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

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

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Rubi [A]  time = 0.15, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 1810, 205} \begin {gather*} \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} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.10, size = 134, normalized size = 1.02 \begin {gather*} \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 {a^2 x (4 a B-3 A b)}{b^5}+\frac {x \left (a^3 A b-a^4 B\right )}{2 b^5 \left (a+b x^2\right )}+\frac {a x^3 (3 a B-2 A b)}{3 b^4}+\frac {x^5 (A b-2 a B)}{5 b^3}+\frac {B x^7}{7 b^2} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 350, normalized size = 2.67 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.42, size = 139, normalized size = 1.06 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.02, size = 155, normalized size = 1.18 \begin {gather*} \frac {B \,x^{7}}{7 b^{2}}+\frac {A \,x^{5}}{5 b^{2}}-\frac {2 B a \,x^{5}}{5 b^{3}}-\frac {2 A a \,x^{3}}{3 b^{3}}+\frac {B \,a^{2} x^{3}}{b^{4}}+\frac {A \,a^{3} x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 A \,a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {B \,a^{4} x}{2 \left (b \,x^{2}+a \right ) b^{5}}+\frac {9 B \,a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{5}}+\frac {3 A \,a^{2} x}{b^{4}}-\frac {4 B \,a^{3} x}{b^{5}} \end {gather*}

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

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maxima [A]  time = 2.39, size = 136, normalized size = 1.04 \begin {gather*} -\frac {{\left (B a^{4} - A a^{3} b\right )} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} + \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 {15 \, B b^{3} x^{7} - 21 \, {\left (2 \, B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3} - 105 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} x}{105 \, b^{5}} \end {gather*}

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]

-1/2*(B*a^4 - A*a^3*b)*x/(b^6*x^2 + a*b^5) + 1/2*(9*B*a^4 - 7*A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) +
1/105*(15*B*b^3*x^7 - 21*(2*B*a*b^2 - A*b^3)*x^5 + 35*(3*B*a^2*b - 2*A*a*b^2)*x^3 - 105*(4*B*a^3 - 3*A*a^2*b)
*x)/b^5

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mupad [B]  time = 0.05, size = 203, normalized size = 1.55 \begin {gather*} x\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}\right )+x^5\,\left (\frac {A}{5\,b^2}-\frac {2\,B\,a}{5\,b^3}\right )-x^3\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{3\,b}+\frac {B\,a^2}{3\,b^4}\right )+\frac {B\,x^7}{7\,b^2}-\frac {x\,\left (\frac {B\,a^4}{2}-\frac {A\,a^3\,b}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (7\,A\,b-9\,B\,a\right )}{9\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-9\,B\,a\right )}{2\,b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((2*a*((2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/b - (a^2*(A/b^2 - (2*B*a)/b^3))/b^2) + x^5*(A/(5*b^2) -
(2*B*a)/(5*b^3)) - x^3*((2*a*(A/b^2 - (2*B*a)/b^3))/(3*b) + (B*a^2)/(3*b^4)) + (B*x^7)/(7*b^2) - (x*((B*a^4)/
2 - (A*a^3*b)/2))/(a*b^5 + b^6*x^2) + (a^(5/2)*atan((a^(5/2)*b^(1/2)*x*(7*A*b - 9*B*a))/(9*B*a^4 - 7*A*a^3*b))
*(7*A*b - 9*B*a))/(2*b^(11/2))

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sympy [A]  time = 0.79, size = 238, normalized size = 1.82 \begin {gather*} \frac {B x^{7}}{7 b^{2}} + x^{5} \left (\frac {A}{5 b^{2}} - \frac {2 B a}{5 b^{3}}\right ) + x^{3} \left (- \frac {2 A a}{3 b^{3}} + \frac {B a^{2}}{b^{4}}\right ) + x \left (\frac {3 A a^{2}}{b^{4}} - \frac {4 B a^{3}}{b^{5}}\right ) + \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} \end {gather*}

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

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