3.463 \(\int \frac{x^{5/2}}{(a+b x)^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0600618, 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 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}}-\frac{5 x^{3/2}}{4 b^2 (a+b x)}-\frac{x^{5/2}}{2 b (a+b x)^2}+\frac{15 \sqrt{x}}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.422, size = 73, normalized size = 0.89 \[ - \frac{15 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} - \frac{x^{\frac{5}{2}}}{2 b \left (a + b x\right )^{2}} - \frac{5 x^{\frac{3}{2}}}{4 b^{2} \left (a + b x\right )} + \frac{15 \sqrt{x}}{4 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0596256, size = 70, normalized size = 0.85 \[ \frac{\sqrt{x} \left (15 a^2+25 a b x+8 b^2 x^2\right )}{4 b^3 (a+b x)^2}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(a + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.017, size = 66, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{{b}^{3}}}+{\frac{9\,a}{4\,{b}^{2} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{7\,{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{15\,a}{4\,{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x+a)^3,x)

[Out]

2*x^(1/2)/b^3+9/4/b^2*a/(b*x+a)^2*x^(3/2)+7/4/b^3*a^2/(b*x+a)^2*x^(1/2)-15/4/b^3
*a/(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(x^(5/2)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240404, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (8 \, b^{2} x^{2} + 25 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(b*x + a)^3,x, algorithm="fricas")

[Out]

[1/8*(15*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b)
- a)/(b*x + a)) + 2*(8*b^2*x^2 + 25*a*b*x + 15*a^2)*sqrt(x))/(b^5*x^2 + 2*a*b^4*
x + a^2*b^3), -1/4*(15*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a
/b)) - (8*b^2*x^2 + 25*a*b*x + 15*a^2)*sqrt(x))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)]

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Sympy [A]  time = 15.4799, size = 672, normalized size = 8.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x+a)**3,x)

[Out]

-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**(2
7/2) + 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) + 1
2*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**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
**(23/2)*x**15/(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)) + 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))

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GIAC/XCAS [A]  time = 0.206419, size = 80, normalized size = 0.98 \[ -\frac{15 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3}} + \frac{2 \, \sqrt{x}}{b^{3}} + \frac{9 \, a b x^{\frac{3}{2}} + 7 \, a^{2} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(b*x + a)^3,x, algorithm="giac")

[Out]

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