3.567 \(\int \frac{A+B x^2}{x^7 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{7/2}}-\frac{b \sqrt{a+b x^2} (5 A b-6 a B)}{16 a^3 x^2}+\frac{\sqrt{a+b x^2} (5 A b-6 a B)}{24 a^2 x^4}-\frac{A \sqrt{a+b x^2}}{6 a x^6} \]

[Out]

-(A*Sqrt[a + b*x^2])/(6*a*x^6) + ((5*A*b - 6*a*B)*Sqrt[a + b*x^2])/(24*a^2*x^4)
- (b*(5*A*b - 6*a*B)*Sqrt[a + b*x^2])/(16*a^3*x^2) + (b^2*(5*A*b - 6*a*B)*ArcTan
h[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.251277, antiderivative size = 123, 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 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{7/2}}-\frac{b \sqrt{a+b x^2} (5 A b-6 a B)}{16 a^3 x^2}+\frac{\sqrt{a+b x^2} (5 A b-6 a B)}{24 a^2 x^4}-\frac{A \sqrt{a+b x^2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^7*Sqrt[a + b*x^2]),x]

[Out]

-(A*Sqrt[a + b*x^2])/(6*a*x^6) + ((5*A*b - 6*a*B)*Sqrt[a + b*x^2])/(24*a^2*x^4)
- (b*(5*A*b - 6*a*B)*Sqrt[a + b*x^2])/(16*a^3*x^2) + (b^2*(5*A*b - 6*a*B)*ArcTan
h[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.9943, size = 114, normalized size = 0.93 \[ - \frac{A \sqrt{a + b x^{2}}}{6 a x^{6}} + \frac{\sqrt{a + b x^{2}} \left (5 A b - 6 B a\right )}{24 a^{2} x^{4}} - \frac{b \sqrt{a + b x^{2}} \left (5 A b - 6 B a\right )}{16 a^{3} x^{2}} + \frac{b^{2} \left (5 A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**7/(b*x**2+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.166233, size = 128, normalized size = 1.04 \[ \frac{b^2 (5 A b-6 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{16 a^{7/2}}-\frac{b^2 \log (x) (5 A b-6 a B)}{16 a^{7/2}}+\sqrt{a+b x^2} \left (\frac{b (6 a B-5 A b)}{16 a^3 x^2}+\frac{5 A b-6 a B}{24 a^2 x^4}-\frac{A}{6 a x^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^7*Sqrt[a + b*x^2]),x]

[Out]

(-A/(6*a*x^6) + (5*A*b - 6*a*B)/(24*a^2*x^4) + (b*(-5*A*b + 6*a*B))/(16*a^3*x^2)
)*Sqrt[a + b*x^2] - (b^2*(5*A*b - 6*a*B)*Log[x])/(16*a^(7/2)) + (b^2*(5*A*b - 6*
a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(16*a^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 161, normalized size = 1.3 \[ -{\frac{A}{6\,a{x}^{6}}\sqrt{b{x}^{2}+a}}+{\frac{5\,Ab}{24\,{a}^{2}{x}^{4}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{b}^{2}A}{16\,{a}^{3}{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{B}{4\,a{x}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{3\,Bb}{8\,{a}^{2}{x}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{3\,B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^7/(b*x^2+a)^(1/2),x)

[Out]

-1/6*A*(b*x^2+a)^(1/2)/a/x^6+5/24*A*b/a^2/x^4*(b*x^2+a)^(1/2)-5/16*A*b^2/a^3/x^2
*(b*x^2+a)^(1/2)+5/16*A*b^3/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/4*B/
a/x^4*(b*x^2+a)^(1/2)+3/8*B*b/a^2/x^2*(b*x^2+a)^(1/2)-3/8*B*b^2/a^(5/2)*ln((2*a+
2*a^(1/2)*(b*x^2+a)^(1/2))/x)

_______________________________________________________________________________________

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)/(sqrt(b*x^2 + a)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.262967, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (6 \, B a b - 5 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, a^{\frac{7}{2}} x^{6}}, -\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (6 \, B a b - 5 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x^7),x, algorithm="fricas")

[Out]

[-1/96*(3*(6*B*a*b^2 - 5*A*b^3)*x^6*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 +
a)*a)/x^2) - 2*(3*(6*B*a*b - 5*A*b^2)*x^4 - 8*A*a^2 - 2*(6*B*a^2 - 5*A*a*b)*x^2
)*sqrt(b*x^2 + a)*sqrt(a))/(a^(7/2)*x^6), -1/48*(3*(6*B*a*b^2 - 5*A*b^3)*x^6*arc
tan(sqrt(-a)/sqrt(b*x^2 + a)) - (3*(6*B*a*b - 5*A*b^2)*x^4 - 8*A*a^2 - 2*(6*B*a^
2 - 5*A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^6)]

_______________________________________________________________________________________

Sympy [A]  time = 86.287, size = 235, normalized size = 1.91 \[ - \frac{A}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A \sqrt{b}}{24 a x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{\frac{3}{2}}}{48 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{\frac{5}{2}}}{16 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{7}{2}}} - \frac{B}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B \sqrt{b}}{8 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 B b^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**7/(b*x**2+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.238615, size = 213, normalized size = 1.73 \[ \frac{\frac{3 \,{\left (6 \, B a b^{3} - 5 \, A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{18 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 30 \, \sqrt{b x^{2} + a} B a^{3} b^{3} - 15 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} + 40 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} - 33 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{a^{3} b^{3} x^{6}}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x^7),x, algorithm="giac")

[Out]

1/48*(3*(6*B*a*b^3 - 5*A*b^4)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) +
(18*(b*x^2 + a)^(5/2)*B*a*b^3 - 48*(b*x^2 + a)^(3/2)*B*a^2*b^3 + 30*sqrt(b*x^2 +
a)*B*a^3*b^3 - 15*(b*x^2 + a)^(5/2)*A*b^4 + 40*(b*x^2 + a)^(3/2)*A*a*b^4 - 33*s
qrt(b*x^2 + a)*A*a^2*b^4)/(a^3*b^3*x^6))/b