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

Optimal. Leaf size=123 \[ \frac{3 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2}}-\frac{3 \sqrt{x} (A b-5 a B)}{4 a b^3}+\frac{x^{3/2} (A b-5 a B)}{4 a b^2 (a+b x)}+\frac{x^{5/2} (A b-a B)}{2 a b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.137369, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{3 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2}}-\frac{3 \sqrt{x} (A b-5 a B)}{4 a b^3}+\frac{x^{3/2} (A b-5 a B)}{4 a b^2 (a+b x)}+\frac{x^{5/2} (A b-a B)}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.5841, size = 109, normalized size = 0.89 \[ \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{2 a b \left (a + b x\right )^{2}} + \frac{x^{\frac{3}{2}} \left (A b - 5 B a\right )}{4 a b^{2} \left (a + b x\right )} - \frac{3 \sqrt{x} \left (A b - 5 B a\right )}{4 a b^{3}} + \frac{3 \left (A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 \sqrt{a} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.136131, size = 91, normalized size = 0.74 \[ \frac{\sqrt{x} \left (15 a^2 B+a (25 b B x-3 A b)+b^2 x (8 B x-5 A)\right )}{4 b^3 (a+b x)^2}+\frac{3 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(15*a^2*B + b^2*x*(-5*A + 8*B*x) + a*(-3*A*b + 25*b*B*x)))/(4*b^3*(a +
b*x)^2) + (3*(A*b - 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*Sqrt[a]*b^(7/2)
)

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Maple [A]  time = 0.022, size = 125, normalized size = 1. \[ 2\,{\frac{B\sqrt{x}}{{b}^{3}}}-{\frac{5\,A}{4\,b \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{9\,Ba}{4\,{b}^{2} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{3\,Aa}{4\,{b}^{2} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{7\,B{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{3\,A}{4\,{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,Ba}{4\,{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227286, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 3 \, A a b + 5 \,{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 3 \,{\left (5 \, B a^{3} - A a^{2} b +{\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{-a b}}, \frac{{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 3 \, A a b + 5 \,{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 3 \,{\left (5 \, B a^{3} - A a^{2} b +{\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*(8*B*b^2*x^2 + 15*B*a^2 - 3*A*a*b + 5*(5*B*a*b - A*b^2)*x)*sqrt(-a*b)*sq
rt(x) - 3*(5*B*a^3 - A*a^2*b + (5*B*a*b^2 - A*b^3)*x^2 + 2*(5*B*a^2*b - A*a*b^2)
*x)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((b^5*x^2 + 2*a*b^4*x
+ a^2*b^3)*sqrt(-a*b)), 1/4*((8*B*b^2*x^2 + 15*B*a^2 - 3*A*a*b + 5*(5*B*a*b - A
*b^2)*x)*sqrt(a*b)*sqrt(x) + 3*(5*B*a^3 - A*a^2*b + (5*B*a*b^2 - A*b^3)*x^2 + 2*
(5*B*a^2*b - A*a*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((b^5*x^2 + 2*a*b^4*x +
a^2*b^3)*sqrt(a*b))]

