3.466 \(\int \frac{1}{\sqrt{x} (a+b x)^3} \, dx\)

Optimal. Leaf size=70 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}+\frac{3 \sqrt{x}}{4 a^2 (a+b x)}+\frac{\sqrt{x}}{2 a (a+b x)^2} \]

[Out]

Sqrt[x]/(2*a*(a + b*x)^2) + (3*Sqrt[x])/(4*a^2*(a + b*x)) + (3*ArcTan[(Sqrt[b]*S
qrt[x])/Sqrt[a]])/(4*a^(5/2)*Sqrt[b])

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Rubi [A]  time = 0.0490012, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}+\frac{3 \sqrt{x}}{4 a^2 (a+b x)}+\frac{\sqrt{x}}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(a + b*x)^3),x]

[Out]

Sqrt[x]/(2*a*(a + b*x)^2) + (3*Sqrt[x])/(4*a^2*(a + b*x)) + (3*ArcTan[(Sqrt[b]*S
qrt[x])/Sqrt[a]])/(4*a^(5/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 9.62725, size = 61, normalized size = 0.87 \[ \frac{\sqrt{x}}{2 a \left (a + b x\right )^{2}} + \frac{3 \sqrt{x}}{4 a^{2} \left (a + b x\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/(2*a*(a + b*x)**2) + 3*sqrt(x)/(4*a**2*(a + b*x)) + 3*atan(sqrt(b)*sqrt(
x)/sqrt(a))/(4*a**(5/2)*sqrt(b))

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Mathematica [A]  time = 0.0452043, size = 59, normalized size = 0.84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}+\frac{\sqrt{x} (5 a+3 b x)}{4 a^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 53, normalized size = 0.8 \[{\frac{1}{2\,a \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{3}{4\,{a}^{2} \left ( bx+a \right ) }\sqrt{x}}+{\frac{3}{4\,{a}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^3/x^(1/2),x)

[Out]

1/2*x^(1/2)/a/(b*x+a)^2+3/4*x^(1/2)/a^2/(b*x+a)+3/4/a^2/(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*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221775, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b}{\left (3 \, b x + 5 \, a\right )} \sqrt{x} + 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \sqrt{-a b}}, \frac{\sqrt{a b}{\left (3 \, b x + 5 \, a\right )} \sqrt{x} - 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*sqrt(-a*b)*(3*b*x + 5*a)*sqrt(x) + 3*(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^2*b^2*x^2 + 2*a^3*b*x + a^4)*s
qrt(-a*b)), 1/4*(sqrt(a*b)*(3*b*x + 5*a)*sqrt(x) - 3*(b^2*x^2 + 2*a*b*x + a^2)*a
rctan(a/(sqrt(a*b)*sqrt(x))))/((a^2*b^2*x^2 + 2*a^3*b*x + a^4)*sqrt(a*b))]

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Sympy [A]  time = 10.1117, size = 590, normalized size = 8.43 \[ \frac{3 a^{\frac{11}{2}} \sqrt{x} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{9 a^{\frac{9}{2}} b x^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{9 a^{\frac{7}{2}} b^{2} x^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{3 a^{\frac{5}{2}} b^{3} x^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{5 a^{5} \sqrt{b} x}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{8 a^{4} b^{\frac{3}{2}} x^{2}}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{3 a^{3} b^{\frac{5}{2}} x^{3}}{4 a^{8} \sqrt{b} \sqrt{x} + 12 a^{7} b^{\frac{3}{2}} x^{\frac{3}{2}} + 12 a^{6} b^{\frac{5}{2}} x^{\frac{5}{2}} + 4 a^{5} b^{\frac{7}{2}} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**(11/2)*sqrt(x)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**8*sqrt(b)*sqrt(x) + 12*a
**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2) + 4*a**5*b**(7/2)*x**(7/2)) +
9*a**(9/2)*b*x**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**8*sqrt(b)*sqrt(x) + 12
*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2) + 4*a**5*b**(7/2)*x**(7/2))
+ 9*a**(7/2)*b**2*x**(5/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**8*sqrt(b)*sqrt(x)
+ 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2) + 4*a**5*b**(7/2)*x**(7
/2)) + 3*a**(5/2)*b**3*x**(7/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**8*sqrt(b)*sq
rt(x) + 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2) + 4*a**5*b**(7/2)*
x**(7/2)) + 5*a**5*sqrt(b)*x/(4*a**8*sqrt(b)*sqrt(x) + 12*a**7*b**(3/2)*x**(3/2)
+ 12*a**6*b**(5/2)*x**(5/2) + 4*a**5*b**(7/2)*x**(7/2)) + 8*a**4*b**(3/2)*x**2/
(4*a**8*sqrt(b)*sqrt(x) + 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2)
+ 4*a**5*b**(7/2)*x**(7/2)) + 3*a**3*b**(5/2)*x**3/(4*a**8*sqrt(b)*sqrt(x) + 12*
a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2) + 4*a**5*b**(7/2)*x**(7/2))

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GIAC/XCAS [A]  time = 0.204984, size = 63, normalized size = 0.9 \[ \frac{3 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{2}} + \frac{3 \, b x^{\frac{3}{2}} + 5 \, a \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/4*(3*b*x^(3/2) + 5*a*sqrt(x)
)/((b*x + a)^2*a^2)