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

Optimal. Leaf size=101 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b} \]

[Out]

(5*a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b^3) - (5*a*x^(3/2)*Sqrt[a + b*x])/(12*b^2) + (
x^(5/2)*Sqrt[a + b*x])/(3*b) - (5*a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/
(8*b^(7/2))

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Rubi [A]  time = 0.0749691, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/Sqrt[a + b*x],x]

[Out]

(5*a^2*Sqrt[x]*Sqrt[a + b*x])/(8*b^3) - (5*a*x^(3/2)*Sqrt[a + b*x])/(12*b^2) + (
x^(5/2)*Sqrt[a + b*x])/(3*b) - (5*a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/
(8*b^(7/2))

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Rubi in Sympy [A]  time = 11.2735, size = 94, normalized size = 0.93 \[ - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{8 b^{\frac{7}{2}}} + \frac{5 a^{2} \sqrt{x} \sqrt{a + b x}}{8 b^{3}} - \frac{5 a x^{\frac{3}{2}} \sqrt{a + b x}}{12 b^{2}} + \frac{x^{\frac{5}{2}} \sqrt{a + b x}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(b*x+a)**(1/2),x)

[Out]

-5*a**3*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(8*b**(7/2)) + 5*a**2*sqrt(x)*sqrt(
a + b*x)/(8*b**3) - 5*a*x**(3/2)*sqrt(a + b*x)/(12*b**2) + x**(5/2)*sqrt(a + b*x
)/(3*b)

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Mathematica [A]  time = 0.0765191, size = 77, normalized size = 0.76 \[ \frac{\sqrt{x} \sqrt{a+b x} \left (15 a^2-10 a b x+8 b^2 x^2\right )}{24 b^3}-\frac{5 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{8 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[a + b*x]*(15*a^2 - 10*a*b*x + 8*b^2*x^2))/(24*b^3) - (5*a^3*Log[b*
Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(8*b^(7/2))

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Maple [A]  time = 0.009, size = 102, normalized size = 1. \[{\frac{1}{3\,b}{x}^{{\frac{5}{2}}}\sqrt{bx+a}}-{\frac{5\,a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{2}}{8\,{b}^{3}}\sqrt{x}\sqrt{bx+a}}-{\frac{5\,{a}^{3}}{16}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x+a)^(1/2),x)

[Out]

1/3*x^(5/2)*(b*x+a)^(1/2)/b-5/12*a*x^(3/2)*(b*x+a)^(1/2)/b^2+5/8*a^2*x^(1/2)*(b*
x+a)^(1/2)/b^3-5/16*a^3/b^(7/2)*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)*ln((1/2*
a+b*x)/b^(1/2)+(b*x^2+a*x)^(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)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245577, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{2} - 10 \, a b x + 15 \, a^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{7}{2}}}, -\frac{15 \, a^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, b^{2} x^{2} - 10 \, a b x + 15 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*a^3*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(8*b^2*x
^2 - 10*a*b*x + 15*a^2)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/b^(7/2), -1/24*(15*a^3*ar
ctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (8*b^2*x^2 - 10*a*b*x + 15*a^2)*sqrt(
b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 70.1071, size = 128, normalized size = 1.27 \[ \frac{5 a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{5 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 + \frac{b x}{a}}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x+a)**(1/2),x)

[Out]

5*a**(5/2)*sqrt(x)/(8*b**3*sqrt(1 + b*x/a)) + 5*a**(3/2)*x**(3/2)/(24*b**2*sqrt(
1 + b*x/a)) - sqrt(a)*x**(5/2)/(12*b*sqrt(1 + b*x/a)) - 5*a**3*asinh(sqrt(b)*sqr
t(x)/sqrt(a))/(8*b**(7/2)) + x**(7/2)/(3*sqrt(a)*sqrt(1 + b*x/a))

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GIAC/XCAS [A]  time = 12.4633, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x