3.468 \(\int \frac{1}{x^{5/2} (a+b x)^3} \, dx\)
Optimal. Leaf size=95 \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{35}{12 a^3 x^{3/2}}+\frac{7}{4 a^2 x^{3/2} (a+b x)}+\frac{1}{2 a x^{3/2} (a+b x)^2} \]
[Out]
-35/(12*a^3*x^(3/2)) + (35*b)/(4*a^4*Sqrt[x]) + 1/(2*a*x^(3/2)*(a + b*x)^2) + 7/
(4*a^2*x^(3/2)*(a + b*x)) + (35*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^
(9/2))
_______________________________________________________________________________________
Rubi [A] time = 0.0765092, antiderivative size = 95, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308
\[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{35 b}{4 a^4 \sqrt{x}}-\frac{35}{12 a^3 x^{3/2}}+\frac{7}{4 a^2 x^{3/2} (a+b x)}+\frac{1}{2 a x^{3/2} (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/2)*(a + b*x)^3),x]
[Out]
-35/(12*a^3*x^(3/2)) + (35*b)/(4*a^4*Sqrt[x]) + 1/(2*a*x^(3/2)*(a + b*x)^2) + 7/
(4*a^2*x^(3/2)*(a + b*x)) + (35*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^
(9/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.4907, size = 88, normalized size = 0.93 \[ \frac{1}{2 a x^{\frac{3}{2}} \left (a + b x\right )^{2}} + \frac{7}{4 a^{2} x^{\frac{3}{2}} \left (a + b x\right )} - \frac{35}{12 a^{3} x^{\frac{3}{2}}} + \frac{35 b}{4 a^{4} \sqrt{x}} + \frac{35 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/2)/(b*x+a)**3,x)
[Out]
1/(2*a*x**(3/2)*(a + b*x)**2) + 7/(4*a**2*x**(3/2)*(a + b*x)) - 35/(12*a**3*x**(
3/2)) + 35*b/(4*a**4*sqrt(x)) + 35*b**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**
(9/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.0693925, size = 81, normalized size = 0.85 \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{-8 a^3+56 a^2 b x+175 a b^2 x^2+105 b^3 x^3}{12 a^4 x^{3/2} (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/2)*(a + b*x)^3),x]
[Out]
(-8*a^3 + 56*a^2*b*x + 175*a*b^2*x^2 + 105*b^3*x^3)/(12*a^4*x^(3/2)*(a + b*x)^2)
+ (35*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))
_______________________________________________________________________________________
Maple [A] time = 0.024, size = 79, normalized size = 0.8 \[ -{\frac{2}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{b}{{a}^{4}\sqrt{x}}}+{\frac{11\,{b}^{3}}{4\,{a}^{4} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,{b}^{2}}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,{b}^{2}}{4\,{a}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/2)/(b*x+a)^3,x)
[Out]
-2/3/a^3/x^(3/2)+6*b/a^4/x^(1/2)+11/4/a^4*b^3/(b*x+a)^2*x^(3/2)+13/4/a^3*b^2/(b*
x+a)^2*x^(1/2)+35/4/a^4*b^2/(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(1/((b*x + a)^3*x^(5/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.225675, size = 1, normalized size = 0.01 \[ \left [\frac{210 \, b^{3} x^{3} + 350 \, a b^{2} x^{2} + 112 \, a^{2} b x - 16 \, a^{3} + 105 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right )}{24 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )} \sqrt{x}}, \frac{105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3} - 105 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right )}{12 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^3*x^(5/2)),x, algorithm="fricas")
[Out]
[1/24*(210*b^3*x^3 + 350*a*b^2*x^2 + 112*a^2*b*x - 16*a^3 + 105*(b^3*x^3 + 2*a*b
^2*x^2 + a^2*b*x)*sqrt(x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x
+ a)))/((a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)*sqrt(x)), 1/12*(105*b^3*x^3 + 175*a
*b^2*x^2 + 56*a^2*b*x - 8*a^3 - 105*(b^3*x^3 + 2*a*b^2*x^2 + a^2*b*x)*sqrt(x)*sq
rt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))))/((a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)*sq
rt(x))]
_______________________________________________________________________________________
Sympy [A] time = 24.8237, size = 3177, normalized size = 33.44 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(5/2)/(b*x+a)**3,x)
[Out]
105*a**(59/2)*b**2*x**2*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**34*sqrt(b)*x**2 + 1
08*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 15
12*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 +
432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11
) + 945*a**(57/2)*b**3*x**3*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**34*sqrt(b)*x**2
+ 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5
+ 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x*
*8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x
**11) + 3780*a**(55/2)*b**4*x**4*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**34*sqrt(b)
*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*
x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/
2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19
/2)*x**11) + 8820*a**(53/2)*b**5*x**5*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**34*sq
rt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(
