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

Optimal. Leaf size=78 \[ \frac{15}{4} b^2 \sqrt{a+b x}-\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{2 x^2}-\frac{5 b (a+b x)^{3/2}}{4 x} \]

[Out]

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

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Rubi [A]  time = 0.0678662, antiderivative size = 78, 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 \[ \frac{15}{4} b^2 \sqrt{a+b x}-\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{2 x^2}-\frac{5 b (a+b x)^{3/2}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.47812, size = 70, normalized size = 0.9 \[ - \frac{15 \sqrt{a} b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4} + \frac{15 b^{2} \sqrt{a + b x}}{4} - \frac{5 b \left (a + b x\right )^{\frac{3}{2}}}{4 x} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0643627, size = 64, normalized size = 0.82 \[ \left (-\frac{a^2}{2 x^2}-\frac{9 a b}{4 x}+2 b^2\right ) \sqrt{a+b x}-\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*b^2 - a^2/(2*x^2) - (9*a*b)/(4*x))*Sqrt[a + b*x] - (15*Sqrt[a]*b^2*ArcTanh[Sq
rt[a + b*x]/Sqrt[a]])/4

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Maple [A]  time = 0.018, size = 61, normalized size = 0.8 \[ 2\,{b}^{2} \left ( \sqrt{bx+a}+a \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -{\frac{9\, \left ( bx+a \right ) ^{3/2}}{8}}+{\frac{7\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{15}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x^3,x)

[Out]

2*b^2*((b*x+a)^(1/2)+a*((-9/8*(b*x+a)^(3/2)+7/8*a*(b*x+a)^(1/2))/x^2/b^2-15/8*ar
ctanh((b*x+a)^(1/2)/a^(1/2))/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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241503, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (8 \, b^{2} x^{2} - 9 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, x^{2}}, -\frac{15 \, \sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (8 \, b^{2} x^{2} - 9 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.3024, size = 126, normalized size = 1.62 \[ - \frac{15 \sqrt{a} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4} - \frac{a^{3}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{11 a^{2} \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{a b^{\frac{3}{2}}}{4 \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{5}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211816, size = 108, normalized size = 1.38 \[ \frac{\frac{15 \, a b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 8 \, \sqrt{b x + a} b^{3} - \frac{9 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{3} - 7 \, \sqrt{b x + a} a^{2} b^{3}}{b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(15*a*b^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 8*sqrt(b*x + a)*b^3 - (9
*(b*x + a)^(3/2)*a*b^3 - 7*sqrt(b*x + a)*a^2*b^3)/(b^2*x^2))/b