3.72 \(\int \frac{x^8 (A+B x^2)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{a^3 x (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}+\frac{x^5 (A b-2 a B)}{5 b^3}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{B x^7}{7 b^2} \]

[Out]

(a^2*(3*A*b - 4*a*B)*x)/b^5 - (a*(2*A*b - 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x^5)/(5*b^3) + (B*x^7)/(7*b^2)
+ (a^3*(A*b - a*B)*x)/(2*b^5*(a + b*x^2)) - (a^(5/2)*(7*A*b - 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Rubi [A]  time = 0.150482, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {455, 1810, 205} \[ \frac{a^3 x (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}+\frac{x^5 (A b-2 a B)}{5 b^3}-\frac{a x^3 (2 A b-3 a B)}{3 b^4}+\frac{B x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

(a^2*(3*A*b - 4*a*B)*x)/b^5 - (a*(2*A*b - 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x^5)/(5*b^3) + (B*x^7)/(7*b^2)
+ (a^3*(A*b - a*B)*x)/(2*b^5*(a + b*x^2)) - (a^(5/2)*(7*A*b - 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^8 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{a^3 (A b-a B) x}{2 b^5 \left (a+b x^2\right )}-\frac{\int \frac{a^3 (A b-a B)-2 a^2 b (A b-a B) x^2+2 a b^2 (A b-a B) x^4-2 b^3 (A b-a B) x^6-2 b^4 B x^8}{a+b x^2} \, dx}{2 b^5}\\ &=\frac{a^3 (A b-a B) x}{2 b^5 \left (a+b x^2\right )}-\frac{\int \left (-2 a^2 (3 A b-4 a B)+2 a b (2 A b-3 a B) x^2-2 b^2 (A b-2 a B) x^4-2 b^3 B x^6+\frac{7 a^3 A b-9 a^4 B}{a+b x^2}\right ) \, dx}{2 b^5}\\ &=\frac{a^2 (3 A b-4 a B) x}{b^5}-\frac{a (2 A b-3 a B) x^3}{3 b^4}+\frac{(A b-2 a B) x^5}{5 b^3}+\frac{B x^7}{7 b^2}+\frac{a^3 (A b-a B) x}{2 b^5 \left (a+b x^2\right )}-\frac{\left (a^3 (7 A b-9 a B)\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^5}\\ &=\frac{a^2 (3 A b-4 a B) x}{b^5}-\frac{a (2 A b-3 a B) x^3}{3 b^4}+\frac{(A b-2 a B) x^5}{5 b^3}+\frac{B x^7}{7 b^2}+\frac{a^3 (A b-a B) x}{2 b^5 \left (a+b x^2\right )}-\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.103465, size = 134, normalized size = 1.02 \[ \frac{x \left (a^3 A b-a^4 B\right )}{2 b^5 \left (a+b x^2\right )}-\frac{a^2 x (4 a B-3 A b)}{b^5}+\frac{a^{5/2} (9 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}+\frac{x^5 (A b-2 a B)}{5 b^3}+\frac{a x^3 (3 a B-2 A b)}{3 b^4}+\frac{B x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

-((a^2*(-3*A*b + 4*a*B)*x)/b^5) + (a*(-2*A*b + 3*a*B)*x^3)/(3*b^4) + ((A*b - 2*a*B)*x^5)/(5*b^3) + (B*x^7)/(7*
b^2) + ((a^3*A*b - a^4*B)*x)/(2*b^5*(a + b*x^2)) + (a^(5/2)*(-7*A*b + 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b
^(11/2))

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Maple [A]  time = 0.009, size = 155, normalized size = 1.2 \begin{align*}{\frac{B{x}^{7}}{7\,{b}^{2}}}+{\frac{A{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,B{x}^{5}a}{5\,{b}^{3}}}-{\frac{2\,aA{x}^{3}}{3\,{b}^{3}}}+{\frac{B{x}^{3}{a}^{2}}{{b}^{4}}}+3\,{\frac{{a}^{2}Ax}{{b}^{4}}}-4\,{\frac{B{a}^{3}x}{{b}^{5}}}+{\frac{{a}^{3}xA}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{4}xB}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,A{a}^{3}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,B{a}^{4}}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

