3.363 \(\int \frac{A+B x}{x^5 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=122 \[ -\frac{3 A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 A c \sqrt{a+c x^2}}{8 a^2 x^2}+\frac{2 B c \sqrt{a+c x^2}}{3 a^2 x}-\frac{A \sqrt{a+c x^2}}{4 a x^4}-\frac{B \sqrt{a+c x^2}}{3 a x^3} \]

[Out]

-(A*Sqrt[a + c*x^2])/(4*a*x^4) - (B*Sqrt[a + c*x^2])/(3*a*x^3) + (3*A*c*Sqrt[a +
c*x^2])/(8*a^2*x^2) + (2*B*c*Sqrt[a + c*x^2])/(3*a^2*x) - (3*A*c^2*ArcTanh[Sqrt
[a + c*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.326854, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 A c \sqrt{a+c x^2}}{8 a^2 x^2}+\frac{2 B c \sqrt{a+c x^2}}{3 a^2 x}-\frac{A \sqrt{a+c x^2}}{4 a x^4}-\frac{B \sqrt{a+c x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^5*Sqrt[a + c*x^2]),x]

[Out]

-(A*Sqrt[a + c*x^2])/(4*a*x^4) - (B*Sqrt[a + c*x^2])/(3*a*x^3) + (3*A*c*Sqrt[a +
c*x^2])/(8*a^2*x^2) + (2*B*c*Sqrt[a + c*x^2])/(3*a^2*x) - (3*A*c^2*ArcTanh[Sqrt
[a + c*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 35.9297, size = 112, normalized size = 0.92 \[ - \frac{A \sqrt{a + c x^{2}}}{4 a x^{4}} + \frac{3 A c \sqrt{a + c x^{2}}}{8 a^{2} x^{2}} - \frac{3 A c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} - \frac{B \sqrt{a + c x^{2}}}{3 a x^{3}} + \frac{2 B c \sqrt{a + c x^{2}}}{3 a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**5/(c*x**2+a)**(1/2),x)

[Out]

-A*sqrt(a + c*x**2)/(4*a*x**4) + 3*A*c*sqrt(a + c*x**2)/(8*a**2*x**2) - 3*A*c**2
*atanh(sqrt(a + c*x**2)/sqrt(a))/(8*a**(5/2)) - B*sqrt(a + c*x**2)/(3*a*x**3) +
2*B*c*sqrt(a + c*x**2)/(3*a**2*x)

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Mathematica [A]  time = 0.120689, size = 94, normalized size = 0.77 \[ \frac{\sqrt{a} \sqrt{a+c x^2} \left (-6 a A-8 a B x+9 A c x^2+16 B c x^3\right )-9 A c^2 x^4 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+9 A c^2 x^4 \log (x)}{24 a^{5/2} x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^5*Sqrt[a + c*x^2]),x]

[Out]

(Sqrt[a]*Sqrt[a + c*x^2]*(-6*a*A - 8*a*B*x + 9*A*c*x^2 + 16*B*c*x^3) + 9*A*c^2*x
^4*Log[x] - 9*A*c^2*x^4*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(24*a^(5/2)*x^4)

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Maple [A]  time = 0.013, size = 108, normalized size = 0.9 \[ -{\frac{A}{4\,a{x}^{4}}\sqrt{c{x}^{2}+a}}+{\frac{3\,Ac}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{3\,A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{3\,a{x}^{3}}\sqrt{c{x}^{2}+a}}+{\frac{2\,Bc}{3\,{a}^{2}x}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^5/(c*x^2+a)^(1/2),x)

[Out]

-1/4*A*(c*x^2+a)^(1/2)/a/x^4+3/8*A*c*(c*x^2+a)^(1/2)/a^2/x^2-3/8*A/a^(5/2)*c^2*l
n((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-1/3*B*(c*x^2+a)^(1/2)/a/x^3+2/3*B*c*(c*x^2+
a)^(1/2)/a^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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31281, size = 1, normalized size = 0.01 \[ \left [\frac{9 \, A c^{2} x^{4} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (16 \, B c x^{3} + 9 \, A c x^{2} - 8 \, B a x - 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{48 \, a^{\frac{5}{2}} x^{4}}, -\frac{9 \, A c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (16 \, B c x^{3} + 9 \, A c x^{2} - 8 \, B a x - 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{24 \, \sqrt{-a} a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*x^5),x, algorithm="fricas")

[Out]

[1/48*(9*A*c^2*x^4*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) + 2*(
16*B*c*x^3 + 9*A*c*x^2 - 8*B*a*x - 6*A*a)*sqrt(c*x^2 + a)*sqrt(a))/(a^(5/2)*x^4)
, -1/24*(9*A*c^2*x^4*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (16*B*c*x^3 + 9*A*c*x^2
- 8*B*a*x - 6*A*a)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^4)]

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Sympy [A]  time = 16.9973, size = 153, normalized size = 1.25 \[ - \frac{A}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A \sqrt{c}}{8 a x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 A c^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{5}{2}}} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a x^{2}} + \frac{2 B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**5/(c*x**2+a)**(1/2),x)

[Out]

-A/(4*sqrt(c)*x**5*sqrt(a/(c*x**2) + 1)) + A*sqrt(c)/(8*a*x**3*sqrt(a/(c*x**2) +
1)) + 3*A*c**(3/2)/(8*a**2*x*sqrt(a/(c*x**2) + 1)) - 3*A*c**2*asinh(sqrt(a)/(sq
rt(c)*x))/(8*a**(5/2)) - B*sqrt(c)*sqrt(a/(c*x**2) + 1)/(3*a*x**2) + 2*B*c**(3/2
)*sqrt(a/(c*x**2) + 1)/(3*a**2)

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GIAC/XCAS [A]  time = 0.276733, size = 325, normalized size = 2.66 \[ \frac{3 \, A c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{2}} - \frac{9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A c^{2} - 33 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a c^{2} - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{2} c^{\frac{3}{2}} - 33 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 64 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{3} c^{\frac{3}{2}} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{3} c^{2} - 16 \, B a^{4} c^{\frac{3}{2}}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*x^5),x, algorithm="giac")

[Out]

3/4*A*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/12*
(9*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*c^2 - 33*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*
a*c^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^4*B*a^2*c^(3/2) - 33*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*A*a^2*c^2 + 64*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^3*c^(3/2) + 9*(
sqrt(c)*x - sqrt(c*x^2 + a))*A*a^3*c^2 - 16*B*a^4*c^(3/2))/(((sqrt(c)*x - sqrt(c
*x^2 + a))^2 - a)^4*a^2)