3.319 \(\int \frac{(a+b x)^{9/2}}{x^3} \, dx\)

Optimal. Leaf size=114 \[ -\frac{63}{4} a^{5/2} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{63}{4} a^2 b^2 \sqrt{a+b x}+\frac{63}{20} b^2 (a+b x)^{5/2}+\frac{21}{4} a b^2 (a+b x)^{3/2}-\frac{(a+b x)^{9/2}}{2 x^2}-\frac{9 b (a+b x)^{7/2}}{4 x} \]

[Out]

(63*a^2*b^2*Sqrt[a + b*x])/4 + (21*a*b^2*(a + b*x)^(3/2))/4 + (63*b^2*(a + b*x)^
(5/2))/20 - (9*b*(a + b*x)^(7/2))/(4*x) - (a + b*x)^(9/2)/(2*x^2) - (63*a^(5/2)*
b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4

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Rubi [A]  time = 0.110132, antiderivative size = 114, 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 \[ -\frac{63}{4} a^{5/2} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{63}{4} a^2 b^2 \sqrt{a+b x}+\frac{63}{20} b^2 (a+b x)^{5/2}+\frac{21}{4} a b^2 (a+b x)^{3/2}-\frac{(a+b x)^{9/2}}{2 x^2}-\frac{9 b (a+b x)^{7/2}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^3,x]

[Out]

(63*a^2*b^2*Sqrt[a + b*x])/4 + (21*a*b^2*(a + b*x)^(3/2))/4 + (63*b^2*(a + b*x)^
(5/2))/20 - (9*b*(a + b*x)^(7/2))/(4*x) - (a + b*x)^(9/2)/(2*x^2) - (63*a^(5/2)*
b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4

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Rubi in Sympy [A]  time = 14.6117, size = 105, normalized size = 0.92 \[ - \frac{63 a^{\frac{5}{2}} b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4} + \frac{63 a^{2} b^{2} \sqrt{a + b x}}{4} + \frac{21 a b^{2} \left (a + b x\right )^{\frac{3}{2}}}{4} + \frac{63 b^{2} \left (a + b x\right )^{\frac{5}{2}}}{20} - \frac{9 b \left (a + b x\right )^{\frac{7}{2}}}{4 x} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**3,x)

[Out]

-63*a**(5/2)*b**2*atanh(sqrt(a + b*x)/sqrt(a))/4 + 63*a**2*b**2*sqrt(a + b*x)/4
+ 21*a*b**2*(a + b*x)**(3/2)/4 + 63*b**2*(a + b*x)**(5/2)/20 - 9*b*(a + b*x)**(7
/2)/(4*x) - (a + b*x)**(9/2)/(2*x**2)

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Mathematica [A]  time = 0.0851843, size = 86, normalized size = 0.75 \[ \frac{\sqrt{a+b x} \left (-10 a^4-85 a^3 b x+288 a^2 b^2 x^2+56 a b^3 x^3+8 b^4 x^4\right )}{20 x^2}-\frac{63}{4} a^{5/2} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^3,x]

[Out]

(Sqrt[a + b*x]*(-10*a^4 - 85*a^3*b*x + 288*a^2*b^2*x^2 + 56*a*b^3*x^3 + 8*b^4*x^
4))/(20*x^2) - (63*a^(5/2)*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4

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Maple [A]  time = 0.019, size = 86, normalized size = 0.8 \[ 2\,{b}^{2} \left ( 1/5\, \left ( bx+a \right ) ^{5/2}+a \left ( bx+a \right ) ^{3/2}+6\,{a}^{2}\sqrt{bx+a}+{a}^{3} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -{\frac{17\, \left ( bx+a \right ) ^{3/2}}{8}}+{\frac{15\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{63}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^3,x)

[Out]

2*b^2*(1/5*(b*x+a)^(5/2)+a*(b*x+a)^(3/2)+6*a^2*(b*x+a)^(1/2)+a^3*((-17/8*(b*x+a)
^(3/2)+15/8*a*(b*x+a)^(1/2))/x^2/b^2-63/8*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221428, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{\frac{5}{2}} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (8 \, b^{4} x^{4} + 56 \, a b^{3} x^{3} + 288 \, a^{2} b^{2} x^{2} - 85 \, a^{3} b x - 10 \, a^{4}\right )} \sqrt{b x + a}}{40 \, x^{2}}, -\frac{315 \, \sqrt{-a} a^{2} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (8 \, b^{4} x^{4} + 56 \, a b^{3} x^{3} + 288 \, a^{2} b^{2} x^{2} - 85 \, a^{3} b x - 10 \, a^{4}\right )} \sqrt{b x + a}}{20 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^3,x, algorithm="fricas")

[Out]

[1/40*(315*a^(5/2)*b^2*x^2*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(8*b
^4*x^4 + 56*a*b^3*x^3 + 288*a^2*b^2*x^2 - 85*a^3*b*x - 10*a^4)*sqrt(b*x + a))/x^
2, -1/20*(315*sqrt(-a)*a^2*b^2*x^2*arctan(sqrt(b*x + a)/sqrt(-a)) - (8*b^4*x^4 +
56*a*b^3*x^3 + 288*a^2*b^2*x^2 - 85*a^3*b*x - 10*a^4)*sqrt(b*x + a))/x^2]

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Sympy [A]  time = 34.7085, size = 184, normalized size = 1.61 \[ - \frac{63 a^{\frac{5}{2}} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4} - \frac{a^{5}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{19 a^{4} \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{203 a^{3} b^{\frac{3}{2}}}{20 \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{86 a^{2} b^{\frac{5}{2}} \sqrt{x}}{5 \sqrt{\frac{a}{b x} + 1}} + \frac{16 a b^{\frac{7}{2}} x^{\frac{3}{2}}}{5 \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{9}{2}} x^{\frac{5}{2}}}{5 \sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**3,x)

[Out]

-63*a**(5/2)*b**2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/4 - a**5/(2*sqrt(b)*x**(5/2)*
sqrt(a/(b*x) + 1)) - 19*a**4*sqrt(b)/(4*x**(3/2)*sqrt(a/(b*x) + 1)) + 203*a**3*b
**(3/2)/(20*sqrt(x)*sqrt(a/(b*x) + 1)) + 86*a**2*b**(5/2)*sqrt(x)/(5*sqrt(a/(b*x
) + 1)) + 16*a*b**(7/2)*x**(3/2)/(5*sqrt(a/(b*x) + 1)) + 2*b**(9/2)*x**(5/2)/(5*
sqrt(a/(b*x) + 1))

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GIAC/XCAS [A]  time = 0.210761, size = 151, normalized size = 1.32 \[ \frac{\frac{315 \, a^{3} b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 8 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{3} + 40 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{3} + 240 \, \sqrt{b x + a} a^{2} b^{3} - \frac{5 \,{\left (17 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{3} - 15 \, \sqrt{b x + a} a^{4} b^{3}\right )}}{b^{2} x^{2}}}{20 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^3,x, algorithm="giac")

[Out]

1/20*(315*a^3*b^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 8*(b*x + a)^(5/2)*b^
3 + 40*(b*x + a)^(3/2)*a*b^3 + 240*sqrt(b*x + a)*a^2*b^3 - 5*(17*(b*x + a)^(3/2)
*a^3*b^3 - 15*sqrt(b*x + a)*a^4*b^3)/(b^2*x^2))/b