3.99 \(\int \frac{x^8 (A+B x^2)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=138 \[ -\frac{a^2 x (13 A b-17 a B)}{8 b^5 \left (a+b x^2\right )}+\frac{a^3 x (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}+\frac{7 a^{3/2} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}+\frac{x^3 (A b-3 a B)}{3 b^4}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{B x^5}{5 b^3} \]

[Out]

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

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Rubi [A]  time = 0.217875, antiderivative size = 138, 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} \[ -\frac{a^2 x (13 A b-17 a B)}{8 b^5 \left (a+b x^2\right )}+\frac{a^3 x (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}+\frac{7 a^{3/2} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}+\frac{x^3 (A b-3 a B)}{3 b^4}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{B x^5}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.102357, size = 133, normalized size = 0.96 \[ \frac{x \left (7 a^2 b^2 x^2 \left (72 B x^2-125 A\right )-525 a^3 b \left (A-3 B x^2\right )+945 a^4 B-8 a b^3 x^4 \left (35 A+9 B x^2\right )+8 b^4 x^6 \left (5 A+3 B x^2\right )\right )}{120 b^5 \left (a+b x^2\right )^2}-\frac{7 a^{3/2} (9 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(x*(945*a^4*B - 525*a^3*b*(A - 3*B*x^2) + 8*b^4*x^6*(5*A + 3*B*x^2) - 8*a*b^3*x^4*(35*A + 9*B*x^2) + 7*a^2*b^2
*x^2*(-125*A + 72*B*x^2)))/(120*b^5*(a + b*x^2)^2) - (7*a^(3/2)*(-5*A*b + 9*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/
(8*b^(11/2))

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Maple [A]  time = 0.01, size = 174, normalized size = 1.3 \begin{align*}{\frac{B{x}^{5}}{5\,{b}^{3}}}+{\frac{A{x}^{3}}{3\,{b}^{3}}}-{\frac{B{x}^{3}a}{{b}^{4}}}-3\,{\frac{aAx}{{b}^{4}}}+6\,{\frac{{a}^{2}Bx}{{b}^{5}}}-{\frac{13\,{a}^{2}A{x}^{3}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{17\,B{a}^{3}{x}^{3}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,A{a}^{3}x}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,B{a}^{4}x}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,A{a}^{2}}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,B{a}^{3}}{8\,{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)^3,x)

[Out]

1/5*B*x^5/b^3+1/3/b^3*A*x^3-1/b^4*B*x^3*a-3/b^4*a*A*x+6/b^5*a^2*B*x-13/8*a^2/b^3/(b*x^2+a)^2*A*x^3+17/8*a^3/b^
4/(b*x^2+a)^2*B*x^3-11/8*a^3/b^4/(b*x^2+a)^2*A*x+15/8*a^4/b^5/(b*x^2+a)^2*B*x+35/8*a^2/b^4/(a*b)^(1/2)*arctan(
b*x/(a*b)^(1/2))*A-63/8*a^3/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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31963, size = 892, normalized size = 6.46 \begin{align*} \left [\frac{48 \, B b^{4} x^{9} - 16 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{7} + 112 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{5} + 350 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{3} - 105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{4} + 2 \,{\left (9 \, B a^{3} b - 5 \, 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} - 5 \, A a^{3} b\right )} x}{240 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac{24 \, B b^{4} x^{9} - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{7} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{5} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{3} - 105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{4} + 2 \,{\left (9 \, B a^{3} b - 5 \, 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} - 5 \, A a^{3} b\right )} x}{120 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} 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)^3,x, algorithm="fricas")

[Out]

