3.1.41 \(\int \frac {x^{10} (A+B x^2)}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=119 \[ -\frac {b^{7/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{11/2}}+\frac {b^3 x (b B-A c)}{c^5}-\frac {b^2 x^3 (b B-A c)}{3 c^4}+\frac {b x^5 (b B-A c)}{5 c^3}-\frac {x^7 (b B-A c)}{7 c^2}+\frac {B x^9}{9 c} \]

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Rubi [A]  time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 459, 302, 205} \begin {gather*} -\frac {b^2 x^3 (b B-A c)}{3 c^4}+\frac {b^3 x (b B-A c)}{c^5}-\frac {b^{7/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{11/2}}-\frac {x^7 (b B-A c)}{7 c^2}+\frac {b x^5 (b B-A c)}{5 c^3}+\frac {B x^9}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^10*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

(b^3*(b*B - A*c)*x)/c^5 - (b^2*(b*B - A*c)*x^3)/(3*c^4) + (b*(b*B - A*c)*x^5)/(5*c^3) - ((b*B - A*c)*x^7)/(7*c
^2) + (B*x^9)/(9*c) - (b^(7/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x^{10} \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {x^8 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac {B x^9}{9 c}-\frac {(9 b B-9 A c) \int \frac {x^8}{b+c x^2} \, dx}{9 c}\\ &=\frac {B x^9}{9 c}-\frac {(9 b B-9 A c) \int \left (-\frac {b^3}{c^4}+\frac {b^2 x^2}{c^3}-\frac {b x^4}{c^2}+\frac {x^6}{c}+\frac {b^4}{c^4 \left (b+c x^2\right )}\right ) \, dx}{9 c}\\ &=\frac {b^3 (b B-A c) x}{c^5}-\frac {b^2 (b B-A c) x^3}{3 c^4}+\frac {b (b B-A c) x^5}{5 c^3}-\frac {(b B-A c) x^7}{7 c^2}+\frac {B x^9}{9 c}-\frac {\left (b^4 (b B-A c)\right ) \int \frac {1}{b+c x^2} \, dx}{c^5}\\ &=\frac {b^3 (b B-A c) x}{c^5}-\frac {b^2 (b B-A c) x^3}{3 c^4}+\frac {b (b B-A c) x^5}{5 c^3}-\frac {(b B-A c) x^7}{7 c^2}+\frac {B x^9}{9 c}-\frac {b^{7/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 119, normalized size = 1.00 \begin {gather*} -\frac {b^{7/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{11/2}}+\frac {b^3 x (b B-A c)}{c^5}-\frac {b^2 x^3 (b B-A c)}{3 c^4}+\frac {b x^5 (b B-A c)}{5 c^3}+\frac {x^7 (A c-b B)}{7 c^2}+\frac {B x^9}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{10} \left (A+B x^2\right )}{b x^2+c x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^10*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

IntegrateAlgebraic[(x^10*(A + B*x^2))/(b*x^2 + c*x^4), x]

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fricas [A]  time = 0.41, size = 274, normalized size = 2.30 \begin {gather*} \left [\frac {70 \, B c^{4} x^{9} - 90 \, {\left (B b c^{3} - A c^{4}\right )} x^{7} + 126 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{5} - 210 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x^{3} - 315 \, {\left (B b^{4} - A b^{3} c\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} + 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) + 630 \, {\left (B b^{4} - A b^{3} c\right )} x}{630 \, c^{5}}, \frac {35 \, B c^{4} x^{9} - 45 \, {\left (B b c^{3} - A c^{4}\right )} x^{7} + 63 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{5} - 105 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x^{3} - 315 \, {\left (B b^{4} - A b^{3} c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) + 315 \, {\left (B b^{4} - A b^{3} c\right )} x}{315 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/630*(70*B*c^4*x^9 - 90*(B*b*c^3 - A*c^4)*x^7 + 126*(B*b^2*c^2 - A*b*c^3)*x^5 - 210*(B*b^3*c - A*b^2*c^2)*x^
3 - 315*(B*b^4 - A*b^3*c)*sqrt(-b/c)*log((c*x^2 + 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) + 630*(B*b^4 - A*b^3*c)*x
)/c^5, 1/315*(35*B*c^4*x^9 - 45*(B*b*c^3 - A*c^4)*x^7 + 63*(B*b^2*c^2 - A*b*c^3)*x^5 - 105*(B*b^3*c - A*b^2*c^
2)*x^3 - 315*(B*b^4 - A*b^3*c)*sqrt(b/c)*arctan(c*x*sqrt(b/c)/b) + 315*(B*b^4 - A*b^3*c)*x)/c^5]

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giac [A]  time = 0.17, size = 133, normalized size = 1.12 \begin {gather*} -\frac {{\left (B b^{5} - A b^{4} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{5}} + \frac {35 \, B c^{8} x^{9} - 45 \, B b c^{7} x^{7} + 45 \, A c^{8} x^{7} + 63 \, B b^{2} c^{6} x^{5} - 63 \, A b c^{7} x^{5} - 105 \, B b^{3} c^{5} x^{3} + 105 \, A b^{2} c^{6} x^{3} + 315 \, B b^{4} c^{4} x - 315 \, A b^{3} c^{5} x}{315 \, c^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

