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

Optimal. Leaf size=80 \[ -\frac{a^2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{\left (a+c x^2\right )^{3/2} (4 A+3 B x)}{12 c}-\frac{a B x \sqrt{a+c x^2}}{8 c} \]

[Out]

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

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Rubi [A]  time = 0.0771735, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{a^2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{\left (a+c x^2\right )^{3/2} (4 A+3 B x)}{12 c}-\frac{a B x \sqrt{a+c x^2}}{8 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.42434, size = 70, normalized size = 0.88 \[ - \frac{B a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{3}{2}}} - \frac{B a x \sqrt{a + c x^{2}}}{8 c} + \frac{\left (4 A + 3 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{12 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0667974, size = 79, normalized size = 0.99 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (8 a A+3 a B x+8 A c x^2+6 B c x^3\right )-3 a^2 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{24 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 75, normalized size = 0.9 \[{\frac{A}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bx}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aBx}{8\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{2}B}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308671, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B a^{2} \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (6 \, B c x^{3} + 8 \, A c x^{2} + 3 \, B a x + 8 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{c}}{48 \, c^{\frac{3}{2}}}, -\frac{3 \, B a^{2} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (6 \, B c x^{3} + 8 \, A c x^{2} + 3 \, B a x + 8 \, A a\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{24 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.0031, size = 124, normalized size = 1.55 \[ A \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + \frac{B a^{\frac{3}{2}} x}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{3}{2}}} + \frac{B c x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273671, size = 92, normalized size = 1.15 \[ \frac{B a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{3}{2}}} + \frac{1}{24} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \, B x + 4 \, A\right )} x + \frac{3 \, B a}{c}\right )} x + \frac{8 \, A a}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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