3.465 \(\int \frac{\sqrt{x}}{(a+b x)^3} \, dx\)
Optimal. Leaf size=73 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a+b x)}-\frac{\sqrt{x}}{2 b (a+b x)^2} \]
[Out]
-Sqrt[x]/(2*b*(a + b*x)^2) + Sqrt[x]/(4*a*b*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]]/(4*a^(3/2)*b^(3/2))
_______________________________________________________________________________________
Rubi [A] time = 0.050461, antiderivative size = 73, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308
\[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a+b x)}-\frac{\sqrt{x}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(a + b*x)^3,x]
[Out]
-Sqrt[x]/(2*b*(a + b*x)^2) + Sqrt[x]/(4*a*b*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]]/(4*a^(3/2)*b^(3/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.84267, size = 58, normalized size = 0.79 \[ - \frac{\sqrt{x}}{2 b \left (a + b x\right )^{2}} + \frac{\sqrt{x}}{4 a b \left (a + b x\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(b*x+a)**3,x)
[Out]
-sqrt(x)/(2*b*(a + b*x)**2) + sqrt(x)/(4*a*b*(a + b*x)) + atan(sqrt(b)*sqrt(x)/s
qrt(a))/(4*a**(3/2)*b**(3/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.0500181, size = 62, normalized size = 0.85 \[ \frac{\frac{\sqrt{a} \sqrt{b} \sqrt{x} (b x-a)}{(a+b x)^2}+\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(a + b*x)^3,x]
[Out]
((Sqrt[a]*Sqrt[b]*Sqrt[x]*(-a + b*x))/(a + b*x)^2 + ArcTan[(Sqrt[b]*Sqrt[x])/Sqr
t[a]])/(4*a^(3/2)*b^(3/2))
_______________________________________________________________________________________
Maple [A] time = 0.018, size = 52, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( 1/8\,{\frac{{x}^{3/2}}{a}}-1/8\,{\frac{\sqrt{x}}{b}} \right ) }+{\frac{1}{4\,ab}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(b*x+a)^3,x)
[Out]
2*(1/8/a*x^(3/2)-1/8*x^(1/2)/b)/(b*x+a)^2+1/4/b/a/(a*b)^(1/2)*arctan(x^(1/2)*b/(
a*b)^(1/2))
_______________________________________________________________________________________
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(sqrt(x)/(b*x + a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.223944, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b}{\left (b x - a\right )} \sqrt{x} +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{-a b}}, \frac{\sqrt{a b}{\left (b x - a\right )} \sqrt{x} -{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(b*x + a)^3,x, algorithm="fricas")
[Out]
[1/8*(2*sqrt(-a*b)*(b*x - a)*sqrt(x) + (b^2*x^2 + 2*a*b*x + a^2)*log((2*a*b*sqrt
(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a*b^3*x^2 + 2*a^2*b^2*x + a^3*b)*sqrt(
-a*b)), 1/4*(sqrt(a*b)*(b*x - a)*sqrt(x) - (b^2*x^2 + 2*a*b*x + a^2)*arctan(a/(s
qrt(a*b)*sqrt(x))))/((a*b^3*x^2 + 2*a^2*b^2*x + a^3*b)*sqrt(a*b))]
_______________________________________________________________________________________
Sympy [A] time = 7.64371, size = 2732, normalized size = 37.42 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(b*x+a)**3,x)
[Out]
a**(47/2)*b**2*x**(7/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**25*b**(7/2)*x**(7/2)
+ 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13
/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) +
112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)
*x**(23/2)) + 8*a**(45/2)*b**3*x**(9/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**25*b
**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) +
224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17
/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4
*a**17*b**(23/2)*x**(23/2)) + 28*a**(43/2)*b**4*x**(11/2)*atan(sqrt(b)*sqrt(x)/s
qrt(a))/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(
11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2)
+ 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(2
1/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) + 56*a**(41/2)*b**5*x**(13/2)*atan
(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2
) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b*
*(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2
) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) + 70*a**(39/2)*b
**6*x**(15/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**25*b**(7/2)*x**(7/2) + 32*a**2
4*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13
/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*
b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)
) + 56*a**(37/2)*b**7*x**(17/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**25*b**(7/2)*
x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**2
2*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(
17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b
**(23/2)*x**(23/2)) + 28*a**(35/2)*b**8*x**(19/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/
(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x*
*(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a*
*20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**
(21/2) + 4*a**17*b**(23/2)*x**(23/2)) + 8*a**(33/2)*b**9*x**(21/2)*atan(sqrt(b)*
sqrt(x)/sqrt(a))/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a
**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x
**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a*
*18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) + a**(31/2)*b**10*x**(23/
2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*
x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a
**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x
**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) - a**23*b
**(5/2)*x**4/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23
*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(1
5/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*
b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) - 5*a**22*b**(7/2)*x**5/(4*a*
*25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/
2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b
**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2
) + 4*a**17*b**(23/2)*x**(23/2)) - 9*a**21*b**(9/2)*x**6/(4*a**25*b**(7/2)*x**(7
/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**
(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2)
+ 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23
/2)*x**(23/2)) - 5*a**20*b**(11/2)*x**7/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b*
*(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2)
+ 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(
19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) +
5*a**19*b**(13/2)*x**8/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) +
112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(1
5/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) +
32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) + 9*a**18*b**(15/2)
*x**9/(4*a**25*b**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11
/2)*x**(11/2) + 224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) +
224*a**20*b**(17/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/
2)*x**(21/2) + 4*a**17*b**(23/2)*x**(23/2)) + 5*a**17*b**(17/2)*x**10/(4*a**25*b
**(7/2)*x**(7/2) + 32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) +
224*a**22*b**(13/2)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17
/2)*x**(17/2) + 112*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4
*a**17*b**(23/2)*x**(23/2)) + a**16*b**(19/2)*x**11/(4*a**25*b**(7/2)*x**(7/2) +
32*a**24*b**(9/2)*x**(9/2) + 112*a**23*b**(11/2)*x**(11/2) + 224*a**22*b**(13/2
)*x**(13/2) + 280*a**21*b**(15/2)*x**(15/2) + 224*a**20*b**(17/2)*x**(17/2) + 11
2*a**19*b**(19/2)*x**(19/2) + 32*a**18*b**(21/2)*x**(21/2) + 4*a**17*b**(23/2)*x
**(23/2))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.204152, size = 70, normalized size = 0.96 \[ \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b} + \frac{b x^{\frac{3}{2}} - a \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(b*x + a)^3,x, algorithm="giac")
[Out]
1/4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a*b) + 1/4*(b*x^(3/2) - a*sqrt(x))/((
b*x + a)^2*a*b)