3.335 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=111 \[ -\frac{c \sqrt{a+c x^2} (3 A+8 B x)}{8 x^2}-\frac{\left (a+c x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}-\frac{3 A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}+B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.264941, 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 \[ -\frac{c \sqrt{a+c x^2} (3 A+8 B x)}{8 x^2}-\frac{\left (a+c x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}-\frac{3 A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}+B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(3/2))/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.6342, size = 104, normalized size = 0.94 \[ - \frac{3 A c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} + B c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )} - \frac{c \left (6 A + 16 B x\right ) \sqrt{a + c x^{2}}}{16 x^{2}} - \frac{\left (3 A + 4 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**5,x)

[Out]

-3*A*c**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(8*sqrt(a)) + B*c**(3/2)*atanh(sqrt(c)
*x/sqrt(a + c*x**2)) - c*(6*A + 16*B*x)*sqrt(a + c*x**2)/(16*x**2) - (3*A + 4*B*
x)*(a + c*x**2)**(3/2)/(12*x**4)

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Mathematica [A]  time = 0.350929, size = 120, normalized size = 1.08 \[ -\frac{\sqrt{a+c x^2} \left (a (6 A+8 B x)+c x^2 (15 A+32 B x)\right )}{24 x^4}-\frac{3 A c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{8 \sqrt{a}}+\frac{3 A c^2 \log (x)}{8 \sqrt{a}}+B c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^5,x]

[Out]

-(Sqrt[a + c*x^2]*(a*(6*A + 8*B*x) + c*x^2*(15*A + 32*B*x)))/(24*x^4) + (3*A*c^2
*Log[x])/(8*Sqrt[a]) - (3*A*c^2*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(8*Sqrt[a]) +
B*c^(3/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]]

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Maple [B]  time = 0.015, size = 202, normalized size = 1.8 \[ -{\frac{A}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ac}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,A{c}^{2}}{8\,a}\sqrt{c{x}^{2}+a}}-{\frac{B}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Bc}{3\,{a}^{2}x} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,B{c}^{2}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{c}^{2}x}{a}\sqrt{c{x}^{2}+a}}+B{c}^{{\frac{3}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(3/2)/x^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.309644, size = 1, normalized size = 0.01 \[ \left [\frac{24 \, B \sqrt{a} c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 9 \, A c^{2} x^{4} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{48 \, \sqrt{a} x^{4}}, \frac{48 \, B \sqrt{a} \sqrt{-c} c x^{4} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 9 \, A c^{2} x^{4} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{a}}{48 \, \sqrt{a} x^{4}}, -\frac{9 \, A c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - 12 \, B \sqrt{-a} c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) +{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{24 \, \sqrt{-a} x^{4}}, \frac{24 \, B \sqrt{-a} \sqrt{-c} c x^{4} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 9 \, A c^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (32 \, B c x^{3} + 15 \, A c x^{2} + 8 \, B a x + 6 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{24 \, \sqrt{-a} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^5,x, algorithm="fricas")

[Out]

[1/48*(24*B*sqrt(a)*c^(3/2)*x^4*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a)
+ 9*A*c^2*x^4*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) - 2*(32*B*
c*x^3 + 15*A*c*x^2 + 8*B*a*x + 6*A*a)*sqrt(c*x^2 + a)*sqrt(a))/(sqrt(a)*x^4), 1/
48*(48*B*sqrt(a)*sqrt(-c)*c*x^4*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) + 9*A*c^2
*x^4*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) - 2*(32*B*c*x^3 + 1
5*A*c*x^2 + 8*B*a*x + 6*A*a)*sqrt(c*x^2 + a)*sqrt(a))/(sqrt(a)*x^4), -1/24*(9*A*
c^2*x^4*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - 12*B*sqrt(-a)*c^(3/2)*x^4*log(-2*c*x^
2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (32*B*c*x^3 + 15*A*c*x^2 + 8*B*a*x + 6*A*
a)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^4), 1/24*(24*B*sqrt(-a)*sqrt(-c)*c*x^4*
arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 9*A*c^2*x^4*arctan(sqrt(-a)/sqrt(c*x^2
+ a)) - (32*B*c*x^3 + 15*A*c*x^2 + 8*B*a*x + 6*A*a)*sqrt(c*x^2 + a)*sqrt(-a))/(s
qrt(-a)*x^4)]

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Sympy [A]  time = 24.3675, size = 236, normalized size = 2.13 \[ - \frac{A a^{2}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A a \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{A c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 \sqrt{a}} - \frac{B \sqrt{a} c}{x \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3} + B c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{B c^{2} x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(3/2)/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.288447, size = 385, normalized size = 3.47 \[ \frac{3 \, A c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} - B c^{\frac{3}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A c^{2} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a c^{\frac{3}{2}} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a c^{2} - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{2} c^{\frac{3}{2}} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{3} c^{\frac{3}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{3} c^{2} - 32 \, B a^{4} c^{\frac{3}{2}}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + 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) - B*c^(3/2)*l
n(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 1/12*(15*(sqrt(c)*x - sqrt(c*x^2 + a))^7*
A*c^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))^6*B*a*c^(3/2) + 9*(sqrt(c)*x - sqrt(c*x
^2 + a))^5*A*a*c^2 - 96*(sqrt(c)*x - sqrt(c*x^2 + a))^4*B*a^2*c^(3/2) + 9*(sqrt(
c)*x - sqrt(c*x^2 + a))^3*A*a^2*c^2 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^3*c
^(3/2) + 15*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^3*c^2 - 32*B*a^4*c^(3/2))/((sqrt(c
)*x - sqrt(c*x^2 + a))^2 - a)^4