(43/2)*atan(sqrt(b)*sqrt(x)/
sqrt(a))/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x**(43/2)) - 15*a**30*
b**(35/2)*x**21/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x**(43/2)) - 10
*a**29*b**(37/2)*x**22/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x**(43/2
)) + 2*a**28*b**(39/2)*x**23/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x*
*(43/2)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220508, size = 128, normalized size = 1.19 \[ \frac{{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{B a^{2} \sqrt{x} - A a b \sqrt{x}}{{\left (b x + a\right )} b^{3}} + \frac{2 \,{\left (B b^{4} x^{\frac{3}{2}} - 6 \, B a b^{3} \sqrt{x} + 3 \, A b^{4} \sqrt{x}\right )}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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