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

Optimal. Leaf size=99 \[ \frac{A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.189962, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{A c \sqrt{a+c x^2}}{8 a x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{4 a x^4}-\frac{B \left (a+c x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.3769, size = 85, normalized size = 0.86 \[ \frac{A c \sqrt{a + c x^{2}}}{8 a x^{2}} - \frac{A \left (a + c x^{2}\right )^{\frac{3}{2}}}{4 a x^{4}} + \frac{A c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} - \frac{B \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.110423, size = 95, normalized size = 0.96 \[ \frac{-\sqrt{a} \sqrt{a+c x^2} \left (6 a A+8 a B x+3 A c x^2+8 B c x^3\right )+3 A c^2 x^4 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-3 A c^2 x^4 \log (x)}{24 a^{3/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 107, normalized size = 1.1 \[ -{\frac{A}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ac}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A{c}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{B}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*A*(c*x^2+a)^(3/2)/a/x^4+1/8*A/a^2*c/x^2*(c*x^2+a)^(3/2)+1/8*A/a^(3/2)*c^2*l
n((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-1/8*A/a^2*c^2*(c*x^2+a)^(1/2)-1/3*B*(c*x^2+
a)^(3/2)/a/x^3

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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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.375055, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, A c^{2} x^{4} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (8 \, B c x^{3} + 3 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{48 \, a^{\frac{3}{2}} x^{4}}, \frac{3 \, A c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (8 \, B c x^{3} + 3 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{24 \, \sqrt{-a} a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.2488, size = 144, normalized size = 1.45 \[ - \frac{A a}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{3}{2}}} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27725, size = 360, normalized size = 3.64 \[ -\frac{A c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A c^{2} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a c^{\frac{3}{2}} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a c^{2} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{2} c^{\frac{3}{2}} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{3} c^{\frac{3}{2}} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{3} c^{2} - 8 \, B a^{4} c^{\frac{3}{2}}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*A*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/12*(
3*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*c^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^6*B*a
*c^(3/2) + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a*c^2 - 24*(sqrt(c)*x - sqrt(c*x
^2 + a))^4*B*a^2*c^(3/2) + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^3*A*a^2*c^2 + 8*(sqr
t(c)*x - sqrt(c*x^2 + a))^2*B*a^3*c^(3/2) + 3*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^
3*c^2 - 8*B*a^4*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^4*a)