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Sympy [A]  time = 77.0144, size = 5435, normalized size = 44.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(3*a**(83/2)*b**12*x**(29/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*
x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a
**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)
*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 6
60*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2
)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) + 33*a**(81/2)*b**13*x**(31/2)*atan(s
qrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31
/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38
*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**
(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a*
*33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(5
1/2)) + 165*a**(79/2)*b**14*x**(33/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**
(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2)
+ 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b*
*(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43
/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b
**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) + 495*a**(77/2)*b**15*x**(35/2
)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2
)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 13
20*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(4
1/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2)
+ 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/
2)*x**(51/2)) + 990*a**(75/2)*b**16*x**(37/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a
**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x*
*(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*
a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2
)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44
*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) + 1386*a**(73/2)*b**17
*x**(39/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41
*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(3
5/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a*
*36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x
**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**3
1*b**(51/2)*x**(51/2)) + 1386*a**(71/2)*b**18*x**(41/2)*atan(sqrt(b)*sqrt(x)/sqr
t(a))/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b*
*(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/
2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**3
5*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(
47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) + 990*a**(69
/2)*b**19*x**(43/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*x**(29/2) +
44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35
/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2)
+ 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b*
*(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2)
+ 4*a**31*b**(51/2)*x**(51/2)) + 495*a**(67/2)*b**20*x**(45/2)*atan(sqrt(b)*sqr
t(x)/sqrt(a))/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*
a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)
*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1
320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47
/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) + 16
5*a**(65/2)*b**21*x**(47/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*x**
(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**3
9*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x*
*(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*
a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x
**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) + 33*a**(63/2)*b**22*x**(49/2)*atan(sqrt
(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2)
+ 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b*
*(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41
/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33
*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2
)) + 3*a**(61/2)*b**23*x**(51/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**42*b**(29/2
)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660
*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/
2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) +
660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49
/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 3*a**41*b**(25/2)*x**15/(4*a**42*
b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/
2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37
*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**
(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**3
2*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 32*a**40*b**(27/2)*x**16/
(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2
)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1
848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(
43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2)
+ 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 153*a**39*b**(29
/2)*x**17/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**4
0*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**
(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*
a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*
x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 432*a*
*38*b**(31/2)*x**18/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2)
+ 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**
(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/
2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*
b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)
) - 798*a**37*b**(33/2)*x**19/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*
x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320
*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/
2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) +
220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)
*x**(51/2)) - 1008*a**36*b**(35/2)*x**20/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41
*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(3
5/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a*
*36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x
**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**3
1*b**(51/2)*x**(51/2)) - 882*a**35*b**(37/2)*x**21/(4*a**42*b**(29/2)*x**(29/2)
+ 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a**39*b**(3
5/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)*x**(39/2)
+ 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 660*a**34*b
**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2)*x**(49/2
) + 4*a**31*b**(51/2)*x**(51/2)) - 528*a**34*b**(39/2)*x**22/(4*a**42*b**(29/2)*
x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/2) + 660*a
**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37*b**(39/2)
*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**(43/2) + 6
60*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**32*b**(49/2
)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 207*a**33*b**(41/2)*x**23/(4*a**42*
b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2)*x**(33/
2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1848*a**37
*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(43/2)*x**
(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2) + 44*a**3
2*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 48*a**32*b**(43/2)*x**24/
(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*b**(33/2
)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(37/2) + 1
848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a**35*b**(
43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x**(47/2)
+ 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2)) - 5*a**31*b**(45/2
)*x**25/(4*a**42*b**(29/2)*x**(29/2) + 44*a**41*b**(31/2)*x**(31/2) + 220*a**40*
b**(33/2)*x**(33/2) + 660*a**39*b**(35/2)*x**(35/2) + 1320*a**38*b**(37/2)*x**(3
7/2) + 1848*a**37*b**(39/2)*x**(39/2) + 1848*a**36*b**(41/2)*x**(41/2) + 1320*a*
*35*b**(43/2)*x**(43/2) + 660*a**34*b**(45/2)*x**(45/2) + 220*a**33*b**(47/2)*x*
*(47/2) + 44*a**32*b**(49/2)*x**(49/2) + 4*a**31*b**(51/2)*x**(51/2))) + B*(-15*
a**(53/2)*b**9*x**(25/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**26*b**(25/2)*x**(25
/2) + 12*a**25*b**(27/2)*x**(27/2) + 12*a**24*b**(29/2)*x**(29/2) + 4*a**23*b**(
31/2)*x**(31/2)) - 45*a**(51/2)*b**10*x**(27/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4
*a**26*b**(25/2)*x**(25/2) + 12*a**25*b**(27/2)*x**(27/2) + 12*a**24*b**(29/2)*x
**(29/2) + 4*a**23*b**(31/2)*x**(31/2)) - 45*a**(49/2)*b**11*x**(29/2)*atan(sqrt
(b)*sqrt(x)/sqrt(a))/(4*a**26*b**(25/2)*x**(25/2) + 12*a**25*b**(27/2)*x**(27/2)
+ 12*a**24*b**(29/2)*x**(29/2) + 4*a**23*b**(31/2)*x**(31/2)) - 15*a**(47/2)*b*
*12*x**(31/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**26*b**(25/2)*x**(25/2) + 12*a*
*25*b**(27/2)*x**(27/2) + 12*a**24*b**(29/2)*x**(29/2) + 4*a**23*b**(31/2)*x**(3
1/2)) + 15*a**26*b**(19/2)*x**13/(4*a**26*b**(25/2)*x**(25/2) + 12*a**25*b**(27/
2)*x**(27/2) + 12*a**24*b**(29/2)*x**(29/2) + 4*a**23*b**(31/2)*x**(31/2)) + 40*
a**25*b**(21/2)*x**14/(4*a**26*b**(25/2)*x**(25/2) + 12*a**25*b**(27/2)*x**(27/2
) + 12*a**24*b**(29/2)*x**(29/2) + 4*a**23*b**(31/2)*x**(31/2)) + 33*a**24*b**(2
3/2)*x**15/(4*a**26*b**(25/2)*x**(25/2) + 12*a**25*b**(27/2)*x**(27/2) + 12*a**2
4*b**(29/2)*x**(29/2) + 4*a**23*b**(31/2)*x**(31/2)) + 8*a**23*b**(25/2)*x**16/(
4*a**26*b**(25/2)*x**(25/2) + 12*a**25*b**(27/2)*x**(27/2) + 12*a**24*b**(29/2)*
x**(29/2) + 4*a**23*b**(31/2)*x**(31/2)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216167, size = 117, normalized size = 0.95 \[ \frac{2 \, B \sqrt{x}}{b^{3}} - \frac{3 \,{\left (5 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3}} + \frac{9 \, B a b x^{\frac{3}{2}} - 5 \, A b^{2} x^{\frac{3}{2}} + 7 \, B a^{2} \sqrt{x} - 3 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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