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

Optimal. Leaf size=64 \[ 2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{a+b x}}{\sqrt{x}} \]

[Out]

(-2*b*Sqrt[a + b*x])/Sqrt[x] - (2*(a + b*x)^(3/2))/(3*x^(3/2)) + 2*b^(3/2)*ArcTa
nh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]]

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Rubi [A]  time = 0.0485427, antiderivative size = 64, 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 \[ 2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/x^(5/2),x]

[Out]

(-2*b*Sqrt[a + b*x])/Sqrt[x] - (2*(a + b*x)^(3/2))/(3*x^(3/2)) + 2*b^(3/2)*ArcTa
nh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]]

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Rubi in Sympy [A]  time = 8.03064, size = 60, normalized size = 0.94 \[ 2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} - \frac{2 b \sqrt{a + b x}}{\sqrt{x}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0482045, size = 56, normalized size = 0.88 \[ 2 b^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x} (a+4 b x)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.029, size = 67, normalized size = 1.1 \[ -{\frac{8\,bx+2\,a}{3}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}}+{1{b}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220613, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b^{\frac{3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (4 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}, \frac{2 \,{\left (3 \, \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) -{\left (4 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 27.8618, size = 71, normalized size = 1.11 \[ - \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*sqrt(b)*sqrt(a/(b*x) + 1)/(3*x) - 8*b**(3/2)*sqrt(a/(b*x) + 1)/3 - b**(3/2)
*log(a/(b*x)) + 2*b**(3/2)*log(sqrt(a/(b*x) + 1) + 1)

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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)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError