3.98 \(\int \frac{x^{10} (A+B x^2)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=158 \[ \frac{a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}-\frac{a^4 x (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{2 a^2 x (3 A b-5 a B)}{b^6}-\frac{9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}+\frac{x^5 (A b-3 a B)}{5 b^4}-\frac{a x^3 (A b-2 a B)}{b^5}+\frac{B x^7}{7 b^3} \]

[Out]

(2*a^2*(3*A*b - 5*a*B)*x)/b^6 - (a*(A*b - 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3) - (a
^4*(A*b - a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) - (9*a^(5/2)*(7*A*b -
11*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Rubi [A]  time = 0.283153, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {455, 1814, 1810, 205} \[ \frac{a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}-\frac{a^4 x (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{2 a^2 x (3 A b-5 a B)}{b^6}-\frac{9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}+\frac{x^5 (A b-3 a B)}{5 b^4}-\frac{a x^3 (A b-2 a B)}{b^5}+\frac{B x^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^10*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(2*a^2*(3*A*b - 5*a*B)*x)/b^6 - (a*(A*b - 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3) - (a
^4*(A*b - a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) - (9*a^(5/2)*(7*A*b -
11*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

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^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac{a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}-\frac{\int \frac{-a^4 (A b-a B)+4 a^3 b (A b-a B) x^2-4 a^2 b^2 (A b-a B) x^4+4 a b^3 (A b-a B) x^6-4 b^4 (A b-a B) x^8-4 b^5 B x^{10}}{\left (a+b x^2\right )^2} \, dx}{4 b^6}\\ &=-\frac{a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac{\int \frac{-a^4 (15 A b-19 a B)+8 a^3 b (3 A b-4 a B) x^2-8 a^2 b^2 (2 A b-3 a B) x^4+8 a b^3 (A b-2 a B) x^6+8 a b^4 B x^8}{a+b x^2} \, dx}{8 a b^6}\\ &=-\frac{a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac{\int \left (16 a^3 (3 A b-5 a B)-24 a^2 b (A b-2 a B) x^2+8 a b^2 (A b-3 a B) x^4+8 a b^3 B x^6+\frac{9 \left (-7 a^4 A b+11 a^5 B\right )}{a+b x^2}\right ) \, dx}{8 a b^6}\\ &=\frac{2 a^2 (3 A b-5 a B) x}{b^6}-\frac{a (A b-2 a B) x^3}{b^5}+\frac{(A b-3 a B) x^5}{5 b^4}+\frac{B x^7}{7 b^3}-\frac{a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac{\left (9 a^3 (7 A b-11 a B)\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^6}\\ &=\frac{2 a^2 (3 A b-5 a B) x}{b^6}-\frac{a (A b-2 a B) x^3}{b^5}+\frac{(A b-3 a B) x^5}{5 b^4}+\frac{B x^7}{7 b^3}-\frac{a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac{9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.0915596, size = 158, normalized size = 1. \[ \frac{a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}+\frac{a^4 x (a B-A b)}{4 b^6 \left (a+b x^2\right )^2}-\frac{2 a^2 x (5 a B-3 A b)}{b^6}+\frac{9 a^{5/2} (11 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}+\frac{x^5 (A b-3 a B)}{5 b^4}+\frac{a x^3 (2 a B-A b)}{b^5}+\frac{B x^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^10*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(-2*a^2*(-3*A*b + 5*a*B)*x)/b^6 + (a*(-(A*b) + 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3)
+ (a^4*(-(A*b) + a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) + (9*a^(5/2)*(
-7*A*b + 11*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Maple [A]  time = 0.011, size = 198, normalized size = 1.3 \begin{align*}{\frac{B{x}^{7}}{7\,{b}^{3}}}+{\frac{A{x}^{5}}{5\,{b}^{3}}}-{\frac{3\,B{x}^{5}a}{5\,{b}^{4}}}-{\frac{aA{x}^{3}}{{b}^{4}}}+2\,{\frac{B{x}^{3}{a}^{2}}{{b}^{5}}}+6\,{\frac{{a}^{2}Ax}{{b}^{5}}}-10\,{\frac{B{a}^{3}x}{{b}^{6}}}+{\frac{17\,A{a}^{3}{x}^{3}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{21\,B{a}^{4}{x}^{3}}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,A{a}^{4}x}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{19\,{a}^{5}Bx}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,A{a}^{3}}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{99\,B{a}^{4}}{8\,{b}^{6}}\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^10*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