[1/240*(48*B*b^4*x^9 - 16*(9*B*a*b^3 - 5*A*b^4)*x^7 + 112*(9*B*a^2*b^2 - 5*A*a*b^3)*x^5 + 350*(9*B*a^3*b - 5*A
*a^2*b^2)*x^3 - 105*(9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a*b^3)*x^4 + 2*(9*B*a^3*b - 5*A*a^2*b^2)*x^2)*sq
rt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(9*B*a^4 - 5*A*a^3*b)*x)/(b^7*x^4 + 2*a*b^6*x^2
+ a^2*b^5), 1/120*(24*B*b^4*x^9 - 8*(9*B*a*b^3 - 5*A*b^4)*x^7 + 56*(9*B*a^2*b^2 - 5*A*a*b^3)*x^5 + 175*(9*B*a
^3*b - 5*A*a^2*b^2)*x^3 - 105*(9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a*b^3)*x^4 + 2*(9*B*a^3*b - 5*A*a^2*b^
2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 105*(9*B*a^4 - 5*A*a^3*b)*x)/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5)]

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Sympy [A]  time = 1.60939, size = 250, normalized size = 1.81 \begin{align*} \frac{B x^{5}}{5 b^{3}} + \frac{7 \sqrt{- \frac{a^{3}}{b^{11}}} \left (- 5 A b + 9 B a\right ) \log{\left (- \frac{7 b^{5} \sqrt{- \frac{a^{3}}{b^{11}}} \left (- 5 A b + 9 B a\right )}{- 35 A a b + 63 B a^{2}} + x \right )}}{16} - \frac{7 \sqrt{- \frac{a^{3}}{b^{11}}} \left (- 5 A b + 9 B a\right ) \log{\left (\frac{7 b^{5} \sqrt{- \frac{a^{3}}{b^{11}}} \left (- 5 A b + 9 B a\right )}{- 35 A a b + 63 B a^{2}} + x \right )}}{16} + \frac{x^{3} \left (- 13 A a^{2} b^{2} + 17 B a^{3} b\right ) + x \left (- 11 A a^{3} b + 15 B a^{4}\right )}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} - \frac{x^{3} \left (- A b + 3 B a\right )}{3 b^{4}} + \frac{x \left (- 3 A a b + 6 B a^{2}\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)**3,x)

[Out]

B*x**5/(5*b**3) + 7*sqrt(-a**3/b**11)*(-5*A*b + 9*B*a)*log(-7*b**5*sqrt(-a**3/b**11)*(-5*A*b + 9*B*a)/(-35*A*a
*b + 63*B*a**2) + x)/16 - 7*sqrt(-a**3/b**11)*(-5*A*b + 9*B*a)*log(7*b**5*sqrt(-a**3/b**11)*(-5*A*b + 9*B*a)/(
-35*A*a*b + 63*B*a**2) + x)/16 + (x**3*(-13*A*a**2*b**2 + 17*B*a**3*b) + x*(-11*A*a**3*b + 15*B*a**4))/(8*a**2
*b**5 + 16*a*b**6*x**2 + 8*b**7*x**4) - x**3*(-A*b + 3*B*a)/(3*b**4) + x*(-3*A*a*b + 6*B*a**2)/b**5

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Giac [A]  time = 1.6717, size = 186, normalized size = 1.35 \begin{align*} -\frac{7 \,{\left (9 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{5}} + \frac{17 \, B a^{3} b x^{3} - 13 \, A a^{2} b^{2} x^{3} + 15 \, B a^{4} x - 11 \, A a^{3} b x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{5}} + \frac{3 \, B b^{12} x^{5} - 15 \, B a b^{11} x^{3} + 5 \, A b^{12} x^{3} + 90 \, B a^{2} b^{10} x - 45 \, A a b^{11} x}{15 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-7/8*(9*B*a^3 - 5*A*a^2*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/8*(17*B*a^3*b*x^3 - 13*A*a^2*b^2*x^3 + 15
*B*a^4*x - 11*A*a^3*b*x)/((b*x^2 + a)^2*b^5) + 1/15*(3*B*b^12*x^5 - 15*B*a*b^11*x^3 + 5*A*b^12*x^3 + 90*B*a^2*
b^10*x - 45*A*a*b^11*x)/b^15