3.318 \(\int \frac{(a+b x)^{9/2}}{x^2} \, dx\)

Optimal. Leaf size=98 \[ -9 a^{7/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+9 a^3 b \sqrt{a+b x}+3 a^2 b (a+b x)^{3/2}-\frac{(a+b x)^{9/2}}{x}+\frac{9}{7} b (a+b x)^{7/2}+\frac{9}{5} a b (a+b x)^{5/2} \]

[Out]

9*a^3*b*Sqrt[a + b*x] + 3*a^2*b*(a + b*x)^(3/2) + (9*a*b*(a + b*x)^(5/2))/5 + (9
*b*(a + b*x)^(7/2))/7 - (a + b*x)^(9/2)/x - 9*a^(7/2)*b*ArcTanh[Sqrt[a + b*x]/Sq
rt[a]]

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Rubi [A]  time = 0.106727, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -9 a^{7/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+9 a^3 b \sqrt{a+b x}+3 a^2 b (a+b x)^{3/2}-\frac{(a+b x)^{9/2}}{x}+\frac{9}{7} b (a+b x)^{7/2}+\frac{9}{5} a b (a+b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^2,x]

[Out]

9*a^3*b*Sqrt[a + b*x] + 3*a^2*b*(a + b*x)^(3/2) + (9*a*b*(a + b*x)^(5/2))/5 + (9
*b*(a + b*x)^(7/2))/7 - (a + b*x)^(9/2)/x - 9*a^(7/2)*b*ArcTanh[Sqrt[a + b*x]/Sq
rt[a]]

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Rubi in Sympy [A]  time = 13.8358, size = 92, normalized size = 0.94 \[ - 9 a^{\frac{7}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 9 a^{3} b \sqrt{a + b x} + 3 a^{2} b \left (a + b x\right )^{\frac{3}{2}} + \frac{9 a b \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{9 b \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**2,x)

[Out]

-9*a**(7/2)*b*atanh(sqrt(a + b*x)/sqrt(a)) + 9*a**3*b*sqrt(a + b*x) + 3*a**2*b*(
a + b*x)**(3/2) + 9*a*b*(a + b*x)**(5/2)/5 + 9*b*(a + b*x)**(7/2)/7 - (a + b*x)*
*(9/2)/x

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Mathematica [A]  time = 0.0832858, size = 84, normalized size = 0.86 \[ \sqrt{a+b x} \left (-\frac{a^4}{x}+\frac{388 a^3 b}{35}+\frac{156}{35} a^2 b^2 x+\frac{58}{35} a b^3 x^2+\frac{2 b^4 x^3}{7}\right )-9 a^{7/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^2,x]

[Out]

Sqrt[a + b*x]*((388*a^3*b)/35 - a^4/x + (156*a^2*b^2*x)/35 + (58*a*b^3*x^2)/35 +
(2*b^4*x^3)/7) - 9*a^(7/2)*b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.016, size = 84, normalized size = 0.9 \[ 2\,b \left ( 1/7\, \left ( bx+a \right ) ^{7/2}+2/5\,a \left ( bx+a \right ) ^{5/2}+{a}^{2} \left ( bx+a \right ) ^{3/2}+4\,\sqrt{bx+a}{a}^{3}+{a}^{4} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-9/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^2,x)

[Out]

2*b*(1/7*(b*x+a)^(7/2)+2/5*a*(b*x+a)^(5/2)+a^2*(b*x+a)^(3/2)+4*(b*x+a)^(1/2)*a^3
+a^4*(-1/2*(b*x+a)^(1/2)/x/b-9/2*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)^(9/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219487, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{\frac{7}{2}} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (10 \, b^{4} x^{4} + 58 \, a b^{3} x^{3} + 156 \, a^{2} b^{2} x^{2} + 388 \, a^{3} b x - 35 \, a^{4}\right )} \sqrt{b x + a}}{70 \, x}, -\frac{315 \, \sqrt{-a} a^{3} b x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (10 \, b^{4} x^{4} + 58 \, a b^{3} x^{3} + 156 \, a^{2} b^{2} x^{2} + 388 \, a^{3} b x - 35 \, a^{4}\right )} \sqrt{b x + a}}{35 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^2,x, algorithm="fricas")

[Out]

[1/70*(315*a^(7/2)*b*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(10*b^4*
x^4 + 58*a*b^3*x^3 + 156*a^2*b^2*x^2 + 388*a^3*b*x - 35*a^4)*sqrt(b*x + a))/x, -
1/35*(315*sqrt(-a)*a^3*b*x*arctan(sqrt(b*x + a)/sqrt(-a)) - (10*b^4*x^4 + 58*a*b
^3*x^3 + 156*a^2*b^2*x^2 + 388*a^3*b*x - 35*a^4)*sqrt(b*x + a))/x]

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Sympy [A]  time = 36.3084, size = 150, normalized size = 1.53 \[ - \frac{a^{\frac{9}{2}} \sqrt{1 + \frac{b x}{a}}}{x} + \frac{388 a^{\frac{7}{2}} b \sqrt{1 + \frac{b x}{a}}}{35} + \frac{9 a^{\frac{7}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} - 9 a^{\frac{7}{2}} b \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{156 a^{\frac{5}{2}} b^{2} x \sqrt{1 + \frac{b x}{a}}}{35} + \frac{58 a^{\frac{3}{2}} b^{3} x^{2} \sqrt{1 + \frac{b x}{a}}}{35} + \frac{2 \sqrt{a} b^{4} x^{3} \sqrt{1 + \frac{b x}{a}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**2,x)

[Out]

-a**(9/2)*sqrt(1 + b*x/a)/x + 388*a**(7/2)*b*sqrt(1 + b*x/a)/35 + 9*a**(7/2)*b*l
og(b*x/a)/2 - 9*a**(7/2)*b*log(sqrt(1 + b*x/a) + 1) + 156*a**(5/2)*b**2*x*sqrt(1
+ b*x/a)/35 + 58*a**(3/2)*b**3*x**2*sqrt(1 + b*x/a)/35 + 2*sqrt(a)*b**4*x**3*sq
rt(1 + b*x/a)/7

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GIAC/XCAS [A]  time = 0.211838, size = 140, normalized size = 1.43 \[ \frac{\frac{315 \, a^{4} b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 10 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{2} + 28 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{2} + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{2} + 280 \, \sqrt{b x + a} a^{3} b^{2} - \frac{35 \, \sqrt{b x + a} a^{4} b}{x}}{35 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^2,x, algorithm="giac")

[Out]

1/35*(315*a^4*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 10*(b*x + a)^(7/2)*b
^2 + 28*(b*x + a)^(5/2)*a*b^2 + 70*(b*x + a)^(3/2)*a^2*b^2 + 280*sqrt(b*x + a)*a
^3*b^2 - 35*sqrt(b*x + a)*a^4*b/x)/b