3.307 \(\int \frac{(a+b x)^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=81 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{(a+b x)^{5/2}}{3 x^3}-\frac{5 b (a+b x)^{3/2}}{12 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0690277, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{(a+b x)^{5/2}}{3 x^3}-\frac{5 b (a+b x)^{3/2}}{12 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.6991, size = 75, normalized size = 0.93 \[ - \frac{5 b^{2} \sqrt{a + b x}}{8 x} - \frac{5 b \left (a + b x\right )^{\frac{3}{2}}}{12 x^{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{3 x^{3}} - \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0677906, size = 64, normalized size = 0.79 \[ -\frac{\sqrt{a+b x} \left (8 a^2+26 a b x+33 b^2 x^2\right )}{24 x^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/x^4,x]

[Out]

-(Sqrt[a + b*x]*(8*a^2 + 26*a*b*x + 33*b^2*x^2))/(24*x^3) - (5*b^3*ArcTanh[Sqrt[
a + b*x]/Sqrt[a]])/(8*Sqrt[a])

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Maple [A]  time = 0.016, size = 63, normalized size = 0.8 \[ 2\,{b}^{3} \left ({\frac{1}{{b}^{3}{x}^{3}} \left ( -{\frac{11\, \left ( bx+a \right ) ^{5/2}}{16}}+5/6\,a \left ( bx+a \right ) ^{3/2}-{\frac{5\,{a}^{2}\sqrt{bx+a}}{16}} \right ) }-{\frac{5}{16\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x^4,x)

[Out]

2*b^3*((-11/16*(b*x+a)^(5/2)+5/6*a*(b*x+a)^(3/2)-5/16*a^2*(b*x+a)^(1/2))/x^3/b^3
-5/16*arctanh((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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220229, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (33 \, b^{2} x^{2} + 26 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{a}}{48 \, \sqrt{a} x^{3}}, \frac{15 \, b^{3} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (33 \, b^{2} x^{2} + 26 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(15*b^3*x^3*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) - 2*(33*b^2*x
^2 + 26*a*b*x + 8*a^2)*sqrt(b*x + a)*sqrt(a))/(sqrt(a)*x^3), 1/24*(15*b^3*x^3*ar
ctan(a/(sqrt(b*x + a)*sqrt(-a))) - (33*b^2*x^2 + 26*a*b*x + 8*a^2)*sqrt(b*x + a)
*sqrt(-a))/(sqrt(-a)*x^3)]

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Sympy [A]  time = 15.01, size = 104, normalized size = 1.28 \[ - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x^{\frac{5}{2}}} - \frac{13 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{12 x^{\frac{3}{2}}} - \frac{11 b^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}}{8 \sqrt{x}} - \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)/x**4,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x) + 1)/(3*x**(5/2)) - 13*a*b**(3/2)*sqrt(a/(b*x) + 1)/(
12*x**(3/2)) - 11*b**(5/2)*sqrt(a/(b*x) + 1)/(8*sqrt(x)) - 5*b**3*asinh(sqrt(a)/
(sqrt(b)*sqrt(x)))/(8*sqrt(a))

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GIAC/XCAS [A]  time = 0.209423, size = 107, normalized size = 1.32 \[ \frac{\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{33 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} - 40 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} + 15 \, \sqrt{b x + a} a^{2} b^{4}}{b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/x^4,x, algorithm="giac")

[Out]

1/24*(15*b^4*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - (33*(b*x + a)^(5/2)*b^4 -
40*(b*x + a)^(3/2)*a*b^4 + 15*sqrt(b*x + a)*a^2*b^4)/(b^3*x^3))/b