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

Optimal. Leaf size=108 \[ a^{3/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{\left (a+b x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac{1}{2} \sqrt{a+b x^2} (2 a B+3 A b x)+\frac{3}{2} a A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.305782, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ a^{3/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{\left (a+b x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac{1}{2} \sqrt{a+b x^2} (2 a B+3 A b x)+\frac{3}{2} a A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.4541, size = 97, normalized size = 0.9 \[ \frac{3 A a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2} - B a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + \frac{\sqrt{a + b x^{2}} \left (6 A b x + 4 B a\right )}{4} - \frac{\left (3 A - B x\right ) \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*A*a*sqrt(b)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/2 - B*a**(3/2)*atanh(sqrt(a + b*
x**2)/sqrt(a)) + sqrt(a + b*x**2)*(6*A*b*x + 4*B*a)/4 - (3*A - B*x)*(a + b*x**2)
**(3/2)/(3*x)

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Mathematica [A]  time = 0.154832, size = 113, normalized size = 1.05 \[ -a^{3/2} B \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{3/2} B \log (x)+\sqrt{a+b x^2} \left (-\frac{a A}{x}+\frac{4 a B}{3}+\frac{A b x}{2}+\frac{1}{3} b B x^2\right )+\frac{3}{2} a A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((4*a*B)/3 - (a*A)/x + (A*b*x)/2 + (b*B*x^2)/3) + a^(3/2)*B*Log[
x] - a^(3/2)*B*Log[a + Sqrt[a]*Sqrt[a + b*x^2]] + (3*a*A*Sqrt[b]*Log[b*x + Sqrt[
b]*Sqrt[a + b*x^2]])/2

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Maple [A]  time = 0.011, size = 126, normalized size = 1.2 \[ -{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Axb}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Axb}{2}\sqrt{b{x}^{2}+a}}+{\frac{3\,Aa}{2}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }+{\frac{B}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-B{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +B\sqrt{b{x}^{2}+a}a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283154, size = 1, normalized size = 0.01 \[ \left [\frac{9 \, A a \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 6 \, B a^{\frac{3}{2}} x \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt{b x^{2} + a}}{12 \, x}, \frac{9 \, A a \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) + 3 \, B a^{\frac{3}{2}} x \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) +{\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt{b x^{2} + a}}{6 \, x}, -\frac{12 \, B \sqrt{-a} a x \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) - 9 \, A a \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt{b x^{2} + a}}{12 \, x}, \frac{9 \, A a \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) - 6 \, B \sqrt{-a} a x \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) +{\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt{b x^{2} + a}}{6 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(9*A*a*sqrt(b)*x*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 6*B*a^(
3/2)*x*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(2*B*b*x^3 + 3*A*
b*x^2 + 8*B*a*x - 6*A*a)*sqrt(b*x^2 + a))/x, 1/6*(9*A*a*sqrt(-b)*x*arctan(b*x/(s
qrt(b*x^2 + a)*sqrt(-b))) + 3*B*a^(3/2)*x*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a
) + 2*a)/x^2) + (2*B*b*x^3 + 3*A*b*x^2 + 8*B*a*x - 6*A*a)*sqrt(b*x^2 + a))/x, -1
/12*(12*B*sqrt(-a)*a*x*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) - 9*A*a*sqrt(b)*x*lo
g(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(2*B*b*x^3 + 3*A*b*x^2 + 8*B*a
*x - 6*A*a)*sqrt(b*x^2 + a))/x, 1/6*(9*A*a*sqrt(-b)*x*arctan(b*x/(sqrt(b*x^2 + a
)*sqrt(-b))) - 6*B*sqrt(-a)*a*x*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) + (2*B*b*x^
3 + 3*A*b*x^2 + 8*B*a*x - 6*A*a)*sqrt(b*x^2 + a))/x]

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Sympy [A]  time = 7.99014, size = 184, normalized size = 1.7 \[ - \frac{A a^{\frac{3}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{A \sqrt{a} b x \sqrt{1 + \frac{b x^{2}}{a}}}{2} - \frac{A \sqrt{a} b x}{\sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} - B a^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{B a^{2}}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B a \sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} + B b \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(3/2)/(x*sqrt(1 + b*x**2/a)) + A*sqrt(a)*b*x*sqrt(1 + b*x**2/a)/2 - A*sqrt
(a)*b*x/sqrt(1 + b*x**2/a) + 3*A*a*sqrt(b)*asinh(sqrt(b)*x/sqrt(a))/2 - B*a**(3/
2)*asinh(sqrt(a)/(sqrt(b)*x)) + B*a**2/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) + B*a*sq
rt(b)*x/sqrt(a/(b*x**2) + 1) + B*b*Piecewise((sqrt(a)*x**2/2, Eq(b, 0)), ((a + b
*x**2)**(3/2)/(3*b), True))

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GIAC/XCAS [A]  time = 0.229309, size = 167, normalized size = 1.55 \[ \frac{2 \, B a^{2} \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3}{2} \, A a \sqrt{b}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right ) + \frac{2 \, A a^{2} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} + \frac{1}{6} \, \sqrt{b x^{2} + a}{\left (8 \, B a +{\left (2 \, B b x + 3 \, A b\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*a^2*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - 3/2*A*a*sqrt(
b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) + 2*A*a^2*sqrt(b)/((sqrt(b)*x - sqrt(b*
x^2 + a))^2 - a) + 1/6*sqrt(b*x^2 + a)*(8*B*a + (2*B*b*x + 3*A*b)*x)