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

Optimal. Leaf size=76 \[ -a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{1}{3} A \left (a+b x^2\right )^{3/2}+a A \sqrt{a+b x^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b} \]

[Out]

a*A*Sqrt[a + b*x^2] + (A*(a + b*x^2)^(3/2))/3 + (B*(a + b*x^2)^(5/2))/(5*b) - a^
(3/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]

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Rubi [A]  time = 0.164721, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{1}{3} A \left (a+b x^2\right )^{3/2}+a A \sqrt{a+b x^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

a*A*Sqrt[a + b*x^2] + (A*(a + b*x^2)^(3/2))/3 + (B*(a + b*x^2)^(5/2))/(5*b) - a^
(3/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]

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Rubi in Sympy [A]  time = 16.0131, size = 65, normalized size = 0.86 \[ - A a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + A a \sqrt{a + b x^{2}} + \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(3/2)*atanh(sqrt(a + b*x**2)/sqrt(a)) + A*a*sqrt(a + b*x**2) + A*(a + b*x*
*2)**(3/2)/3 + B*(a + b*x**2)**(5/2)/(5*b)

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Mathematica [A]  time = 0.151229, size = 93, normalized size = 1.22 \[ -a^{3/2} A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{3/2} A \log (x)+\sqrt{a+b x^2} \left (\frac{1}{15} x^2 (6 a B+5 A b)+\frac{a (3 a B+20 A b)}{15 b}+\frac{1}{5} b B x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a*(20*A*b + 3*a*B))/(15*b) + ((5*A*b + 6*a*B)*x^2)/15 + (b*B*x
^4)/5) + a^(3/2)*A*Log[x] - a^(3/2)*A*Log[a + Sqrt[a]*Sqrt[a + b*x^2]]

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Maple [A]  time = 0.01, size = 70, normalized size = 0.9 \[{\frac{A}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +aA\sqrt{b{x}^{2}+a}+{\frac{B}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*A*(b*x^2+a)^(3/2)-A*a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+a*A*(b*x^2
+a)^(1/2)+1/5*B*(b*x^2+a)^(5/2)/b

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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)*(b*x^2 + a)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231901, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{\frac{3}{2}} b \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \, B b^{2} x^{4} + 3 \, B a^{2} + 20 \, A a b +{\left (6 \, B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{30 \, b}, -\frac{15 \, A \sqrt{-a} a b \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (3 \, B b^{2} x^{4} + 3 \, B a^{2} + 20 \, A a b +{\left (6 \, B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*A*a^(3/2)*b*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(3
*B*b^2*x^4 + 3*B*a^2 + 20*A*a*b + (6*B*a*b + 5*A*b^2)*x^2)*sqrt(b*x^2 + a))/b, -
1/15*(15*A*sqrt(-a)*a*b*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) - (3*B*b^2*x^4 + 3*
B*a^2 + 20*A*a*b + (6*B*a*b + 5*A*b^2)*x^2)*sqrt(b*x^2 + a))/b]

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Sympy [A]  time = 20.0003, size = 134, normalized size = 1.76 \[ - A a^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x^{2} \wedge - a < 0 \end{cases}\right ) + A a \sqrt{a + b x^{2}} + \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**2*Piecewise((-atan(sqrt(a + b*x**2)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sq
rt(a + b*x**2)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x**2)), (atanh(sqrt(a + b
*x**2)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*x**2))) + A*a*sqrt(a + b*x**2) +
A*(a + b*x**2)**(3/2)/3 + B*(a + b*x**2)**(5/2)/(5*b)

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GIAC/XCAS [A]  time = 0.247243, size = 107, normalized size = 1.41 \[ \frac{A a^{2} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B b^{4} + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{5} + 15 \, \sqrt{b x^{2} + a} A a b^{5}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a^2*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 1/15*(3*(b*x^2 + a)^(5/2)*B*b^
4 + 5*(b*x^2 + a)^(3/2)*A*b^5 + 15*sqrt(b*x^2 + a)*A*a*b^5)/b^5