3.594 \(\int \frac{A+B x^2}{x^3 \left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A b-2 a B}{2 a^3 \sqrt{a+b x^2}}-\frac{5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.240657, antiderivative size = 113, 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 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A b-2 a B}{2 a^3 \sqrt{a+b x^2}}-\frac{5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.3942, size = 99, normalized size = 0.88 \[ - \frac{A}{2 a x^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{\frac{5 A b}{2} - B a}{3 a^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{\frac{5 A b}{2} - B a}{a^{3} \sqrt{a + b x^{2}}} + \frac{\left (\frac{5 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**3/(b*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.318537, size = 114, normalized size = 1.01 \[ \frac{\frac{\sqrt{a} \left (-3 a^2 A+8 a^2 B x^2-20 a A b x^2+6 a b B x^4-15 A b^2 x^4\right )}{x^2 \left (a+b x^2\right )^{3/2}}+3 (5 A b-2 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\log (x) (6 a B-15 A b)}{6 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)),x]

[Out]

((Sqrt[a]*(-3*a^2*A - 20*a*A*b*x^2 + 8*a^2*B*x^2 - 15*A*b^2*x^4 + 6*a*b*B*x^4))/
(x^2*(a + b*x^2)^(3/2)) + (-15*A*b + 6*a*B)*Log[x] + 3*(5*A*b - 2*a*B)*Log[a + S
qrt[a]*Sqrt[a + b*x^2]])/(6*a^(7/2))

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Maple [A]  time = 0.013, size = 140, normalized size = 1.2 \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ab}{6\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ab}{2\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}+{\frac{B}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{B}{{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^3/(b*x^2+a)^(5/2),x)

[Out]

-1/2*A/a/x^2/(b*x^2+a)^(3/2)-5/6*A*b/a^2/(b*x^2+a)^(3/2)-5/2*A*b/a^3/(b*x^2+a)^(
1/2)+5/2*A*b/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/3*B/a/(b*x^2+a)^(3/
2)+B/a^2/(b*x^2+a)^(1/2)-B/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251177, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} + 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} - 3 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{12 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{a}}, \frac{{\left (3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} + 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[1/12*(2*(3*(2*B*a*b - 5*A*b^2)*x^4 - 3*A*a^2 + 4*(2*B*a^2 - 5*A*a*b)*x^2)*sqrt(
b*x^2 + a)*sqrt(a) - 3*((2*B*a*b^2 - 5*A*b^3)*x^6 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^
4 + (2*B*a^3 - 5*A*a^2*b)*x^2)*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a)*a
)/x^2))/((a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*sqrt(a)), 1/6*((3*(2*B*a*b - 5*A*
b^2)*x^4 - 3*A*a^2 + 4*(2*B*a^2 - 5*A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a) - 3*((2
*B*a*b^2 - 5*A*b^3)*x^6 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^4 + (2*B*a^3 - 5*A*a^2*b)*
x^2)*arctan(sqrt(-a)/sqrt(b*x^2 + a)))/((a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*sq
rt(-a))]

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Sympy [A]  time = 119.557, size = 1608, normalized size = 14.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**3/(b*x**2+a)**(5/2),x)

[Out]

A*(-6*a**17*sqrt(1 + b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**
(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 46*a**16*b*x**2*sqrt(1 + b*x**2/a)/
(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)
*b**3*x**8) - 15*a**16*b*x**2*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*
x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 30*a**16*b*x**2*log(sq
rt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b*
*2*x**6 + 12*a**(33/2)*b**3*x**8) - 70*a**15*b**2*x**4*sqrt(1 + b*x**2/a)/(12*a*
*(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*
x**8) - 45*a**15*b**2*x**4*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**
4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 90*a**15*b**2*x**4*log(sq
rt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b*
*2*x**6 + 12*a**(33/2)*b**3*x**8) - 30*a**14*b**3*x**6*sqrt(1 + b*x**2/a)/(12*a*
*(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*
x**8) - 45*a**14*b**3*x**6*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**
4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 90*a**14*b**3*x**6*log(sq
rt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b*
*2*x**6 + 12*a**(33/2)*b**3*x**8) - 15*a**13*b**4*x**8*log(b*x**2/a)/(12*a**(39/
2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8)
+ 30*a**13*b**4*x**8*log(sqrt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37
/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8)) + B*(8*a**7*sqrt(
1 + b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a*
*(13/2)*b**3*x**6) + 3*a**7*log(b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 1
8*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 6*a**7*log(sqrt(1 + b*x**2/a) +
1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b*
*3*x**6) + 14*a**6*b*x**2*sqrt(1 + b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2
+ 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 9*a**6*b*x**2*log(b*x**2/a)/
(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x
**6) - 18*a**6*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*
x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 6*a**5*b**2*x**4*sqrt(1
+ b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**
(13/2)*b**3*x**6) + 9*a**5*b**2*x**4*log(b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b
*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 18*a**5*b**2*x**4*log(
sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x
**4 + 6*a**(13/2)*b**3*x**6) + 3*a**4*b**3*x**6*log(b*x**2/a)/(6*a**(19/2) + 18*
a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 6*a**4*b**3
*x**6*log(sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15
/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6))

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GIAC/XCAS [A]  time = 0.235652, size = 136, normalized size = 1.2 \[ \frac{{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a^{3}} + \frac{3 \,{\left (b x^{2} + a\right )} B a + B a^{2} - 6 \,{\left (b x^{2} + a\right )} A b - A a b}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} - \frac{\sqrt{b x^{2} + a} A}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="giac")

[Out]

1/2*(2*B*a - 5*A*b)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) + 1/3*(3*(b*
x^2 + a)*B*a + B*a^2 - 6*(b*x^2 + a)*A*b - A*a*b)/((b*x^2 + a)^(3/2)*a^3) - 1/2*
sqrt(b*x^2 + a)*A/(a^3*x^2)