3.333 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{\left (a+c x^2\right )^{3/2} (A-B x)}{2 x^2}-\frac{3 \sqrt{a+c x^2} (a B-A c x)}{2 x}-\frac{3}{2} \sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{3}{2} a B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.249336, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (a+c x^2\right )^{3/2} (A-B x)}{2 x^2}-\frac{3 \sqrt{a+c x^2} (a B-A c x)}{2 x}-\frac{3}{2} \sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{3}{2} a B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(3/2))/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.9127, size = 109, normalized size = 0.98 \[ - \frac{3 A \sqrt{a} c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2} + \frac{3 B a \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2} - \frac{3 \sqrt{a + c x^{2}} \left (- 4 A c x + 4 B a\right )}{8 x} - \frac{\left (2 A - 2 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.175302, size = 113, normalized size = 1.02 \[ \frac{1}{2} \left (\frac{\sqrt{a+c x^2} \left (c x^2 (2 A+B x)-a (A+2 B x)\right )}{x^2}-3 \sqrt{a} A c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+3 \sqrt{a} A c \log (x)+3 a B \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^3,x]

[Out]

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

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Maple [A]  time = 0.013, size = 150, normalized size = 1.4 \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ac}{2\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Ac}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{3\,Ac}{2}\sqrt{c{x}^{2}+a}}-{\frac{B}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bcx}{a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bcx}{2}\sqrt{c{x}^{2}+a}}+{\frac{3\,Ba}{2}\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(3/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307308, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B a \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 3 \, A \sqrt{a} c x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{4 \, x^{2}}, \frac{6 \, B a \sqrt{-c} x^{2} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 3 \, A \sqrt{a} c x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{4 \, x^{2}}, -\frac{6 \, A \sqrt{-a} c x^{2} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 3 \, B a \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{4 \, x^{2}}, \frac{3 \, B a \sqrt{-c} x^{2} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 3 \, A \sqrt{-a} c x^{2} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{2 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

[1/4*(3*B*a*sqrt(c)*x^2*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*A*sq
rt(a)*c*x^2*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(B*c*x^3 + 2
*A*c*x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2, 1/4*(6*B*a*sqrt(-c)*x^2*arctan(c
*x/(sqrt(c*x^2 + a)*sqrt(-c))) + 3*A*sqrt(a)*c*x^2*log(-(c*x^2 - 2*sqrt(c*x^2 +
a)*sqrt(a) + 2*a)/x^2) + 2*(B*c*x^3 + 2*A*c*x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a)
)/x^2, -1/4*(6*A*sqrt(-a)*c*x^2*arctan(a/(sqrt(c*x^2 + a)*sqrt(-a))) - 3*B*a*sqr
t(c)*x^2*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(B*c*x^3 + 2*A*c*x^
2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2, 1/2*(3*B*a*sqrt(-c)*x^2*arctan(c*x/(sqr
t(c*x^2 + a)*sqrt(-c))) - 3*A*sqrt(-a)*c*x^2*arctan(a/(sqrt(c*x^2 + a)*sqrt(-a))
) + (B*c*x^3 + 2*A*c*x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2]

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Sympy [A]  time = 19.9443, size = 182, normalized size = 1.64 \[ - \frac{3 A \sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2} - \frac{A a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} + \frac{A a \sqrt{c}}{x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{\frac{3}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B a^{\frac{3}{2}}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{B \sqrt{a} c x \sqrt{1 + \frac{c x^{2}}{a}}}{2} - \frac{B \sqrt{a} c x}{\sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B a \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(3/2)/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.289827, size = 258, normalized size = 2.32 \[ \frac{3 \, A a c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3}{2} \, B a \sqrt{c}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{1}{2} \,{\left (B c x + 2 \, A c\right )} \sqrt{c x^{2} + a} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{2} \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{2} c - 2 \, B a^{3} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^3,x, algorithm="giac")

[Out]

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