3.364 \(\int \frac{A+B x}{x^6 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4} \]

[Out]

-(A*Sqrt[a + c*x^2])/(5*a*x^5) - (B*Sqrt[a + c*x^2])/(4*a*x^4) + (4*A*c*Sqrt[a +
c*x^2])/(15*a^2*x^3) + (3*B*c*Sqrt[a + c*x^2])/(8*a^2*x^2) - (8*A*c^2*Sqrt[a +
c*x^2])/(15*a^3*x) - (3*B*c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.418289, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^6*Sqrt[a + c*x^2]),x]

[Out]

-(A*Sqrt[a + c*x^2])/(5*a*x^5) - (B*Sqrt[a + c*x^2])/(4*a*x^4) + (4*A*c*Sqrt[a +
c*x^2])/(15*a^2*x^3) + (3*B*c*Sqrt[a + c*x^2])/(8*a^2*x^2) - (8*A*c^2*Sqrt[a +
c*x^2])/(15*a^3*x) - (3*B*c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 49.7688, size = 138, normalized size = 0.94 \[ - \frac{A \sqrt{a + c x^{2}}}{5 a x^{5}} + \frac{4 A c \sqrt{a + c x^{2}}}{15 a^{2} x^{3}} - \frac{8 A c^{2} \sqrt{a + c x^{2}}}{15 a^{3} x} - \frac{B \sqrt{a + c x^{2}}}{4 a x^{4}} + \frac{3 B c \sqrt{a + c x^{2}}}{8 a^{2} x^{2}} - \frac{3 B c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**6/(c*x**2+a)**(1/2),x)

[Out]

-A*sqrt(a + c*x**2)/(5*a*x**5) + 4*A*c*sqrt(a + c*x**2)/(15*a**2*x**3) - 8*A*c**
2*sqrt(a + c*x**2)/(15*a**3*x) - B*sqrt(a + c*x**2)/(4*a*x**4) + 3*B*c*sqrt(a +
c*x**2)/(8*a**2*x**2) - 3*B*c**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(8*a**(5/2))

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Mathematica [A]  time = 0.196119, size = 104, normalized size = 0.71 \[ \frac{\frac{\sqrt{a+c x^2} \left (-6 a^2 (4 A+5 B x)+a c x^2 (32 A+45 B x)-64 A c^2 x^4\right )}{x^5}-45 \sqrt{a} B c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+45 \sqrt{a} B c^2 \log (x)}{120 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^6*Sqrt[a + c*x^2]),x]

[Out]

((Sqrt[a + c*x^2]*(-64*A*c^2*x^4 - 6*a^2*(4*A + 5*B*x) + a*c*x^2*(32*A + 45*B*x)
))/x^5 + 45*Sqrt[a]*B*c^2*Log[x] - 45*Sqrt[a]*B*c^2*Log[a + Sqrt[a]*Sqrt[a + c*x
^2]])/(120*a^3)

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Maple [A]  time = 0.017, size = 129, normalized size = 0.9 \[ -{\frac{A}{5\,a{x}^{5}}\sqrt{c{x}^{2}+a}}+{\frac{4\,Ac}{15\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+a}}-{\frac{8\,A{c}^{2}}{15\,{a}^{3}x}\sqrt{c{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}}\sqrt{c{x}^{2}+a}}+{\frac{3\,Bc}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{3\,B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^6/(c*x^2+a)^(1/2),x)

[Out]

-1/5*A*(c*x^2+a)^(1/2)/a/x^5+4/15*A*c*(c*x^2+a)^(1/2)/a^2/x^3-8/15*A*c^2*(c*x^2+
a)^(1/2)/a^3/x-1/4*B*(c*x^2+a)^(1/2)/a/x^4+3/8*B*c*(c*x^2+a)^(1/2)/a^2/x^2-3/8*B
/a^(5/2)*c^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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305991, size = 1, normalized size = 0.01 \[ \left [\frac{45 \, B a c^{2} x^{5} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{240 \, a^{\frac{7}{2}} x^{5}}, -\frac{45 \, B a c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{120 \, \sqrt{-a} a^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*x^6),x, algorithm="fricas")

[Out]

[1/240*(45*B*a*c^2*x^5*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) -
2*(64*A*c^2*x^4 - 45*B*a*c*x^3 - 32*A*a*c*x^2 + 30*B*a^2*x + 24*A*a^2)*sqrt(c*x
^2 + a)*sqrt(a))/(a^(7/2)*x^5), -1/120*(45*B*a*c^2*x^5*arctan(sqrt(-a)/sqrt(c*x^
2 + a)) + (64*A*c^2*x^4 - 45*B*a*c*x^3 - 32*A*a*c*x^2 + 30*B*a^2*x + 24*A*a^2)*s
qrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^5)]

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Sympy [A]  time = 18.3553, size = 408, normalized size = 2.78 \[ - \frac{3 A a^{4} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{2 A a^{3} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{3 A a^{2} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{12 A a c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{8 A c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{B}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B \sqrt{c}}{8 a x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 B c^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**6/(c*x**2+a)**(1/2),x)

[Out]

-3*A*a**4*c**(9/2)*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 +
15*a**3*c**6*x**8) - 2*A*a**3*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4
*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - 3*A*a**2*c**(13/2)*x**4*sqrt(a/
(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - 12*A
*a*c**(15/2)*x**6*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 +
15*a**3*c**6*x**8) - 8*A*c**(17/2)*x**8*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4
+ 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - B/(4*sqrt(c)*x**5*sqrt(a/(c*x**2) + 1
)) + B*sqrt(c)/(8*a*x**3*sqrt(a/(c*x**2) + 1)) + 3*B*c**(3/2)/(8*a**2*x*sqrt(a/(
c*x**2) + 1)) - 3*B*c**2*asinh(sqrt(a)/(sqrt(c)*x))/(8*a**(5/2))

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GIAC/XCAS [A]  time = 0.28169, size = 325, normalized size = 2.21 \[ \frac{3 \, B c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{2}} - \frac{45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B c^{2} - 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a c^{2} - 640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{5}{2}} + 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{3} c^{2} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{3} c^{\frac{5}{2}} - 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c^{2} - 64 \, A a^{4} c^{\frac{5}{2}}}{60 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + a)*x^6),x, algorithm="giac")

[Out]

3/4*B*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/60*
(45*(sqrt(c)*x - sqrt(c*x^2 + a))^9*B*c^2 - 210*(sqrt(c)*x - sqrt(c*x^2 + a))^7*
B*a*c^2 - 640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^2*c^(5/2) + 210*(sqrt(c)*x - s
qrt(c*x^2 + a))^3*B*a^3*c^2 + 320*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^3*c^(5/2)
- 45*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^4*c^2 - 64*A*a^4*c^(5/2))/(((sqrt(c)*x -
sqrt(c*x^2 + a))^2 - a)^5*a^2)