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

Optimal. Leaf size=89 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{5/2}}{\sqrt{x}}+\frac{5}{2} b \sqrt{x} (a+b x)^{3/2}+\frac{15}{4} a b \sqrt{x} \sqrt{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.0662144, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{5/2}}{\sqrt{x}}+\frac{5}{2} b \sqrt{x} (a+b x)^{3/2}+\frac{15}{4} a b \sqrt{x} \sqrt{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.58941, size = 85, normalized size = 0.96 \[ \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4} + \frac{15 a b \sqrt{x} \sqrt{a + b x}}{4} + \frac{5 b \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{2} - \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0651111, size = 73, normalized size = 0.82 \[ \frac{1}{4} \left (\frac{\sqrt{a+b x} \left (-8 a^2+9 a b x+2 b^2 x^2\right )}{\sqrt{x}}+15 a^2 \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a + b*x]*(-8*a^2 + 9*a*b*x + 2*b^2*x^2))/Sqrt[x] + 15*a^2*Sqrt[b]*Log[b*S
qrt[x] + Sqrt[b]*Sqrt[a + b*x]])/4

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Maple [A]  time = 0.026, size = 84, normalized size = 0.9 \[ -{\frac{-2\,{b}^{2}{x}^{2}-9\,abx+8\,{a}^{2}}{4}\sqrt{bx+a}{\frac{1}{\sqrt{x}}}}+{\frac{15\,{a}^{2}}{8}\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x^(3/2),x)

[Out]

-1/4*(b*x+a)^(1/2)*(-2*b^2*x^2-9*a*b*x+8*a^2)/x^(1/2)+15/8*a^2*b^(1/2)*ln((1/2*a
+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(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((b*x + a)^(5/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228409, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{8 \, x}, \frac{15 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{4 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*a^2*sqrt(b)*x*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(2*b
^2*x^2 + 9*a*b*x - 8*a^2)*sqrt(b*x + a)*sqrt(x))/x, 1/4*(15*a^2*sqrt(-b)*x*arcta
n(sqrt(b*x + a)/(sqrt(-b)*sqrt(x))) + (2*b^2*x^2 + 9*a*b*x - 8*a^2)*sqrt(b*x + a
)*sqrt(x))/x]

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Sympy [A]  time = 78.4115, size = 126, normalized size = 1.42 \[ - \frac{2 a^{\frac{5}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + \frac{a^{\frac{3}{2}} b \sqrt{x}}{4 \sqrt{1 + \frac{b x}{a}}} + \frac{11 \sqrt{a} b^{2} x^{\frac{3}{2}}}{4 \sqrt{1 + \frac{b x}{a}}} + \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4} + \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/2)/(sqrt(x)*sqrt(1 + b*x/a)) + a**(3/2)*b*sqrt(x)/(4*sqrt(1 + b*x/a)) +
11*sqrt(a)*b**2*x**(3/2)/(4*sqrt(1 + b*x/a)) + 15*a**2*sqrt(b)*asinh(sqrt(b)*sq
rt(x)/sqrt(a))/4 + b**3*x**(5/2)/(2*sqrt(a)*sqrt(1 + b*x/a))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError