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

Optimal. Leaf size=122 \[ \frac{B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac{A \left (a+c x^2\right )^{3/2}}{5 a x^5}+\frac{B c \sqrt{a+c x^2}}{8 a x^2}-\frac{B \left (a+c x^2\right )^{3/2}}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.26613, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{2 A c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac{A \left (a+c x^2\right )^{3/2}}{5 a x^5}+\frac{B c \sqrt{a+c x^2}}{8 a x^2}-\frac{B \left (a+c x^2\right )^{3/2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.9786, size = 109, normalized size = 0.89 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{3}{2}}}{5 a x^{5}} + \frac{2 A c \left (a + c x^{2}\right )^{\frac{3}{2}}}{15 a^{2} x^{3}} + \frac{B c \sqrt{a + c x^{2}}}{8 a x^{2}} - \frac{B \left (a + c x^{2}\right )^{\frac{3}{2}}}{4 a x^{4}} + \frac{B c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.271394, size = 104, normalized size = 0.85 \[ -\frac{\frac{\sqrt{a+c x^2} \left (6 a^2 (4 A+5 B x)+a c x^2 (8 A+15 B x)-16 A c^2 x^4\right )}{x^5}-15 \sqrt{a} B c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+15 \sqrt{a} B c^2 \log (x)}{120 a^2} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[a + c*x^2]*(-16*A*c^2*x^4 + 6*a^2*(4*A + 5*B*x) + a*c*x^2*(8*A + 15*B*x)
))/x^5 + 15*Sqrt[a]*B*c^2*Log[x] - 15*Sqrt[a]*B*c^2*Log[a + Sqrt[a]*Sqrt[a + c*x
^2]])/(120*a^2)

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Maple [A]  time = 0.015, size = 126, normalized size = 1. \[ -{\frac{A}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{2\,Ac}{15\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bc}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B{c}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.370853, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a c^{2} x^{5} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (16 \, A c^{2} x^{4} - 15 \, B a c x^{3} - 8 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{240 \, a^{\frac{5}{2}} x^{5}}, \frac{15 \, B a c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (16 \, A c^{2} x^{4} - 15 \, B a c x^{3} - 8 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{120 \, \sqrt{-a} a^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/240*(15*B*a*c^2*x^5*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2) +
2*(16*A*c^2*x^4 - 15*B*a*c*x^3 - 8*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt(c*x^
2 + a)*sqrt(a))/(a^(5/2)*x^5), 1/120*(15*B*a*c^2*x^5*arctan(sqrt(-a)/sqrt(c*x^2
+ a)) + (16*A*c^2*x^4 - 15*B*a*c*x^3 - 8*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt
(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^5)]

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Sympy [A]  time = 16.1299, size = 173, normalized size = 1.42 \[ - \frac{A \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a x^{2}} + \frac{2 A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{2}} - \frac{B a}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B c^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*x**4) - A*c**(3/2)*sqrt(a/(c*x**2) + 1)/(15*a
*x**2) + 2*A*c**(5/2)*sqrt(a/(c*x**2) + 1)/(15*a**2) - B*a/(4*sqrt(c)*x**5*sqrt(
a/(c*x**2) + 1)) - 3*B*sqrt(c)/(8*x**3*sqrt(a/(c*x**2) + 1)) - B*c**(3/2)/(8*a*x
*sqrt(a/(c*x**2) + 1)) + B*c**2*asinh(sqrt(a)/(sqrt(c)*x))/(8*a**(3/2))

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GIAC/XCAS [A]  time = 0.282546, size = 360, normalized size = 2.95 \[ -\frac{B c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a} + \frac{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B c^{2} + 90 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a c^{2} + 240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} A a c^{\frac{5}{2}} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{5}{2}} - 90 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{3} c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{3} c^{\frac{5}{2}} - 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c^{2} - 16 \, A a^{4} c^{\frac{5}{2}}}{60 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*B*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/60*(
15*(sqrt(c)*x - sqrt(c*x^2 + a))^9*B*c^2 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*
a*c^2 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^6*A*a*c^(5/2) + 80*(sqrt(c)*x - sqrt(c
*x^2 + a))^4*A*a^2*c^(5/2) - 90*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^3*c^2 + 80*(
sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^3*c^(5/2) - 15*(sqrt(c)*x - sqrt(c*x^2 + a))*
B*a^4*c^2 - 16*A*a^4*c^(5/2))/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^5*a)