1/7*B*x^7/b^3+1/5/b^3*A*x^5-3/5/b^4*B*x^5*a-1/b^4*A*x^3*a+2/b^5*B*x^3*a^2+6/b^5*A*a^2*x-10/b^6*B*a^3*x+17/8*a^
3/b^4/(b*x^2+a)^2*A*x^3-21/8*a^4/b^5/(b*x^2+a)^2*B*x^3+15/8*a^4/b^5/(b*x^2+a)^2*A*x-19/8*a^5/b^6/(b*x^2+a)^2*B
*x-63/8*a^3/b^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A+99/8*a^4/b^6/(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^10*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3479, size = 1025, normalized size = 6.49 \begin{align*} \left [\frac{80 \, B b^{5} x^{11} - 16 \,{\left (11 \, B a b^{4} - 7 \, A b^{5}\right )} x^{9} + 48 \,{\left (11 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} - 336 \,{\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{5} - 1050 \,{\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{3} - 315 \,{\left (11 \, B a^{5} - 7 \, A a^{4} b +{\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + 2 \,{\left (11 \, B a^{4} b - 7 \, A a^{3} 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 ) - 630 \,{\left (11 \, B a^{5} - 7 \, A a^{4} b\right )} x}{560 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac{40 \, B b^{5} x^{11} - 8 \,{\left (11 \, B a b^{4} - 7 \, A b^{5}\right )} x^{9} + 24 \,{\left (11 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} - 168 \,{\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{5} - 525 \,{\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{3} + 315 \,{\left (11 \, B a^{5} - 7 \, A a^{4} b +{\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + 2 \,{\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 315 \,{\left (11 \, B a^{5} - 7 \, A a^{4} b\right )} x}{280 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/560*(80*B*b^5*x^11 - 16*(11*B*a*b^4 - 7*A*b^5)*x^9 + 48*(11*B*a^2*b^3 - 7*A*a*b^4)*x^7 - 336*(11*B*a^3*b^2
- 7*A*a^2*b^3)*x^5 - 1050*(11*B*a^4*b - 7*A*a^3*b^2)*x^3 - 315*(11*B*a^5 - 7*A*a^4*b + (11*B*a^3*b^2 - 7*A*a^2
*b^3)*x^4 + 2*(11*B*a^4*b - 7*A*a^3*b^2)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 630
*(11*B*a^5 - 7*A*a^4*b)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6), 1/280*(40*B*b^5*x^11 - 8*(11*B*a*b^4 - 7*A*b^5)*
x^9 + 24*(11*B*a^2*b^3 - 7*A*a*b^4)*x^7 - 168*(11*B*a^3*b^2 - 7*A*a^2*b^3)*x^5 - 525*(11*B*a^4*b - 7*A*a^3*b^2
)*x^3 + 315*(11*B*a^5 - 7*A*a^4*b + (11*B*a^3*b^2 - 7*A*a^2*b^3)*x^4 + 2*(11*B*a^4*b - 7*A*a^3*b^2)*x^2)*sqrt(
a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(11*B*a^5 - 7*A*a^4*b)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)]

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Sympy [A]  time = 1.71509, size = 274, normalized size = 1.73 \begin{align*} \frac{B x^{7}}{7 b^{3}} - \frac{9 \sqrt{- \frac{a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right ) \log{\left (- \frac{9 b^{6} \sqrt{- \frac{a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right )}{- 63 A a^{2} b + 99 B a^{3}} + x \right )}}{16} + \frac{9 \sqrt{- \frac{a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right ) \log{\left (\frac{9 b^{6} \sqrt{- \frac{a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right )}{- 63 A a^{2} b + 99 B a^{3}} + x \right )}}{16} - \frac{x^{3} \left (- 17 A a^{3} b^{2} + 21 B a^{4} b\right ) + x \left (- 15 A a^{4} b + 19 B a^{5}\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} - \frac{x^{5} \left (- A b + 3 B a\right )}{5 b^{4}} + \frac{x^{3} \left (- A a b + 2 B a^{2}\right )}{b^{5}} - \frac{x \left (- 6 A a^{2} b + 10 B a^{3}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

B*x**7/(7*b**3) - 9*sqrt(-a**5/b**13)*(-7*A*b + 11*B*a)*log(-9*b**6*sqrt(-a**5/b**13)*(-7*A*b + 11*B*a)/(-63*A
*a**2*b + 99*B*a**3) + x)/16 + 9*sqrt(-a**5/b**13)*(-7*A*b + 11*B*a)*log(9*b**6*sqrt(-a**5/b**13)*(-7*A*b + 11
*B*a)/(-63*A*a**2*b + 99*B*a**3) + x)/16 - (x**3*(-17*A*a**3*b**2 + 21*B*a**4*b) + x*(-15*A*a**4*b + 19*B*a**5
))/(8*a**2*b**6 + 16*a*b**7*x**2 + 8*b**8*x**4) - x**5*(-A*b + 3*B*a)/(5*b**4) + x**3*(-A*a*b + 2*B*a**2)/b**5
- x*(-6*A*a**2*b + 10*B*a**3)/b**6

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Giac [A]  time = 1.18908, size = 219, normalized size = 1.39 \begin{align*} \frac{9 \,{\left (11 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{6}} - \frac{21 \, B a^{4} b x^{3} - 17 \, A a^{3} b^{2} x^{3} + 19 \, B a^{5} x - 15 \, A a^{4} b x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{5 \, B b^{18} x^{7} - 21 \, B a b^{17} x^{5} + 7 \, A b^{18} x^{5} + 70 \, B a^{2} b^{16} x^{3} - 35 \, A a b^{17} x^{3} - 350 \, B a^{3} b^{15} x + 210 \, A a^{2} b^{16} x}{35 \, b^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

9/8*(11*B*a^4 - 7*A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) - 1/8*(21*B*a^4*b*x^3 - 17*A*a^3*b^2*x^3 + 19
*B*a^5*x - 15*A*a^4*b*x)/((b*x^2 + a)^2*b^6) + 1/35*(5*B*b^18*x^7 - 21*B*a*b^17*x^5 + 7*A*b^18*x^5 + 70*B*a^2*
b^16*x^3 - 35*A*a*b^17*x^3 - 350*B*a^3*b^15*x + 210*A*a^2*b^16*x)/b^21