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

Optimal. Leaf size=106 \[ -a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{3 a^2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{8} a \sqrt{a+c x^2} (8 A+3 B x)+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 A+3 B x) \]

[Out]

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

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Rubi [A]  time = 0.263617, antiderivative size = 106, 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 \[ -a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{3 a^2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{8} a \sqrt{a+c x^2} (8 A+3 B x)+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 A+3 B x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.8392, size = 97, normalized size = 0.92 \[ - A a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )} + \frac{3 B a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 \sqrt{c}} + \frac{a \left (8 A + 3 B x\right ) \sqrt{a + c x^{2}}}{8} + \frac{\left (4 A + 3 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.171297, size = 112, normalized size = 1.06 \[ -a^{3/2} A \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+a^{3/2} A \log (x)+\frac{3 a^2 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{8 \sqrt{c}}+\frac{1}{24} \sqrt{a+c x^2} \left (32 a A+15 a B x+8 A c x^2+6 B c x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 107, normalized size = 1. \[{\frac{Bx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aBx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}B}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{A}{3} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +A\sqrt{c{x}^{2}+a}a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.344527, size = 1, normalized size = 0.01 \[ \left [\frac{9 \, B a^{2} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 24 \, A a^{\frac{3}{2}} \sqrt{c} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (6 \, B c x^{3} + 8 \, A c x^{2} + 15 \, B a x + 32 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{c}}{48 \, \sqrt{c}}, \frac{9 \, B a^{2} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + 12 \, A a^{\frac{3}{2}} \sqrt{-c} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) +{\left (6 \, B c x^{3} + 8 \, A c x^{2} + 15 \, B a x + 32 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{24 \, \sqrt{-c}}, -\frac{48 \, A \sqrt{-a} a \sqrt{c} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 9 \, B a^{2} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) - 2 \,{\left (6 \, B c x^{3} + 8 \, A c x^{2} + 15 \, B a x + 32 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{c}}{48 \, \sqrt{c}}, \frac{9 \, B a^{2} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 24 \, A \sqrt{-a} a \sqrt{-c} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (6 \, B c x^{3} + 8 \, A c x^{2} + 15 \, B a x + 32 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{24 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 22.1514, size = 218, normalized size = 2.06 \[ - A a^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{A a^{2}}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A a \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + A c \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + \frac{B a^{\frac{3}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{B a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B \sqrt{a} c x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + \frac{B c^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(3/2)*asinh(sqrt(a)/(sqrt(c)*x)) + A*a**2/(sqrt(c)*x*sqrt(a/(c*x**2) + 1))
+ A*a*sqrt(c)*x/sqrt(a/(c*x**2) + 1) + A*c*Piecewise((sqrt(a)*x**2/2, Eq(c, 0))
, ((a + c*x**2)**(3/2)/(3*c), True)) + B*a**(3/2)*x*sqrt(1 + c*x**2/a)/2 + B*a**
(3/2)*x/(8*sqrt(1 + c*x**2/a)) + 3*B*sqrt(a)*c*x**3/(8*sqrt(1 + c*x**2/a)) + 3*B
*a**2*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) + B*c**2*x**5/(4*sqrt(a)*sqrt(1 + c*x
**2/a))

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GIAC/XCAS [A]  time = 0.279631, size = 135, normalized size = 1.27 \[ \frac{2 \, A a^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3 \, B a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{24} \, \sqrt{c x^{2} + a}{\left (32 \, A a +{\left (15 \, B a + 2 \,{\left (3 \, B c x + 4 \, A c\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 3/8*B*a^2*ln(
abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/24*sqrt(c*x^2 + a)*(32*A*a + (15*
B*a + 2*(3*B*c*x + 4*A*c)*x)*x)