3.582 \(\int \frac{A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=153 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{9/2}}-\frac{5 b \sqrt{a+b x^2} (7 A b-6 a B)}{16 a^4 x^2}+\frac{5 \sqrt{a+b x^2} (7 A b-6 a B)}{24 a^3 x^4}-\frac{7 A b-6 a B}{6 a^2 x^4 \sqrt{a+b x^2}}-\frac{A}{6 a x^6 \sqrt{a+b x^2}} \]

[Out]

-A/(6*a*x^6*Sqrt[a + b*x^2]) - (7*A*b - 6*a*B)/(6*a^2*x^4*Sqrt[a + b*x^2]) + (5*
(7*A*b - 6*a*B)*Sqrt[a + b*x^2])/(24*a^3*x^4) - (5*b*(7*A*b - 6*a*B)*Sqrt[a + b*
x^2])/(16*a^4*x^2) + (5*b^2*(7*A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(1
6*a^(9/2))

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Rubi [A]  time = 0.306121, antiderivative size = 153, 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 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{9/2}}-\frac{5 b \sqrt{a+b x^2} (7 A b-6 a B)}{16 a^4 x^2}+\frac{5 \sqrt{a+b x^2} (7 A b-6 a B)}{24 a^3 x^4}-\frac{7 A b-6 a B}{6 a^2 x^4 \sqrt{a+b x^2}}-\frac{A}{6 a x^6 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]

[Out]

-A/(6*a*x^6*Sqrt[a + b*x^2]) - (7*A*b - 6*a*B)/(6*a^2*x^4*Sqrt[a + b*x^2]) + (5*
(7*A*b - 6*a*B)*Sqrt[a + b*x^2])/(24*a^3*x^4) - (5*b*(7*A*b - 6*a*B)*Sqrt[a + b*
x^2])/(16*a^4*x^2) + (5*b^2*(7*A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(1
6*a^(9/2))

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Rubi in Sympy [A]  time = 25.6741, size = 148, normalized size = 0.97 \[ - \frac{A}{6 a x^{6} \sqrt{a + b x^{2}}} - \frac{7 A b - 6 B a}{6 a^{2} x^{4} \sqrt{a + b x^{2}}} + \frac{5 \sqrt{a + b x^{2}} \left (7 A b - 6 B a\right )}{24 a^{3} x^{4}} - \frac{5 b \sqrt{a + b x^{2}} \left (7 A b - 6 B a\right )}{16 a^{4} x^{2}} + \frac{5 b^{2} \left (7 A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**7/(b*x**2+a)**(3/2),x)

[Out]

-A/(6*a*x**6*sqrt(a + b*x**2)) - (7*A*b - 6*B*a)/(6*a**2*x**4*sqrt(a + b*x**2))
+ 5*sqrt(a + b*x**2)*(7*A*b - 6*B*a)/(24*a**3*x**4) - 5*b*sqrt(a + b*x**2)*(7*A*
b - 6*B*a)/(16*a**4*x**2) + 5*b**2*(7*A*b - 6*B*a)*atanh(sqrt(a + b*x**2)/sqrt(a
))/(16*a**(9/2))

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Mathematica [A]  time = 0.336931, size = 143, normalized size = 0.93 \[ \frac{\frac{\sqrt{a} \left (-4 a^3 \left (2 A+3 B x^2\right )+2 a^2 b x^2 \left (7 A+15 B x^2\right )+5 a b^2 x^4 \left (18 B x^2-7 A\right )-105 A b^3 x^6\right )}{x^6 \sqrt{a+b x^2}}+15 b^2 (7 A b-6 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+15 b^2 \log (x) (6 a B-7 A b)}{48 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]

[Out]

((Sqrt[a]*(-105*A*b^3*x^6 - 4*a^3*(2*A + 3*B*x^2) + 2*a^2*b*x^2*(7*A + 15*B*x^2)
+ 5*a*b^2*x^4*(-7*A + 18*B*x^2)))/(x^6*Sqrt[a + b*x^2]) + 15*b^2*(-7*A*b + 6*a*
B)*Log[x] + 15*b^2*(7*A*b - 6*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(48*a^(9/2)
)

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Maple [A]  time = 0.018, size = 197, normalized size = 1.3 \[ -{\frac{A}{6\,a{x}^{6}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{7\,Ab}{24\,{a}^{2}{x}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,{b}^{2}A}{48\,{a}^{3}{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,A{b}^{3}}{16\,{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}-{\frac{B}{4\,a{x}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Bb}{8\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,B{b}^{2}}{8\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^7/(b*x^2+a)^(3/2),x)

[Out]