1/7*B*x^7/b^2+1/5/b^2*A*x^5-2/5/b^3*B*x^5*a-2/3/b^3*A*x^3*a+1/b^4*B*x^3*a^2+3/b^4*A*a^2*x-4/b^5*B*a^3*x+1/2*a^
3/b^4*x/(b*x^2+a)*A-1/2*a^4/b^5*x/(b*x^2+a)*B-7/2*a^3/b^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A+9/2*a^4/b^5/(a
*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*B

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29378, size = 755, normalized size = 5.76 \begin{align*} \left [\frac{60 \, B b^{4} x^{9} - 12 \,{\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{7} + 28 \,{\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{5} - 140 \,{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{3} - 105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b +{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 210 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} x}{420 \,{\left (b^{6} x^{2} + a b^{5}\right )}}, \frac{30 \, B b^{4} x^{9} - 6 \,{\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{7} + 14 \,{\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{5} - 70 \,{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{3} + 105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b +{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} x}{210 \,{\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/420*(60*B*b^4*x^9 - 12*(9*B*a*b^3 - 7*A*b^4)*x^7 + 28*(9*B*a^2*b^2 - 7*A*a*b^3)*x^5 - 140*(9*B*a^3*b - 7*A*
a^2*b^2)*x^3 - 105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a
/b) - a)/(b*x^2 + a)) - 210*(9*B*a^4 - 7*A*a^3*b)*x)/(b^6*x^2 + a*b^5), 1/210*(30*B*b^4*x^9 - 6*(9*B*a*b^3 - 7
*A*b^4)*x^7 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^5 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x^3 + 105*(9*B*a^4 - 7*A*a^3*b +
(9*B*a^3*b - 7*A*a^2*b^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 105*(9*B*a^4 - 7*A*a^3*b)*x)/(b^6*x^2 + a*
b^5)]

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Sympy [A]  time = 0.90571, size = 233, normalized size = 1.78 \begin{align*} \frac{B x^{7}}{7 b^{2}} - \frac{x \left (- A a^{3} b + B a^{4}\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{\sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right ) \log{\left (- \frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right )}{- 7 A a^{2} b + 9 B a^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right ) \log{\left (\frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}} \left (- 7 A b + 9 B a\right )}{- 7 A a^{2} b + 9 B a^{3}} + x \right )}}{4} - \frac{x^{5} \left (- A b + 2 B a\right )}{5 b^{3}} + \frac{x^{3} \left (- 2 A a b + 3 B a^{2}\right )}{3 b^{4}} - \frac{x \left (- 3 A a^{2} b + 4 B a^{3}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**7/(7*b**2) - x*(-A*a**3*b + B*a**4)/(2*a*b**5 + 2*b**6*x**2) - sqrt(-a**5/b**11)*(-7*A*b + 9*B*a)*log(-b*
*5*sqrt(-a**5/b**11)*(-7*A*b + 9*B*a)/(-7*A*a**2*b + 9*B*a**3) + x)/4 + sqrt(-a**5/b**11)*(-7*A*b + 9*B*a)*log
(b**5*sqrt(-a**5/b**11)*(-7*A*b + 9*B*a)/(-7*A*a**2*b + 9*B*a**3) + x)/4 - x**5*(-A*b + 2*B*a)/(5*b**3) + x**3
*(-2*A*a*b + 3*B*a**2)/(3*b**4) - x*(-3*A*a**2*b + 4*B*a**3)/b**5

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Giac [A]  time = 1.12032, size = 188, normalized size = 1.44 \begin{align*} \frac{{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{5}} - \frac{B a^{4} x - A a^{3} b x}{2 \,{\left (b x^{2} + a\right )} b^{5}} + \frac{15 \, B b^{12} x^{7} - 42 \, B a b^{11} x^{5} + 21 \, A b^{12} x^{5} + 105 \, B a^{2} b^{10} x^{3} - 70 \, A a b^{11} x^{3} - 420 \, B a^{3} b^{9} x + 315 \, A a^{2} b^{10} x}{105 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(9*B*a^4 - 7*A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/2*(B*a^4*x - A*a^3*b*x)/((b*x^2 + a)*b^5)
+ 1/105*(15*B*b^12*x^7 - 42*B*a*b^11*x^5 + 21*A*b^12*x^5 + 105*B*a^2*b^10*x^3 - 70*A*a*b^11*x^3 - 420*B*a^3*b^
9*x + 315*A*a^2*b^10*x)/b^14