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

Optimal. Leaf size=104 \[ \frac{a^3 (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^2 (3 A b-4 a B) \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^4 (A b-2 a B)}{4 b^3}+\frac{B x^6}{6 b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.281264, antiderivative size = 104, 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 \[ \frac{a^3 (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^2 (3 A b-4 a B) \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^4 (A b-2 a B)}{4 b^3}+\frac{B x^6}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{6}}{6 b^{2}} + \frac{a^{3} \left (A b - B a\right )}{2 b^{5} \left (a + b x^{2}\right )} + \frac{a^{2} \left (3 A b - 4 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} + \frac{\left (A b - 2 B a\right ) \int ^{x^{2}} x\, dx}{2 b^{3}} - \frac{\left (2 A b - 3 B a\right ) \int ^{x^{2}} a\, dx}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.121785, size = 93, normalized size = 0.89 \[ \frac{\frac{6 a^3 (A b-a B)}{a+b x^2}+6 a^2 (3 A b-4 a B) \log \left (a+b x^2\right )+3 b^2 x^4 (A b-2 a B)+6 a b x^2 (3 a B-2 A b)+2 b^3 B x^6}{12 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 122, normalized size = 1.2 \[{\frac{B{x}^{6}}{6\,{b}^{2}}}+{\frac{A{x}^{4}}{4\,{b}^{2}}}-{\frac{B{x}^{4}a}{2\,{b}^{3}}}-{\frac{aA{x}^{2}}{{b}^{3}}}+{\frac{3\,B{x}^{2}{a}^{2}}{2\,{b}^{4}}}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{4}}}-2\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) B}{{b}^{5}}}+{\frac{{a}^{3}A}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{B{a}^{4}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34831, size = 144, normalized size = 1.38 \[ -\frac{B a^{4} - A a^{3} b}{2 \,{\left (b^{6} x^{2} + a b^{5}\right )}} + \frac{2 \, B b^{2} x^{6} - 3 \,{\left (2 \, B a b - A b^{2}\right )} x^{4} + 6 \,{\left (3 \, B a^{2} - 2 \, A a b\right )} x^{2}}{12 \, b^{4}} - \frac{{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.231588, size = 200, normalized size = 1.92 \[ \frac{2 \, B b^{4} x^{8} -{\left (4 \, B a b^{3} - 3 \, A b^{4}\right )} x^{6} - 6 \, B a^{4} + 6 \, A a^{3} b + 3 \,{\left (4 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + 6 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} - 6 \,{\left (4 \, B a^{4} - 3 \, A a^{3} b +{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{6} x^{2} + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.40407, size = 102, normalized size = 0.98 \[ \frac{B x^{6}}{6 b^{2}} - \frac{a^{2} \left (- 3 A b + 4 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} - \frac{- A a^{3} b + B a^{4}}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{x^{4} \left (- A b + 2 B a\right )}{4 b^{3}} + \frac{x^{2} \left (- 2 A a b + 3 B a^{2}\right )}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224694, size = 182, normalized size = 1.75 \[ -\frac{{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac{2 \, B b^{4} x^{6} - 6 \, B a b^{3} x^{4} + 3 \, A b^{4} x^{4} + 18 \, B a^{2} b^{2} x^{2} - 12 \, A a b^{3} x^{2}}{12 \, b^{6}} + \frac{4 \, B a^{3} b x^{2} - 3 \, A a^{2} b^{2} x^{2} + 3 \, B a^{4} - 2 \, A a^{3} b}{2 \,{\left (b x^{2} + a\right )} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*B*a^3 - 3*A*a^2*b)*ln(abs(b*x^2 + a))/b^5 + 1/12*(2*B*b^4*x^6 - 6*B*a*b^
3*x^4 + 3*A*b^4*x^4 + 18*B*a^2*b^2*x^2 - 12*A*a*b^3*x^2)/b^6 + 1/2*(4*B*a^3*b*x^
2 - 3*A*a^2*b^2*x^2 + 3*B*a^4 - 2*A*a^3*b)/((b*x^2 + a)*b^5)