3.320 \(\int \frac{(a+b x)^{9/2}}{x^4} \, dx\)

Optimal. Leaf size=114 \[ -\frac{105}{8} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{35}{8} b^3 (a+b x)^{3/2}+\frac{105}{8} a b^3 \sqrt{a+b x}-\frac{21 b^2 (a+b x)^{5/2}}{8 x}-\frac{(a+b x)^{9/2}}{3 x^3}-\frac{3 b (a+b x)^{7/2}}{4 x^2} \]

[Out]

(105*a*b^3*Sqrt[a + b*x])/8 + (35*b^3*(a + b*x)^(3/2))/8 - (21*b^2*(a + b*x)^(5/
2))/(8*x) - (3*b*(a + b*x)^(7/2))/(4*x^2) - (a + b*x)^(9/2)/(3*x^3) - (105*a^(3/
2)*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/8

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Rubi [A]  time = 0.109339, 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 \[ -\frac{105}{8} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{35}{8} b^3 (a+b x)^{3/2}+\frac{105}{8} a b^3 \sqrt{a+b x}-\frac{21 b^2 (a+b x)^{5/2}}{8 x}-\frac{(a+b x)^{9/2}}{3 x^3}-\frac{3 b (a+b x)^{7/2}}{4 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^4,x]

[Out]

(105*a*b^3*Sqrt[a + b*x])/8 + (35*b^3*(a + b*x)^(3/2))/8 - (21*b^2*(a + b*x)^(5/
2))/(8*x) - (3*b*(a + b*x)^(7/2))/(4*x^2) - (a + b*x)^(9/2)/(3*x^3) - (105*a^(3/
2)*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/8

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Rubi in Sympy [A]  time = 14.7823, size = 105, normalized size = 0.92 \[ - \frac{105 a^{\frac{3}{2}} b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8} + \frac{105 a b^{3} \sqrt{a + b x}}{8} + \frac{35 b^{3} \left (a + b x\right )^{\frac{3}{2}}}{8} - \frac{21 b^{2} \left (a + b x\right )^{\frac{5}{2}}}{8 x} - \frac{3 b \left (a + b x\right )^{\frac{7}{2}}}{4 x^{2}} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**4,x)

[Out]

-105*a**(3/2)*b**3*atanh(sqrt(a + b*x)/sqrt(a))/8 + 105*a*b**3*sqrt(a + b*x)/8 +
35*b**3*(a + b*x)**(3/2)/8 - 21*b**2*(a + b*x)**(5/2)/(8*x) - 3*b*(a + b*x)**(7
/2)/(4*x**2) - (a + b*x)**(9/2)/(3*x**3)

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Mathematica [A]  time = 0.0833607, size = 85, normalized size = 0.75 \[ \frac{1}{24} \left (\frac{\sqrt{a+b x} \left (-8 a^4-50 a^3 b x-165 a^2 b^2 x^2+208 a b^3 x^3+16 b^4 x^4\right )}{x^3}-315 a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^4,x]

[Out]

((Sqrt[a + b*x]*(-8*a^4 - 50*a^3*b*x - 165*a^2*b^2*x^2 + 208*a*b^3*x^3 + 16*b^4*
x^4))/x^3 - 315*a^(3/2)*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/24

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Maple [A]  time = 0.016, size = 87, normalized size = 0.8 \[ 2\,{b}^{3} \left ( 1/3\, \left ( bx+a \right ) ^{3/2}+4\,a\sqrt{bx+a}+{a}^{2} \left ({\frac{1}{{b}^{3}{x}^{3}} \left ( -{\frac{55\, \left ( bx+a \right ) ^{5/2}}{16}}+{\frac{35\,a \left ( bx+a \right ) ^{3/2}}{6}}-{\frac{41\,{a}^{2}\sqrt{bx+a}}{16}} \right ) }-{\frac{105}{16\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^4,x)

[Out]

2*b^3*(1/3*(b*x+a)^(3/2)+4*a*(b*x+a)^(1/2)+a^2*((-55/16*(b*x+a)^(5/2)+35/6*a*(b*
x+a)^(3/2)-41/16*a^2*(b*x+a)^(1/2))/x^3/b^3-105/16*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)^(9/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21845, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{\frac{3}{2}} b^{3} x^{3} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (16 \, b^{4} x^{4} + 208 \, a b^{3} x^{3} - 165 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x - 8 \, a^{4}\right )} \sqrt{b x + a}}{48 \, x^{3}}, -\frac{315 \, \sqrt{-a} a b^{3} x^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (16 \, b^{4} x^{4} + 208 \, a b^{3} x^{3} - 165 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x - 8 \, a^{4}\right )} \sqrt{b x + a}}{24 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(315*a^(3/2)*b^3*x^3*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(16*
b^4*x^4 + 208*a*b^3*x^3 - 165*a^2*b^2*x^2 - 50*a^3*b*x - 8*a^4)*sqrt(b*x + a))/x
^3, -1/24*(315*sqrt(-a)*a*b^3*x^3*arctan(sqrt(b*x + a)/sqrt(-a)) - (16*b^4*x^4 +
208*a*b^3*x^3 - 165*a^2*b^2*x^2 - 50*a^3*b*x - 8*a^4)*sqrt(b*x + a))/x^3]

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Sympy [A]  time = 31.8207, size = 184, normalized size = 1.61 \[ - \frac{105 a^{\frac{3}{2}} b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8} - \frac{a^{5}}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{29 a^{4} \sqrt{b}}{12 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{215 a^{3} b^{\frac{3}{2}}}{24 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{43 a^{2} b^{\frac{5}{2}}}{24 \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{28 a b^{\frac{7}{2}} \sqrt{x}}{3 \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{9}{2}} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**4,x)

[Out]

-105*a**(3/2)*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/8 - a**5/(3*sqrt(b)*x**(7/2)
*sqrt(a/(b*x) + 1)) - 29*a**4*sqrt(b)/(12*x**(5/2)*sqrt(a/(b*x) + 1)) - 215*a**3
*b**(3/2)/(24*x**(3/2)*sqrt(a/(b*x) + 1)) + 43*a**2*b**(5/2)/(24*sqrt(x)*sqrt(a/
(b*x) + 1)) + 28*a*b**(7/2)*sqrt(x)/(3*sqrt(a/(b*x) + 1)) + 2*b**(9/2)*x**(3/2)/
(3*sqrt(a/(b*x) + 1))

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GIAC/XCAS [A]  time = 0.213985, size = 151, normalized size = 1.32 \[ \frac{\frac{315 \, a^{2} b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 16 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4} + 192 \, \sqrt{b x + a} a b^{4} - \frac{165 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{4} - 280 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{4} + 123 \, \sqrt{b x + a} a^{4} b^{4}}{b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^4,x, algorithm="giac")

[Out]

1/24*(315*a^2*b^4*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 16*(b*x + a)^(3/2)*b
^4 + 192*sqrt(b*x + a)*a*b^4 - (165*(b*x + a)^(5/2)*a^2*b^4 - 280*(b*x + a)^(3/2
)*a^3*b^4 + 123*sqrt(b*x + a)*a^4*b^4)/(b^3*x^3))/b