3.467 \(\int \frac{1}{x^{3/2} (a+b x)^3} \, dx\)

Optimal. Leaf size=82 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15}{4 a^3 \sqrt{x}}+\frac{5}{4 a^2 \sqrt{x} (a+b x)}+\frac{1}{2 a \sqrt{x} (a+b x)^2} \]

[Out]

-15/(4*a^3*Sqrt[x]) + 1/(2*a*Sqrt[x]*(a + b*x)^2) + 5/(4*a^2*Sqrt[x]*(a + b*x))
- (15*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.0615942, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{15}{4 a^3 \sqrt{x}}+\frac{5}{4 a^2 \sqrt{x} (a+b x)}+\frac{1}{2 a \sqrt{x} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + b*x)^3),x]

[Out]

-15/(4*a^3*Sqrt[x]) + 1/(2*a*Sqrt[x]*(a + b*x)^2) + 5/(4*a^2*Sqrt[x]*(a + b*x))
- (15*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 12.3122, size = 75, normalized size = 0.91 \[ \frac{1}{2 a \sqrt{x} \left (a + b x\right )^{2}} + \frac{5}{4 a^{2} \sqrt{x} \left (a + b x\right )} - \frac{15}{4 a^{3} \sqrt{x}} - \frac{15 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(b*x+a)**3,x)

[Out]

1/(2*a*sqrt(x)*(a + b*x)**2) + 5/(4*a**2*sqrt(x)*(a + b*x)) - 15/(4*a**3*sqrt(x)
) - 15*sqrt(b)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(7/2))

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Mathematica [A]  time = 0.0600499, size = 70, normalized size = 0.85 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{8 a^2+25 a b x+15 b^2 x^2}{4 a^3 \sqrt{x} (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(8*a^2 + 25*a*b*x + 15*b^2*x^2)/(4*a^3*Sqrt[x]*(a + b*x)^2) - (15*Sqrt[b]*ArcTa
n[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2))

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Maple [A]  time = 0.019, size = 66, normalized size = 0.8 \[ -2\,{\frac{1}{{a}^{3}\sqrt{x}}}-{\frac{7\,{b}^{2}}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{9\,b}{4\,{a}^{2} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{15\,b}{4\,{a}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x+a)^3,x)

[Out]

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

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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(1/((b*x + a)^3*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225581, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} x^{2} + 50 \, a b x - 15 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 16 \, a^{2}}{8 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \sqrt{x}}, -\frac{15 \, b^{2} x^{2} + 25 \, a b x - 15 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 8 \, a^{2}}{4 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^(3/2)),x, algorithm="fricas")

[Out]

[-1/8*(30*b^2*x^2 + 50*a*b*x - 15*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(x)*sqrt(-b/a)*l
og((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 16*a^2)/((a^3*b^2*x^2 + 2*a^4
*b*x + a^5)*sqrt(x)), -1/4*(15*b^2*x^2 + 25*a*b*x - 15*(b^2*x^2 + 2*a*b*x + a^2)
*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 8*a^2)/((a^3*b^2*x^2 + 2*a^
4*b*x + a^5)*sqrt(x))]

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Sympy [A]  time = 13.0209, size = 2790, normalized size = 34.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(b*x+a)**3,x)

[Out]

-8*a**(45/2)/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2
*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) + 224*a**(
41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 32*a**(37/2)*b**7*x**(15/2
) + 4*a**(35/2)*b**8*x**(17/2)) - 73*a**(43/2)*b*x/(4*a**(51/2)*sqrt(x) + 32*a**
(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) +
280*a**(43/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*
x**(13/2) + 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 285*a**(
41/2)*b**2*x**2/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b
**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) + 224*a
**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 32*a**(37/2)*b**7*x**(1
5/2) + 4*a**(35/2)*b**8*x**(17/2)) - 625*a**(39/2)*b**3*x**3/(4*a**(51/2)*sqrt(x
) + 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x
**(7/2) + 280*a**(43/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(3
9/2)*b**6*x**(13/2) + 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2))
- 845*a**(37/2)*b**4*x**4/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(3/2) + 112*a
**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/
2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 32*a**(37/2)*
b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 723*a**(35/2)*b**5*x**5/(4*a**(51
/2)*sqrt(x) + 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45
/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) +
112*a**(39/2)*b**6*x**(13/2) + 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x
**(17/2)) - 383*a**(33/2)*b**6*x**6/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(3/
2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b
**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 32*
a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 115*a**(31/2)*b**7*x**7
/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) +
224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x
**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35
/2)*b**8*x**(17/2)) - 15*a**(29/2)*b**8*x**8/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)
*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a*
*(43/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13
/2) + 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 15*a**22*sqrt(
b)*sqrt(x)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x
**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43
/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2)
+ 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 120*a**21*b**(3/2)
*x**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x*
*(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/
2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) +
32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 420*a**20*b**(5/2)*
x**(5/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**
(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2
)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) +
32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 840*a**19*b**(7/2)*x
**(7/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(
3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)
*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 3
2*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 1050*a**18*b**(9/2)*x
**(9/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**(
3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)
*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) + 3
2*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 840*a**17*b**(11/2)*x
**(11/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x**
(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2
)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) +
32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 420*a**16*b**(13/2)*
x**(13/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x*
*(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/
2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) +
32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 120*a**15*b**(15/2)
*x**(15/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x
**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43
/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2)
+ 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 15*a**14*b**(17/2)
*x**(17/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(51/2)*sqrt(x) + 32*a**(49/2)*b*x
**(3/2) + 112*a**(47/2)*b**2*x**(5/2) + 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43
/2)*b**4*x**(9/2) + 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2)
+ 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2))

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GIAC/XCAS [A]  time = 0.205264, size = 80, normalized size = 0.98 \[ -\frac{15 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{3}} - \frac{2}{a^{3} \sqrt{x}} - \frac{7 \, b^{2} x^{\frac{3}{2}} + 9 \, a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^(3/2)),x, algorithm="giac")

[Out]

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