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

Optimal. Leaf size=97 \[ -2 a^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a^4 \sqrt{a+b x}+\frac{2}{3} a^3 (a+b x)^{3/2}+\frac{2}{5} a^2 (a+b x)^{5/2}+\frac{2}{7} a (a+b x)^{7/2}+\frac{2}{9} (a+b x)^{9/2} \]

[Out]

2*a^4*Sqrt[a + b*x] + (2*a^3*(a + b*x)^(3/2))/3 + (2*a^2*(a + b*x)^(5/2))/5 + (2
*a*(a + b*x)^(7/2))/7 + (2*(a + b*x)^(9/2))/9 - 2*a^(9/2)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]]

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Rubi [A]  time = 0.102232, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -2 a^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a^4 \sqrt{a+b x}+\frac{2}{3} a^3 (a+b x)^{3/2}+\frac{2}{5} a^2 (a+b x)^{5/2}+\frac{2}{7} a (a+b x)^{7/2}+\frac{2}{9} (a+b x)^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x,x]

[Out]

2*a^4*Sqrt[a + b*x] + (2*a^3*(a + b*x)^(3/2))/3 + (2*a^2*(a + b*x)^(5/2))/5 + (2
*a*(a + b*x)^(7/2))/7 + (2*(a + b*x)^(9/2))/9 - 2*a^(9/2)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]]

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Rubi in Sympy [A]  time = 13.6862, size = 90, normalized size = 0.93 \[ - 2 a^{\frac{9}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 a^{4} \sqrt{a + b x} + \frac{2 a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{2 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{2 \left (a + b x\right )^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(9/2)*atanh(sqrt(a + b*x)/sqrt(a)) + 2*a**4*sqrt(a + b*x) + 2*a**3*(a + b*
x)**(3/2)/3 + 2*a**2*(a + b*x)**(5/2)/5 + 2*a*(a + b*x)**(7/2)/7 + 2*(a + b*x)**
(9/2)/9

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Mathematica [A]  time = 0.0525345, size = 78, normalized size = 0.8 \[ \frac{2}{315} \sqrt{a+b x} \left (563 a^4+506 a^3 b x+408 a^2 b^2 x^2+185 a b^3 x^3+35 b^4 x^4\right )-2 a^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x,x]

[Out]

(2*Sqrt[a + b*x]*(563*a^4 + 506*a^3*b*x + 408*a^2*b^2*x^2 + 185*a*b^3*x^3 + 35*b
^4*x^4))/315 - 2*a^(9/2)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.009, size = 74, normalized size = 0.8 \[{\frac{2\,{a}^{3}}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{a}^{2}}{5} \left ( bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{7} \left ( bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{2}{9} \left ( bx+a \right ) ^{{\frac{9}{2}}}}-2\,{a}^{9/2}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +2\,{a}^{4}\sqrt{bx+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221988, size = 1, normalized size = 0.01 \[ \left [a^{\frac{9}{2}} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \frac{2}{315} \,{\left (35 \, b^{4} x^{4} + 185 \, a b^{3} x^{3} + 408 \, a^{2} b^{2} x^{2} + 506 \, a^{3} b x + 563 \, a^{4}\right )} \sqrt{b x + a}, -2 \, \sqrt{-a} a^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) + \frac{2}{315} \,{\left (35 \, b^{4} x^{4} + 185 \, a b^{3} x^{3} + 408 \, a^{2} b^{2} x^{2} + 506 \, a^{3} b x + 563 \, a^{4}\right )} \sqrt{b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[a^(9/2)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2/315*(35*b^4*x^4 + 185*
a*b^3*x^3 + 408*a^2*b^2*x^2 + 506*a^3*b*x + 563*a^4)*sqrt(b*x + a), -2*sqrt(-a)*
a^4*arctan(sqrt(b*x + a)/sqrt(-a)) + 2/315*(35*b^4*x^4 + 185*a*b^3*x^3 + 408*a^2
*b^2*x^2 + 506*a^3*b*x + 563*a^4)*sqrt(b*x + a)]

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Sympy [A]  time = 35.2355, size = 148, normalized size = 1.53 \[ \frac{1126 a^{\frac{9}{2}} \sqrt{1 + \frac{b x}{a}}}{315} + a^{\frac{9}{2}} \log{\left (\frac{b x}{a} \right )} - 2 a^{\frac{9}{2}} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{1012 a^{\frac{7}{2}} b x \sqrt{1 + \frac{b x}{a}}}{315} + \frac{272 a^{\frac{5}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{105} + \frac{74 a^{\frac{3}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{63} + \frac{2 \sqrt{a} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x,x)

[Out]

1126*a**(9/2)*sqrt(1 + b*x/a)/315 + a**(9/2)*log(b*x/a) - 2*a**(9/2)*log(sqrt(1
+ b*x/a) + 1) + 1012*a**(7/2)*b*x*sqrt(1 + b*x/a)/315 + 272*a**(5/2)*b**2*x**2*s
qrt(1 + b*x/a)/105 + 74*a**(3/2)*b**3*x**3*sqrt(1 + b*x/a)/63 + 2*sqrt(a)*b**4*x
**4*sqrt(1 + b*x/a)/9

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GIAC/XCAS [A]  time = 0.209386, size = 108, normalized size = 1.11 \[ \frac{2 \, a^{5} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2}{9} \,{\left (b x + a\right )}^{\frac{9}{2}} + \frac{2}{7} \,{\left (b x + a\right )}^{\frac{7}{2}} a + \frac{2}{5} \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} + \frac{2}{3} \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} + 2 \, \sqrt{b x + a} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a^5*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2/9*(b*x + a)^(9/2) + 2/7*(b*x +
a)^(7/2)*a + 2/5*(b*x + a)^(5/2)*a^2 + 2/3*(b*x + a)^(3/2)*a^3 + 2*sqrt(b*x + a
)*a^4