-(B*b^5 - A*b^4*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^5) + 1/315*(35*B*c^8*x^9 - 45*B*b*c^7*x^7 + 45*A*c^8*x^7
+ 63*B*b^2*c^6*x^5 - 63*A*b*c^7*x^5 - 105*B*b^3*c^5*x^3 + 105*A*b^2*c^6*x^3 + 315*B*b^4*c^4*x - 315*A*b^3*c^5
*x)/c^9

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maple [A]  time = 0.05, size = 140, normalized size = 1.18 \begin {gather*} \frac {B \,x^{9}}{9 c}+\frac {A \,x^{7}}{7 c}-\frac {B b \,x^{7}}{7 c^{2}}-\frac {A b \,x^{5}}{5 c^{2}}+\frac {B \,b^{2} x^{5}}{5 c^{3}}+\frac {A \,b^{2} x^{3}}{3 c^{3}}-\frac {B \,b^{3} x^{3}}{3 c^{4}}+\frac {A \,b^{4} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{4}}-\frac {B \,b^{5} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{5}}-\frac {A \,b^{3} x}{c^{4}}+\frac {B \,b^{4} x}{c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(B*x^2+A)/(c*x^4+b*x^2),x)

[Out]

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

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maxima [A]  time = 3.00, size = 124, normalized size = 1.04 \begin {gather*} -\frac {{\left (B b^{5} - A b^{4} c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{5}} + \frac {35 \, B c^{4} x^{9} - 45 \, {\left (B b c^{3} - A c^{4}\right )} x^{7} + 63 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{5} - 105 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x^{3} + 315 \, {\left (B b^{4} - A b^{3} c\right )} x}{315 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

-(B*b^5 - A*b^4*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^5) + 1/315*(35*B*c^4*x^9 - 45*(B*b*c^3 - A*c^4)*x^7 + 63
*(B*b^2*c^2 - A*b*c^3)*x^5 - 105*(B*b^3*c - A*b^2*c^2)*x^3 + 315*(B*b^4 - A*b^3*c)*x)/c^5

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mupad [B]  time = 0.17, size = 144, normalized size = 1.21 \begin {gather*} x^7\,\left (\frac {A}{7\,c}-\frac {B\,b}{7\,c^2}\right )+\frac {B\,x^9}{9\,c}+\frac {b^2\,x^3\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{3\,c^2}-\frac {b^{7/2}\,\mathrm {atan}\left (\frac {b^{7/2}\,\sqrt {c}\,x\,\left (A\,c-B\,b\right )}{B\,b^5-A\,b^4\,c}\right )\,\left (A\,c-B\,b\right )}{c^{11/2}}-\frac {b\,x^5\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{5\,c}-\frac {b^3\,x\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^10*(A + B*x^2))/(b*x^2 + c*x^4),x)

[Out]

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

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sympy [A]  time = 0.44, size = 204, normalized size = 1.71 \begin {gather*} \frac {B x^{9}}{9 c} + x^{7} \left (\frac {A}{7 c} - \frac {B b}{7 c^{2}}\right ) + x^{5} \left (- \frac {A b}{5 c^{2}} + \frac {B b^{2}}{5 c^{3}}\right ) + x^{3} \left (\frac {A b^{2}}{3 c^{3}} - \frac {B b^{3}}{3 c^{4}}\right ) + x \left (- \frac {A b^{3}}{c^{4}} + \frac {B b^{4}}{c^{5}}\right ) + \frac {\sqrt {- \frac {b^{7}}{c^{11}}} \left (- A c + B b\right ) \log {\left (- \frac {c^{5} \sqrt {- \frac {b^{7}}{c^{11}}} \left (- A c + B b\right )}{- A b^{3} c + B b^{4}} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{7}}{c^{11}}} \left (- A c + B b\right ) \log {\left (\frac {c^{5} \sqrt {- \frac {b^{7}}{c^{11}}} \left (- A c + B b\right )}{- A b^{3} c + B b^{4}} + x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2),x)

[Out]

B*x**9/(9*c) + x**7*(A/(7*c) - B*b/(7*c**2)) + x**5*(-A*b/(5*c**2) + B*b**2/(5*c**3)) + x**3*(A*b**2/(3*c**3)
- B*b**3/(3*c**4)) + x*(-A*b**3/c**4 + B*b**4/c**5) + sqrt(-b**7/c**11)*(-A*c + B*b)*log(-c**5*sqrt(-b**7/c**1
1)*(-A*c + B*b)/(-A*b**3*c + B*b**4) + x)/2 - sqrt(-b**7/c**11)*(-A*c + B*b)*log(c**5*sqrt(-b**7/c**11)*(-A*c
+ B*b)/(-A*b**3*c + B*b**4) + x)/2

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