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

Optimal. Leaf size=143 \[ -\frac{\left (a+b x^2\right )^{5/2} (4 a B+3 A b)}{8 a x^2}+\frac{5 b \left (a+b x^2\right )^{3/2} (4 a B+3 A b)}{24 a}+\frac{5}{8} b \sqrt{a+b x^2} (4 a B+3 A b)-\frac{5}{8} \sqrt{a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{A \left (a+b x^2\right )^{7/2}}{4 a x^4} \]

[Out]

(5*b*(3*A*b + 4*a*B)*Sqrt[a + b*x^2])/8 + (5*b*(3*A*b + 4*a*B)*(a + b*x^2)^(3/2)
)/(24*a) - ((3*A*b + 4*a*B)*(a + b*x^2)^(5/2))/(8*a*x^2) - (A*(a + b*x^2)^(7/2))
/(4*a*x^4) - (5*Sqrt[a]*b*(3*A*b + 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/8

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Rubi [A]  time = 0.276589, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (a+b x^2\right )^{5/2} (4 a B+3 A b)}{8 a x^2}+\frac{5 b \left (a+b x^2\right )^{3/2} (4 a B+3 A b)}{24 a}+\frac{5}{8} b \sqrt{a+b x^2} (4 a B+3 A b)-\frac{5}{8} \sqrt{a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{A \left (a+b x^2\right )^{7/2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

(5*b*(3*A*b + 4*a*B)*Sqrt[a + b*x^2])/8 + (5*b*(3*A*b + 4*a*B)*(a + b*x^2)^(3/2)
)/(24*a) - ((3*A*b + 4*a*B)*(a + b*x^2)^(5/2))/(8*a*x^2) - (A*(a + b*x^2)^(7/2))
/(4*a*x^4) - (5*Sqrt[a]*b*(3*A*b + 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/8

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Rubi in Sympy [A]  time = 22.6851, size = 134, normalized size = 0.94 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{4 a x^{4}} - \frac{5 \sqrt{a} b \left (3 A b + 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8} + \frac{5 b \sqrt{a + b x^{2}} \left (3 A b + 4 B a\right )}{8} + \frac{5 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 A b + 4 B a\right )}{24 a} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (3 A b + 4 B a\right )}{8 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x**2)**(7/2)/(4*a*x**4) - 5*sqrt(a)*b*(3*A*b + 4*B*a)*atanh(sqrt(a + b
*x**2)/sqrt(a))/8 + 5*b*sqrt(a + b*x**2)*(3*A*b + 4*B*a)/8 + 5*b*(a + b*x**2)**(
3/2)*(3*A*b + 4*B*a)/(24*a) - (a + b*x**2)**(5/2)*(3*A*b + 4*B*a)/(8*a*x**2)

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Mathematica [A]  time = 0.326401, size = 119, normalized size = 0.83 \[ \frac{1}{24} \left (\sqrt{a+b x^2} \left (-\frac{6 a^2 A}{x^4}-\frac{3 a (4 a B+9 A b)}{x^2}+8 b (7 a B+3 A b)+8 b^2 B x^2\right )-15 \sqrt{a} b (4 a B+3 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+15 \sqrt{a} b \log (x) (4 a B+3 A b)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*(8*b*(3*A*b + 7*a*B) - (6*a^2*A)/x^4 - (3*a*(9*A*b + 4*a*B))/x^
2 + 8*b^2*B*x^2) + 15*Sqrt[a]*b*(3*A*b + 4*a*B)*Log[x] - 15*Sqrt[a]*b*(3*A*b + 4
*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/24

