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

Optimal. Leaf size=149 \[ \frac{b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac{b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac{3 b (2 A b-a B)}{2 a^5 x^2}-\frac{b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac{3 A b-a B}{4 a^4 x^4}-\frac{A}{6 a^3 x^6} \]

[Out]

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

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Rubi [A]  time = 0.377389, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac{b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac{3 b (2 A b-a B)}{2 a^5 x^2}-\frac{b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac{3 A b-a B}{4 a^4 x^4}-\frac{A}{6 a^3 x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.4647, size = 143, normalized size = 0.96 \[ - \frac{A}{6 a^{3} x^{6}} - \frac{b^{2} \left (A b - B a\right )}{4 a^{4} \left (a + b x^{2}\right )^{2}} + \frac{3 A b - B a}{4 a^{4} x^{4}} - \frac{b^{2} \left (4 A b - 3 B a\right )}{2 a^{5} \left (a + b x^{2}\right )} - \frac{3 b \left (2 A b - B a\right )}{2 a^{5} x^{2}} - \frac{b^{2} \left (5 A b - 3 B a\right ) \log{\left (x^{2} \right )}}{a^{6}} + \frac{b^{2} \left (5 A b - 3 B a\right ) \log{\left (a + b x^{2} \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(6*a**3*x**6) - b**2*(A*b - B*a)/(4*a**4*(a + b*x**2)**2) + (3*A*b - B*a)/(4*
a**4*x**4) - b**2*(4*A*b - 3*B*a)/(2*a**5*(a + b*x**2)) - 3*b*(2*A*b - B*a)/(2*a
**5*x**2) - b**2*(5*A*b - 3*B*a)*log(x**2)/a**6 + b**2*(5*A*b - 3*B*a)*log(a + b
*x**2)/a**6

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Mathematica [A]  time = 0.223029, size = 135, normalized size = 0.91 \[ \frac{-\frac{2 a^3 A}{x^6}+\frac{3 a^2 b^2 (a B-A b)}{\left (a+b x^2\right )^2}-\frac{3 a^2 (a B-3 A b)}{x^4}+\frac{6 a b^2 (3 a B-4 A b)}{a+b x^2}+12 b^2 (5 A b-3 a B) \log \left (a+b x^2\right )+24 b^2 \log (x) (3 a B-5 A b)+\frac{18 a b (a B-2 A b)}{x^2}}{12 a^6} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a^3*A)/x^6 - (3*a^2*(-3*A*b + a*B))/x^4 + (18*a*b*(-2*A*b + a*B))/x^2 + (3*
a^2*b^2*(-(A*b) + a*B))/(a + b*x^2)^2 + (6*a*b^2*(-4*A*b + 3*a*B))/(a + b*x^2) +
24*b^2*(-5*A*b + 3*a*B)*Log[x] + 12*b^2*(5*A*b - 3*a*B)*Log[a + b*x^2])/(12*a^6
)

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Maple [A]  time = 0.025, size = 180, normalized size = 1.2 \[ -{\frac{A}{6\,{a}^{3}{x}^{6}}}+{\frac{3\,Ab}{4\,{a}^{4}{x}^{4}}}-{\frac{B}{4\,{a}^{3}{x}^{4}}}-3\,{\frac{{b}^{2}A}{{a}^{5}{x}^{2}}}+{\frac{3\,Bb}{2\,{a}^{4}{x}^{2}}}-10\,{\frac{{b}^{3}\ln \left ( x \right ) A}{{a}^{6}}}+6\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{5}}}-{\frac{A{b}^{3}}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B{b}^{2}}{4\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+5\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{a}^{6}}}-3\,{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) B}{{a}^{5}}}-2\,{\frac{A{b}^{3}}{{a}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,B{b}^{2}}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*A/a^3/x^6+3/4/a^4/x^4*A*b-1/4/a^3/x^4*B-3*b^2/a^5/x^2*A+3/2*b/a^4/x^2*B-10*
b^3/a^6*ln(x)*A+6*b^2/a^5*ln(x)*B-1/4/a^4*b^3/(b*x^2+a)^2*A+1/4/a^3*b^2/(b*x^2+a
)^2*B+5/a^6*b^3*ln(b*x^2+a)*A-3/a^5*b^2*ln(b*x^2+a)*B-2/a^5*b^3*A/(b*x^2+a)+3/2/
a^4*b^2/(b*x^2+a)*B

