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

Optimal. Leaf size=128 \[ \frac{105}{16} a^3 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{105}{16} a^2 b^2 x \sqrt{a+b x^2}+\frac{7}{2} b^2 x \left (a+b x^2\right )^{5/2}+\frac{35}{8} a b^2 x \left (a+b x^2\right )^{3/2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{x}-\frac{\left (a+b x^2\right )^{9/2}}{3 x^3} \]

[Out]

(105*a^2*b^2*x*Sqrt[a + b*x^2])/16 + (35*a*b^2*x*(a + b*x^2)^(3/2))/8 + (7*b^2*x
*(a + b*x^2)^(5/2))/2 - (3*b*(a + b*x^2)^(7/2))/x - (a + b*x^2)^(9/2)/(3*x^3) +
(105*a^3*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/16

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Rubi [A]  time = 0.126791, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{105}{16} a^3 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{105}{16} a^2 b^2 x \sqrt{a+b x^2}+\frac{7}{2} b^2 x \left (a+b x^2\right )^{5/2}+\frac{35}{8} a b^2 x \left (a+b x^2\right )^{3/2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{x}-\frac{\left (a+b x^2\right )^{9/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(9/2)/x^4,x]

[Out]

(105*a^2*b^2*x*Sqrt[a + b*x^2])/16 + (35*a*b^2*x*(a + b*x^2)^(3/2))/8 + (7*b^2*x
*(a + b*x^2)^(5/2))/2 - (3*b*(a + b*x^2)^(7/2))/x - (a + b*x^2)^(9/2)/(3*x^3) +
(105*a^3*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/16

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Rubi in Sympy [A]  time = 13.7662, size = 121, normalized size = 0.95 \[ \frac{105 a^{3} b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16} + \frac{105 a^{2} b^{2} x \sqrt{a + b x^{2}}}{16} + \frac{35 a b^{2} x \left (a + b x^{2}\right )^{\frac{3}{2}}}{8} + \frac{7 b^{2} x \left (a + b x^{2}\right )^{\frac{5}{2}}}{2} - \frac{3 b \left (a + b x^{2}\right )^{\frac{7}{2}}}{x} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

105*a**3*b**(3/2)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/16 + 105*a**2*b**2*x*sqrt(a
+ b*x**2)/16 + 35*a*b**2*x*(a + b*x**2)**(3/2)/8 + 7*b**2*x*(a + b*x**2)**(5/2)/
2 - 3*b*(a + b*x**2)**(7/2)/x - (a + b*x**2)**(9/2)/(3*x**3)

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Mathematica [A]  time = 0.103198, size = 95, normalized size = 0.74 \[ \frac{1}{48} \left (315 a^3 b^{3/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\frac{\sqrt{a+b x^2} \left (-16 a^4-208 a^3 b x^2+165 a^2 b^2 x^4+50 a b^3 x^6+8 b^4 x^8\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a + b*x^2]*(-16*a^4 - 208*a^3*b*x^2 + 165*a^2*b^2*x^4 + 50*a*b^3*x^6 + 8*
b^4*x^8))/x^3 + 315*a^3*b^(3/2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/48

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Maple [A]  time = 0.008, size = 146, normalized size = 1.1 \[ -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{8\,b}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{8\,{b}^{2}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+3\,{\frac{{b}^{2}x \left ( b{x}^{2}+a \right ) ^{7/2}}{a}}+{\frac{7\,{b}^{2}x}{2} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{35\,a{b}^{2}x}{8} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{105\,{a}^{2}{b}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{105\,{a}^{3}}{16}{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(9/2)/x^4,x)

[Out]

-1/3/a/x^3*(b*x^2+a)^(11/2)-8/3*b/a^2/x*(b*x^2+a)^(11/2)+8/3*b^2/a^2*x*(b*x^2+a)
^(9/2)+3*b^2/a*x*(b*x^2+a)^(7/2)+7/2*b^2*x*(b*x^2+a)^(5/2)+35/8*a*b^2*x*(b*x^2+a
)^(3/2)+105/16*a^2*b^2*x*(b*x^2+a)^(1/2)+105/16*b^(3/2)*a^3*ln(x*b^(1/2)+(b*x^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^2 + a)^(9/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271192, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{3} b^{\frac{3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (8 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 165 \, a^{2} b^{2} x^{4} - 208 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt{b x^{2} + a}}{96 \, x^{3}}, \frac{315 \, a^{3} \sqrt{-b} b x^{3} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) +{\left (8 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 165 \, a^{2} b^{2} x^{4} - 208 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt{b x^{2} + a}}{48 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(315*a^3*b^(3/2)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(
8*b^4*x^8 + 50*a*b^3*x^6 + 165*a^2*b^2*x^4 - 208*a^3*b*x^2 - 16*a^4)*sqrt(b*x^2
+ a))/x^3, 1/48*(315*a^3*sqrt(-b)*b*x^3*arctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b))) +
(8*b^4*x^8 + 50*a*b^3*x^6 + 165*a^2*b^2*x^4 - 208*a^3*b*x^2 - 16*a^4)*sqrt(b*x^
2 + a))/x^3]

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Sympy [A]  time = 33.113, size = 175, normalized size = 1.37 \[ - \frac{a^{\frac{9}{2}}}{3 x^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{14 a^{\frac{7}{2}} b}{3 x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{43 a^{\frac{5}{2}} b^{2} x}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{215 a^{\frac{3}{2}} b^{3} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 \sqrt{a} b^{4} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{105 a^{3} b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16} + \frac{b^{5} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(9/2)/(3*x**3*sqrt(1 + b*x**2/a)) - 14*a**(7/2)*b/(3*x*sqrt(1 + b*x**2/a)) -
43*a**(5/2)*b**2*x/(48*sqrt(1 + b*x**2/a)) + 215*a**(3/2)*b**3*x**3/(48*sqrt(1
+ b*x**2/a)) + 29*sqrt(a)*b**4*x**5/(24*sqrt(1 + b*x**2/a)) + 105*a**3*b**(3/2)*
asinh(sqrt(b)*x/sqrt(a))/16 + b**5*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.212968, size = 216, normalized size = 1.69 \[ -\frac{105}{32} \, a^{3} b^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{1}{48} \,{\left (165 \, a^{2} b^{2} + 2 \,{\left (4 \, b^{4} x^{2} + 25 \, a b^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x + \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} b^{\frac{3}{2}} - 24 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{5} b^{\frac{3}{2}} + 13 \, a^{6} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-105/32*a^3*b^(3/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 1/48*(165*a^2*b^2 + 2*
(4*b^4*x^2 + 25*a*b^3)*x^2)*sqrt(b*x^2 + a)*x + 2/3*(15*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*a^4*b^(3/2) - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(3/2) + 13*a^6*b^
(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3