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

Optimal. Leaf size=69 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0516152, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{2 \sqrt{x}}{b^2 \sqrt{a+b x}}-\frac{2 x^{3/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.29108, size = 63, normalized size = 0.91 \[ - \frac{2 x^{\frac{3}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \sqrt{x}}{b^{2} \sqrt{a + b x}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(3/2)/(3*b*(a + b*x)**(3/2)) - 2*sqrt(x)/(b**2*sqrt(a + b*x)) + 2*atanh(sq
rt(b)*sqrt(x)/sqrt(a + b*x))/b**(5/2)

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Mathematica [A]  time = 0.104215, size = 61, normalized size = 0.88 \[ \frac{2 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{5/2}}-\frac{2 \sqrt{x} (3 a+4 b x)}{3 b^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x+a)^(5/2),x)

[Out]

int(x^(3/2)/(b*x+a)^(5/2),x)

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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^(3/2)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228741, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (4 \, b x^{2} + 3 \, a x\right )} \sqrt{b}}{3 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (4 \, b x^{2} + 3 \, a x\right )} \sqrt{-b}\right )}}{3 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 30.313, size = 328, normalized size = 4.75 \[ \frac{6 a^{\frac{39}{2}} b^{11} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{6 a^{\frac{37}{2}} b^{12} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{6 a^{19} b^{\frac{23}{2}} x^{14}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{8 a^{18} b^{\frac{25}{2}} x^{15}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*a**(39/2)*b**11*x**(27/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a*
*(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sq
rt(1 + b*x/a)) + 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(
x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(2
9/2)*x**(29/2)*sqrt(1 + b*x/a)) - 6*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)
*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) -
8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**
(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a))

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GIAC/XCAS [A]  time = 0.227009, size = 223, normalized size = 3.23 \[ -\frac{{\left (\frac{3 \,{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}} + \frac{8 \,{\left (3 \, a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b} + 3 \, a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}} + 2 \, a^{3} b^{\frac{5}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3}}\right )}{\left | b \right |}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/sqrt(b) + 8*(3*a
*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b) + 3*a^2*(sqrt(b*x +
a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2) + 2*a^3*b^(5/2))/((sqrt(b*x + a
)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3)*abs(b)/b^3