-1/6*A/a/x^6/(b*x^2+a)^(1/2)+7/24*A*b/a^2/x^4/(b*x^2+a)^(1/2)-35/48*A*b^2/a^3/x^
2/(b*x^2+a)^(1/2)-35/16*A*b^3/a^4/(b*x^2+a)^(1/2)+35/16*A*b^3/a^(9/2)*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)+5/8*B*b/a^2/x^2/(b*x^2+a
)^(1/2)+15/8*B*b^2/a^3/(b*x^2+a)^(1/2)-15/8*B*b^2/a^(7/2)*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)^(3/2)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234418, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 5 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \,{\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} - 15 \,{\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} +{\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{96 \,{\left (a^{4} b x^{8} + a^{5} x^{6}\right )} \sqrt{a}}, \frac{{\left (15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 5 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \,{\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 15 \,{\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} +{\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{48 \,{\left (a^{4} b x^{8} + a^{5} x^{6}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^7),x, algorithm="fricas")

[Out]

[1/96*(2*(15*(6*B*a*b^2 - 7*A*b^3)*x^6 + 5*(6*B*a^2*b - 7*A*a*b^2)*x^4 - 8*A*a^3
- 2*(6*B*a^3 - 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a) - 15*((6*B*a*b^3 - 7*A*b
^4)*x^8 + (6*B*a^2*b^2 - 7*A*a*b^3)*x^6)*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*
x^2 + a)*a)/x^2))/((a^4*b*x^8 + a^5*x^6)*sqrt(a)), 1/48*((15*(6*B*a*b^2 - 7*A*b^
3)*x^6 + 5*(6*B*a^2*b - 7*A*a*b^2)*x^4 - 8*A*a^3 - 2*(6*B*a^3 - 7*A*a^2*b)*x^2)*
sqrt(b*x^2 + a)*sqrt(-a) - 15*((6*B*a*b^3 - 7*A*b^4)*x^8 + (6*B*a^2*b^2 - 7*A*a*
b^3)*x^6)*arctan(sqrt(-a)/sqrt(b*x^2 + a)))/((a^4*b*x^8 + a^5*x^6)*sqrt(-a))]

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Sympy [A]  time = 91.6751, size = 236, normalized size = 1.54 \[ A \left (- \frac{1}{6 a \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{7 \sqrt{b}}{24 a^{2} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{35 b^{\frac{3}{2}}}{48 a^{3} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{35 b^{\frac{5}{2}}}{16 a^{4} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{35 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{9}{2}}}\right ) + B \left (- \frac{1}{4 a \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 \sqrt{b}}{8 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{15 b^{\frac{3}{2}}}{8 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{7}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**7/(b*x**2+a)**(3/2),x)

[Out]

A*(-1/(6*a*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) + 7*sqrt(b)/(24*a**2*x**5*sqrt(a/(
b*x**2) + 1)) - 35*b**(3/2)/(48*a**3*x**3*sqrt(a/(b*x**2) + 1)) - 35*b**(5/2)/(1
6*a**4*x*sqrt(a/(b*x**2) + 1)) + 35*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(9/2)
)) + B*(-1/(4*a*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + 5*sqrt(b)/(8*a**2*x**3*sqrt
(a/(b*x**2) + 1)) + 15*b**(3/2)/(8*a**3*x*sqrt(a/(b*x**2) + 1)) - 15*b**2*asinh(
sqrt(a)/(sqrt(b)*x))/(8*a**(7/2)))

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GIAC/XCAS [A]  time = 0.231239, size = 243, normalized size = 1.59 \[ \frac{5 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{16 \, \sqrt{-a} a^{4}} + \frac{B a b^{2} - A b^{3}}{\sqrt{b x^{2} + a} a^{4}} + \frac{42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{2} - 96 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{2} + 54 \, \sqrt{b x^{2} + a} B a^{3} b^{2} - 57 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{3} + 136 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{3} - 87 \, \sqrt{b x^{2} + a} A a^{2} b^{3}}{48 \, a^{4} b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^7),x, algorithm="giac")

[Out]

5/16*(6*B*a*b^2 - 7*A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) + (B*
a*b^2 - A*b^3)/(sqrt(b*x^2 + a)*a^4) + 1/48*(42*(b*x^2 + a)^(5/2)*B*a*b^2 - 96*(
b*x^2 + a)^(3/2)*B*a^2*b^2 + 54*sqrt(b*x^2 + a)*B*a^3*b^2 - 57*(b*x^2 + a)^(5/2)
*A*b^3 + 136*(b*x^2 + a)^(3/2)*A*a*b^3 - 87*sqrt(b*x^2 + a)*A*a^2*b^3)/(a^4*b^3*
x^6)