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

Optimal. Leaf size=136 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{x \left (a+b x^2\right )^{5/2} (a B+6 A b)}{6 a}+\frac{5}{24} x \left (a+b x^2\right )^{3/2} (a B+6 A b)+\frac{5}{16} a x \sqrt{a+b x^2} (a B+6 A b)-\frac{A \left (a+b x^2\right )^{7/2}}{a x} \]

[Out]

(5*a*(6*A*b + a*B)*x*Sqrt[a + b*x^2])/16 + (5*(6*A*b + a*B)*x*(a + b*x^2)^(3/2))
/24 + ((6*A*b + a*B)*x*(a + b*x^2)^(5/2))/(6*a) - (A*(a + b*x^2)^(7/2))/(a*x) +
(5*a^2*(6*A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])

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Rubi [A]  time = 0.154543, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{x \left (a+b x^2\right )^{5/2} (a B+6 A b)}{6 a}+\frac{5}{24} x \left (a+b x^2\right )^{3/2} (a B+6 A b)+\frac{5}{16} a x \sqrt{a+b x^2} (a B+6 A b)-\frac{A \left (a+b x^2\right )^{7/2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

(5*a*(6*A*b + a*B)*x*Sqrt[a + b*x^2])/16 + (5*(6*A*b + a*B)*x*(a + b*x^2)^(3/2))
/24 + ((6*A*b + a*B)*x*(a + b*x^2)^(5/2))/(6*a) - (A*(a + b*x^2)^(7/2))/(a*x) +
(5*a^2*(6*A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])

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Rubi in Sympy [A]  time = 14.9891, size = 128, normalized size = 0.94 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{a x} + \frac{5 a^{2} \left (6 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 \sqrt{b}} + \frac{5 a x \sqrt{a + b x^{2}} \left (6 A b + B a\right )}{16} + x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (\frac{5 A b}{4} + \frac{5 B a}{24}\right ) + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (6 A b + B a\right )}{6 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x**2)**(7/2)/(a*x) + 5*a**2*(6*A*b + B*a)*atanh(sqrt(b)*x/sqrt(a + b*x
**2))/(16*sqrt(b)) + 5*a*x*sqrt(a + b*x**2)*(6*A*b + B*a)/16 + x*(a + b*x**2)**(
3/2)*(5*A*b/4 + 5*B*a/24) + x*(a + b*x**2)**(5/2)*(6*A*b + B*a)/(6*a)

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Mathematica [A]  time = 0.177886, size = 108, normalized size = 0.79 \[ \sqrt{a+b x^2} \left (-\frac{a^2 A}{x}+\frac{1}{24} b x^3 (13 a B+6 A b)+\frac{1}{16} a x (11 a B+18 A b)+\frac{1}{6} b^2 B x^5\right )+\frac{5 a^2 (a B+6 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*(-((a^2*A)/x) + (a*(18*A*b + 11*a*B)*x)/16 + (b*(6*A*b + 13*a*B)
*x^3)/24 + (b^2*B*x^5)/6) + (5*a^2*(6*A*b + a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^
2]])/(16*Sqrt[b])

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Maple [A]  time = 0.012, size = 158, normalized size = 1.2 \[{\frac{Bx}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Bxa}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Bx{a}^{2}}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}-{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Axb}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Axb}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,Axab}{8}\sqrt{b{x}^{2}+a}}+{\frac{15\,A{a}^{2}}{8}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x*B*(b*x^2+a)^(5/2)+5/24*B*a*x*(b*x^2+a)^(3/2)+5/16*B*a^2*x*(b*x^2+a)^(1/2)+
5/16*B*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-A*(b*x^2+a)^(7/2)/a/x+A*b/a*x*(
b*x^2+a)^(5/2)+5/4*A*b*x*(b*x^2+a)^(3/2)+15/8*A*b*a*x*(b*x^2+a)^(1/2)+15/8*A*b^(
1/2)*a^2*ln(x*b^(1/2)+(b*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((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264099, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, B b^{2} x^{6} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{4} - 48 \, A a^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{96 \, \sqrt{b} x}, \frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, B b^{2} x^{6} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{4} - 48 \, A a^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{48 \, \sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(15*(B*a^3 + 6*A*a^2*b)*x*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(
b)) + 2*(8*B*b^2*x^6 + 2*(13*B*a*b + 6*A*b^2)*x^4 - 48*A*a^2 + 3*(11*B*a^2 + 18*
A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(b))/(sqrt(b)*x), 1/48*(15*(B*a^3 + 6*A*a^2*b)*x
*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (8*B*b^2*x^6 + 2*(13*B*a*b + 6*A*b^2)*x^4
- 48*A*a^2 + 3*(11*B*a^2 + 18*A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*x)
]

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Sympy [A]  time = 57.1551, size = 306, normalized size = 2.25 \[ - \frac{A a^{\frac{5}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + A a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x^{2}}{a}} - \frac{7 A a^{\frac{3}{2}} b x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} b^{2} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 A a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8} + \frac{A b^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 B a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 B a^{\frac{3}{2}} b x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 B \sqrt{a} b^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{B b^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(5/2)/(x*sqrt(1 + b*x**2/a)) + A*a**(3/2)*b*x*sqrt(1 + b*x**2/a) - 7*A*a**
(3/2)*b*x/(8*sqrt(1 + b*x**2/a)) + 3*A*sqrt(a)*b**2*x**3/(8*sqrt(1 + b*x**2/a))
+ 15*A*a**2*sqrt(b)*asinh(sqrt(b)*x/sqrt(a))/8 + A*b**3*x**5/(4*sqrt(a)*sqrt(1 +
b*x**2/a)) + B*a**(5/2)*x*sqrt(1 + b*x**2/a)/2 + 3*B*a**(5/2)*x/(16*sqrt(1 + b*
x**2/a)) + 35*B*a**(3/2)*b*x**3/(48*sqrt(1 + b*x**2/a)) + 17*B*sqrt(a)*b**2*x**5
/(24*sqrt(1 + b*x**2/a)) + 5*B*a**3*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b)) + B*b*
*3*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.250509, size = 197, normalized size = 1.45 \[ \frac{2 \, A a^{3} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} + \frac{1}{48} \,{\left (2 \,{\left (4 \, B b^{2} x^{2} + \frac{13 \, B a b^{5} + 6 \, A b^{6}}{b^{4}}\right )} x^{2} + \frac{3 \,{\left (11 \, B a^{2} b^{4} + 18 \, A a b^{5}\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x - \frac{5 \,{\left (B a^{3} \sqrt{b} + 6 \, A a^{2} b^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{32 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a^3*sqrt(b)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a) + 1/48*(2*(4*B*b^2*x^2 + (
13*B*a*b^5 + 6*A*b^6)/b^4)*x^2 + 3*(11*B*a^2*b^4 + 18*A*a*b^5)/b^4)*sqrt(b*x^2 +
a)*x - 5/32*(B*a^3*sqrt(b) + 6*A*a^2*b^(3/2))*ln((sqrt(b)*x - sqrt(b*x^2 + a))^
2)/b