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Maxima [A]  time = 1.35314, size = 230, normalized size = 1.54 \[ \frac{12 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{8} + 18 \,{\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 4 \,{\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{4} -{\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x^{2}}{12 \,{\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} - \frac{{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{a^{6}} + \frac{{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*(3*B*a*b^3 - 5*A*b^4)*x^8 + 18*(3*B*a^2*b^2 - 5*A*a*b^3)*x^6 - 2*A*a^4
+ 4*(3*B*a^3*b - 5*A*a^2*b^2)*x^4 - (3*B*a^4 - 5*A*a^3*b)*x^2)/(a^5*b^2*x^10 + 2
*a^6*b*x^8 + a^7*x^6) - (3*B*a*b^2 - 5*A*b^3)*log(b*x^2 + a)/a^6 + (3*B*a*b^2 -
5*A*b^3)*log(x^2)/a^6

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Fricas [A]  time = 0.233327, size = 360, normalized size = 2.42 \[ \frac{12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6} - 2 \, A a^{5} + 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{4} -{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x^{2} - 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + 18*(3*B*a^3*b^2 - 5*A*a^2*b^3)*x^6 - 2*
A*a^5 + 4*(3*B*a^4*b - 5*A*a^3*b^2)*x^4 - (3*B*a^5 - 5*A*a^4*b)*x^2 - 12*((3*B*a
*b^4 - 5*A*b^5)*x^10 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + (3*B*a^3*b^2 - 5*A*a^2*
b^3)*x^6)*log(b*x^2 + a) + 24*((3*B*a*b^4 - 5*A*b^5)*x^10 + 2*(3*B*a^2*b^3 - 5*A
*a*b^4)*x^8 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*x^6)*log(x))/(a^6*b^2*x^10 + 2*a^7*b*x
^8 + a^8*x^6)

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Sympy [A]  time = 10.577, size = 165, normalized size = 1.11 \[ \frac{- 2 A a^{4} + x^{8} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{6} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{4} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x^{2} \left (5 A a^{3} b - 3 B a^{4}\right )}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} + \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x \right )}}{a^{6}} - \frac{b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*A*a**4 + x**8*(-60*A*b**4 + 36*B*a*b**3) + x**6*(-90*A*a*b**3 + 54*B*a**2*b*
*2) + x**4*(-20*A*a**2*b**2 + 12*B*a**3*b) + x**2*(5*A*a**3*b - 3*B*a**4))/(12*a
**7*x**6 + 24*a**6*b*x**8 + 12*a**5*b**2*x**10) + 2*b**2*(-5*A*b + 3*B*a)*log(x)
/a**6 - b**2*(-5*A*b + 3*B*a)*log(a/b + x**2)/a**6

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GIAC/XCAS [A]  time = 0.238064, size = 271, normalized size = 1.82 \[ \frac{{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )}{\rm ln}\left (x^{2}\right )}{a^{6}} - \frac{{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{6} b} + \frac{18 \, B a b^{4} x^{4} - 30 \, A b^{5} x^{4} + 42 \, B a^{2} b^{3} x^{2} - 68 \, A a b^{4} x^{2} + 25 \, B a^{3} b^{2} - 39 \, A a^{2} b^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} a^{6}} - \frac{66 \, B a b^{2} x^{6} - 110 \, A b^{3} x^{6} - 18 \, B a^{2} b x^{4} + 36 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 9 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*B*a*b^2 - 5*A*b^3)*ln(x^2)/a^6 - (3*B*a*b^3 - 5*A*b^4)*ln(abs(b*x^2 + a))/(a^
6*b) + 1/4*(18*B*a*b^4*x^4 - 30*A*b^5*x^4 + 42*B*a^2*b^3*x^2 - 68*A*a*b^4*x^2 +
25*B*a^3*b^2 - 39*A*a^2*b^3)/((b*x^2 + a)^2*a^6) - 1/12*(66*B*a*b^2*x^6 - 110*A*
b^3*x^6 - 18*B*a^2*b*x^4 + 36*A*a*b^2*x^4 + 3*B*a^3*x^2 - 9*A*a^2*b*x^2 + 2*A*a^
3)/(a^6*x^6)