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Maple [A]  time = 0.012, size = 213, normalized size = 1.5 \[ -{\frac{A}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Ab}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{b}^{2}A}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{b}^{2}A}{8\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{b}^{2}A}{8}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{15\,{b}^{2}A}{8}\sqrt{b{x}^{2}+a}}-{\frac{B}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bb}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Bb}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bb}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{5\,abB}{2}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24765, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (4 \, B a b + 3 \, A b^{2}\right )} \sqrt{a} x^{4} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (8 \, B b^{2} x^{6} + 8 \,{\left (7 \, B a b + 3 \, A b^{2}\right )} x^{4} - 6 \, A a^{2} - 3 \,{\left (4 \, B a^{2} + 9 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{48 \, x^{4}}, -\frac{15 \,{\left (4 \, B a b + 3 \, A b^{2}\right )} \sqrt{-a} x^{4} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (8 \, B b^{2} x^{6} + 8 \,{\left (7 \, B a b + 3 \, A b^{2}\right )} x^{4} - 6 \, A a^{2} - 3 \,{\left (4 \, B a^{2} + 9 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{24 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*(4*B*a*b + 3*A*b^2)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a
) + 2*a)/x^2) + 2*(8*B*b^2*x^6 + 8*(7*B*a*b + 3*A*b^2)*x^4 - 6*A*a^2 - 3*(4*B*a^
2 + 9*A*a*b)*x^2)*sqrt(b*x^2 + a))/x^4, -1/24*(15*(4*B*a*b + 3*A*b^2)*sqrt(-a)*x
^4*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) - (8*B*b^2*x^6 + 8*(7*B*a*b + 3*A*b^2)*x
^4 - 6*A*a^2 - 3*(4*B*a^2 + 9*A*a*b)*x^2)*sqrt(b*x^2 + a))/x^4]

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Sympy [A]  time = 105.41, size = 279, normalized size = 1.95 \[ - \frac{15 A \sqrt{a} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8} - \frac{A a^{3}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A a^{2} \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{x} + \frac{7 A a b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{5}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 B a^{\frac{3}{2}} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{B a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{2 B a^{2} \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{2 B a b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} + B b^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*A*sqrt(a)*b**2*asinh(sqrt(a)/(sqrt(b)*x))/8 - A*a**3/(4*sqrt(b)*x**5*sqrt(a/
(b*x**2) + 1)) - 3*A*a**2*sqrt(b)/(8*x**3*sqrt(a/(b*x**2) + 1)) - A*a*b**(3/2)*s
qrt(a/(b*x**2) + 1)/x + 7*A*a*b**(3/2)/(8*x*sqrt(a/(b*x**2) + 1)) + A*b**(5/2)*x
/sqrt(a/(b*x**2) + 1) - 5*B*a**(3/2)*b*asinh(sqrt(a)/(sqrt(b)*x))/2 - B*a**2*sqr
t(b)*sqrt(a/(b*x**2) + 1)/(2*x) + 2*B*a**2*sqrt(b)/(x*sqrt(a/(b*x**2) + 1)) + 2*
B*a*b**(3/2)*x/sqrt(a/(b*x**2) + 1) + B*b**2*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 = 0.238546, size = 231, normalized size = 1.62 \[ \frac{8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B b^{2} + 48 \, \sqrt{b x^{2} + a} B a b^{2} + 24 \, \sqrt{b x^{2} + a} A b^{3} + \frac{15 \,{\left (4 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{2} - 4 \, \sqrt{b x^{2} + a} B a^{3} b^{2} + 9 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{3} - 7 \, \sqrt{b x^{2} + a} A a^{2} b^{3}\right )}}{b^{2} x^{4}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(8*(b*x^2 + a)^(3/2)*B*b^2 + 48*sqrt(b*x^2 + a)*B*a*b^2 + 24*sqrt(b*x^2 + a
)*A*b^3 + 15*(4*B*a^2*b^2 + 3*A*a*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a)
- 3*(4*(b*x^2 + a)^(3/2)*B*a^2*b^2 - 4*sqrt(b*x^2 + a)*B*a^3*b^2 + 9*(b*x^2 + a
)^(3/2)*A*a*b^3 - 7*sqrt(b*x^2 + a)*A*a^2*b^3)/(b^2*x^4))/b