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

Optimal. Leaf size=95 \[ -a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+a^2 A \sqrt{a+b x^2}+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]

[Out]

a^2*A*Sqrt[a + b*x^2] + (a*A*(a + b*x^2)^(3/2))/3 + (A*(a + b*x^2)^(5/2))/5 + (B
*(a + b*x^2)^(7/2))/(7*b) - a^(5/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]

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Rubi [A]  time = 0.190377, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+a^2 A \sqrt{a+b x^2}+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

a^2*A*Sqrt[a + b*x^2] + (a*A*(a + b*x^2)^(3/2))/3 + (A*(a + b*x^2)^(5/2))/5 + (B
*(a + b*x^2)^(7/2))/(7*b) - a^(5/2)*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]

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Rubi in Sympy [A]  time = 19.0256, size = 82, normalized size = 0.86 \[ - A a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + A a^{2} \sqrt{a + b x^{2}} + \frac{A a \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{5} + \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(5/2)*atanh(sqrt(a + b*x**2)/sqrt(a)) + A*a**2*sqrt(a + b*x**2) + A*a*(a +
b*x**2)**(3/2)/3 + A*(a + b*x**2)**(5/2)/5 + B*(a + b*x**2)**(7/2)/(7*b)

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Mathematica [A]  time = 0.231555, size = 115, normalized size = 1.21 \[ -a^{5/2} A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{5/2} A \log (x)+\sqrt{a+b x^2} \left (\frac{a^2 (15 a B+161 A b)}{105 b}+\frac{1}{35} b x^4 (15 a B+7 A b)+\frac{1}{105} a x^2 (45 a B+77 A b)+\frac{1}{7} b^2 B x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a^2*(161*A*b + 15*a*B))/(105*b) + (a*(77*A*b + 45*a*B)*x^2)/10
5 + (b*(7*A*b + 15*a*B)*x^4)/35 + (b^2*B*x^6)/7) + a^(5/2)*A*Log[x] - a^(5/2)*A*
Log[a + Sqrt[a]*Sqrt[a + b*x^2]]

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Maple [A]  time = 0.011, size = 85, normalized size = 0.9 \[{\frac{A}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Aa}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +{a}^{2}A\sqrt{b{x}^{2}+a}+{\frac{B}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*A*(b*x^2+a)^(5/2)+1/3*a*A*(b*x^2+a)^(3/2)-A*a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2
+a)^(1/2))/x)+a^2*A*(b*x^2+a)^(1/2)+1/7*B*(b*x^2+a)^(7/2)/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)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239964, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A a^{\frac{5}{2}} b \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (15 \, B b^{3} x^{6} + 3 \,{\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{4} + 15 \, B a^{3} + 161 \, A a^{2} b +{\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{210 \, b}, -\frac{105 \, A \sqrt{-a} a^{2} b \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (15 \, B b^{3} x^{6} + 3 \,{\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{4} + 15 \, B a^{3} + 161 \, A a^{2} b +{\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/210*(105*A*a^(5/2)*b*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*
(15*B*b^3*x^6 + 3*(15*B*a*b^2 + 7*A*b^3)*x^4 + 15*B*a^3 + 161*A*a^2*b + (45*B*a^
2*b + 77*A*a*b^2)*x^2)*sqrt(b*x^2 + a))/b, -1/105*(105*A*sqrt(-a)*a^2*b*arctan(a
/(sqrt(b*x^2 + a)*sqrt(-a))) - (15*B*b^3*x^6 + 3*(15*B*a*b^2 + 7*A*b^3)*x^4 + 15
*B*a^3 + 161*A*a^2*b + (45*B*a^2*b + 77*A*a*b^2)*x^2)*sqrt(b*x^2 + a))/b]

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Sympy [A]  time = 33.3996, size = 151, normalized size = 1.59 \[ - A a^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x^{2} \wedge - a < 0 \end{cases}\right ) + A a^{2} \sqrt{a + b x^{2}} + \frac{A a \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{5} + \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**3*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**2*sqrt(a + b*x**2)
+ A*a*(a + b*x**2)**(3/2)/3 + A*(a + b*x**2)**(5/2)/5 + B*(a + b*x**2)**(7/2)/(
7*b)

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GIAC/XCAS [A]  time = 0.224855, size = 131, normalized size = 1.38 \[ \frac{A a^{3} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B b^{6} + 21 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{7} + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{7} + 105 \, \sqrt{b x^{2} + a} A a^{2} b^{7}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a^3*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 1/105*(15*(b*x^2 + a)^(7/2)*B*
b^6 + 21*(b*x^2 + a)^(5/2)*A*b^7 + 35*(b*x^2 + a)^(3/2)*A*a*b^7 + 105*sqrt(b*x^2
+ a)*A*a^2*b^7)/b^7