7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b*
*(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b
**(19/2)*x**11) + 13230*a**(51/2)*b**6*x**6*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a*
*34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**3
1*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a*
*28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a
**25*b**(19/2)*x**11) + 13230*a**(49/2)*b**7*x**7*atan(sqrt(b)*sqrt(x)/sqrt(a))/
(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 100
8*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1
008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10
+ 12*a**25*b**(19/2)*x**11) + 8820*a**(47/2)*b**8*x**8*atan(sqrt(b)*sqrt(x)/sqrt
(a))/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4
+ 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**
7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x
**10 + 12*a**25*b**(19/2)*x**11) + 3780*a**(45/2)*b**9*x**9*atan(sqrt(b)*sqrt(x)
/sqrt(a))/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*
x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2
)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17
/2)*x**10 + 12*a**25*b**(19/2)*x**11) + 945*a**(43/2)*b**10*x**10*atan(sqrt(b)*s
qrt(x)/sqrt(a))/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**
(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b*
*(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*
b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11) + 105*a**(41/2)*b**11*x**11*atan(sqr
t(b)*sqrt(x)/sqrt(a))/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**
32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a*
*29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*
a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11) - 8*a**31*sqrt(b)*sqrt(x)/(12*
a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a*
*31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*
a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12
*a**25*b**(19/2)*x**11) + 399*a**29*b**(5/2)*x**(5/2)/(12*a**34*sqrt(b)*x**2 + 1
08*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 15
12*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 +
432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11
) + 2226*a**28*b**(7/2)*x**(7/2)/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**
3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**
6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*
x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11) + 6090*a**27*b**(9/
2)*x**(9/2)/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2
)*x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11
/2)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(
17/2)*x**10 + 12*a**25*b**(19/2)*x**11) + 10122*a**26*b**(11/2)*x**(11/2)/(12*a*
*34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**3
1*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a*
*28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a
**25*b**(19/2)*x**11) + 10920*a**25*b**(13/2)*x**(13/2)/(12*a**34*sqrt(b)*x**2 +
108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 +
1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8
+ 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**
11) + 7734*a**24*b**(15/2)*x**(15/2)/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)
*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)
*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15
/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11) + 3486*a**23*b*
*(17/2)*x**(17/2)/(12*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b
**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*
b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**2
6*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x**11) + 910*a**22*b**(19/2)*x**(19/2)/(1
2*a**34*sqrt(b)*x**2 + 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*
a**31*b**(7/2)*x**5 + 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 100
8*a**28*b**(13/2)*x**8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 +
12*a**25*b**(19/2)*x**11) + 105*a**21*b**(21/2)*x**(21/2)/(12*a**34*sqrt(b)*x**2
+ 108*a**33*b**(3/2)*x**3 + 432*a**32*b**(5/2)*x**4 + 1008*a**31*b**(7/2)*x**5
+ 1512*a**30*b**(9/2)*x**6 + 1512*a**29*b**(11/2)*x**7 + 1008*a**28*b**(13/2)*x*
*8 + 432*a**27*b**(15/2)*x**9 + 108*a**26*b**(17/2)*x**10 + 12*a**25*b**(19/2)*x
**11)
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216417, size = 96, normalized size = 1.01 \[ \frac{35 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{4}} + \frac{2 \,{\left (9 \, b x - a\right )}}{3 \, a^{4} x^{\frac{3}{2}}} + \frac{11 \, b^{3} x^{\frac{3}{2}} + 13 \, a b^{2} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^3*x^(5/2)),x, algorithm="giac")
[Out]
35/4*b^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + 2/3*(9*b*x - a)/(a^4*x^(3
/2)) + 1/4*(11*b^3*x^(3/2) + 13*a*b^2*sqrt(x))/((b*x + a)^2*a^4)