3.362 \(\int \frac{A+B x}{x^4 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{2 A c \sqrt{a+c x^2}}{3 a^2 x}-\frac{A \sqrt{a+c x^2}}{3 a x^3}-\frac{B \sqrt{a+c x^2}}{2 a x^2} \]

[Out]

-(A*Sqrt[a + c*x^2])/(3*a*x^3) - (B*Sqrt[a + c*x^2])/(2*a*x^2) + (2*A*c*Sqrt[a +
c*x^2])/(3*a^2*x) + (B*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2))

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Rubi [A]  time = 0.244564, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{2 A c \sqrt{a+c x^2}}{3 a^2 x}-\frac{A \sqrt{a+c x^2}}{3 a x^3}-\frac{B \sqrt{a+c x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^4*Sqrt[a + c*x^2]),x]

[Out]

-(A*Sqrt[a + c*x^2])/(3*a*x^3) - (B*Sqrt[a + c*x^2])/(2*a*x^2) + (2*A*c*Sqrt[a +
c*x^2])/(3*a^2*x) + (B*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2))

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Rubi in Sympy [A]  time = 24.1603, size = 85, normalized size = 0.88 \[ - \frac{A \sqrt{a + c x^{2}}}{3 a x^{3}} + \frac{2 A c \sqrt{a + c x^{2}}}{3 a^{2} x} - \frac{B \sqrt{a + c x^{2}}}{2 a x^{2}} + \frac{B c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**4/(c*x**2+a)**(1/2),x)

[Out]

-A*sqrt(a + c*x**2)/(3*a*x**3) + 2*A*c*sqrt(a + c*x**2)/(3*a**2*x) - B*sqrt(a +
c*x**2)/(2*a*x**2) + B*c*atanh(sqrt(a + c*x**2)/sqrt(a))/(2*a**(3/2))

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Mathematica [A]  time = 0.123897, size = 80, normalized size = 0.82 \[ \frac{\frac{\sqrt{a+c x^2} \left (-2 a A-3 a B x+4 A c x^2\right )}{x^3}+3 \sqrt{a} B c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-3 \sqrt{a} B c \log (x)}{6 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^4*Sqrt[a + c*x^2]),x]

[Out]

((Sqrt[a + c*x^2]*(-2*a*A - 3*a*B*x + 4*A*c*x^2))/x^3 - 3*Sqrt[a]*B*c*Log[x] + 3
*Sqrt[a]*B*c*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(6*a^2)

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Maple [A]  time = 0.011, size = 87, normalized size = 0.9 \[ -{\frac{A}{3\,a{x}^{3}}\sqrt{c{x}^{2}+a}}+{\frac{2\,Ac}{3\,{a}^{2}x}\sqrt{c{x}^{2}+a}}-{\frac{B}{2\,a{x}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{Bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/(c*x^2+a)^(1/2),x)

[Out]

-1/3*A*(c*x^2+a)^(1/2)/a/x^3+2/3*A*c*(c*x^2+a)^(1/2)/a^2/x-1/2*B*(c*x^2+a)^(1/2)
/a/x^2+1/2*B*c/a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)

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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)/(sqrt(c*x^2 + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287796, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B a c x^{3} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (4 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{12 \, a^{\frac{5}{2}} x^{3}}, \frac{3 \, B a c x^{3} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (4 \, A c x^{2} - 3 \, B a x - 2 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{6 \, \sqrt{-a} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*x^4),x, algorithm="fricas")

[Out]

[1/12*(3*B*a*c*x^3*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2) + 2*(
4*A*c*x^2 - 3*B*a*x - 2*A*a)*sqrt(c*x^2 + a)*sqrt(a))/(a^(5/2)*x^3), 1/6*(3*B*a*
c*x^3*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (4*A*c*x^2 - 3*B*a*x - 2*A*a)*sqrt(c*x^
2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^3)]

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Sympy [A]  time = 10.8673, size = 97, normalized size = 1. \[ - \frac{A \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a x^{2}} + \frac{2 A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a^{2}} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 a x} + \frac{B c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**4/(c*x**2+a)**(1/2),x)

[Out]

-A*sqrt(c)*sqrt(a/(c*x**2) + 1)/(3*a*x**2) + 2*A*c**(3/2)*sqrt(a/(c*x**2) + 1)/(
3*a**2) - B*sqrt(c)*sqrt(a/(c*x**2) + 1)/(2*a*x) + B*c*asinh(sqrt(a)/(sqrt(c)*x)
)/(2*a**(3/2))

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GIAC/XCAS [A]  time = 0.27663, size = 204, normalized size = 2.1 \[ -\frac{B c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B c + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{2} c - 4 \, A a^{2} c^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{3} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*x^4),x, algorithm="giac")

[Out]

-B*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/3*(3*(sqrt
(c)*x - sqrt(c*x^2 + a))^5*B*c + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a*c^(3/2)
- 3*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^2*c - 4*A*a^2*c^(3/2))/(((sqrt(c)*x - sqrt
(c*x^2 + a))^2 - a)^3*a)