3.74 \(\int \frac {x^6 (A+B x^2)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=110 \[ \frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}-\frac {a^2 x (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac {a x (2 A b-3 a B)}{b^4}+\frac {x^3 (A b-2 a B)}{3 b^3}+\frac {B x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 110, 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} \[ -\frac {a^2 x (A b-a B)}{2 b^4 \left (a+b x^2\right )}+\frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}+\frac {x^3 (A b-2 a B)}{3 b^3}-\frac {a x (2 A b-3 a B)}{b^4}+\frac {B x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.08, size = 111, normalized size = 1.01 \[ -\frac {a^{3/2} (7 a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}-\frac {x \left (a^2 A b-a^3 B\right )}{2 b^4 \left (a+b x^2\right )}+\frac {a x (3 a B-2 A b)}{b^4}+\frac {x^3 (A b-2 a B)}{3 b^3}+\frac {B x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

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

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fricas [A]  time = 0.50, size = 298, normalized size = 2.71 \[ \left [\frac {12 \, B b^{3} x^{7} - 4 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 20 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 30 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x}{60 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {6 \, B b^{3} x^{7} - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x}{30 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/60*(12*B*b^3*x^7 - 4*(7*B*a*b^2 - 5*A*b^3)*x^5 + 20*(7*B*a^2*b - 5*A*a*b^2)*x^3 - 15*(7*B*a^3 - 5*A*a^2*b +
(7*B*a^2*b - 5*A*a*b^2)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 30*(7*B*a^3 - 5*A*a
^2*b)*x)/(b^5*x^2 + a*b^4), 1/30*(6*B*b^3*x^7 - 2*(7*B*a*b^2 - 5*A*b^3)*x^5 + 10*(7*B*a^2*b - 5*A*a*b^2)*x^3 -
15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 15*(7*B*a^3 - 5*A*
a^2*b)*x)/(b^5*x^2 + a*b^4)]

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giac [A]  time = 0.27, size = 115, normalized size = 1.05 \[ -\frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {B a^{3} x - A a^{2} b x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, B b^{8} x^{5} - 10 \, B a b^{7} x^{3} + 5 \, A b^{8} x^{3} + 45 \, B a^{2} b^{6} x - 30 \, A a b^{7} x}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(7*B*a^3 - 5*A*a^2*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/2*(B*a^3*x - A*a^2*b*x)/((b*x^2 + a)*b^4)
+ 1/15*(3*B*b^8*x^5 - 10*B*a*b^7*x^3 + 5*A*b^8*x^3 + 45*B*a^2*b^6*x - 30*A*a*b^7*x)/b^10

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maple [A]  time = 0.01, size = 132, normalized size = 1.20 \[ \frac {B \,x^{5}}{5 b^{2}}+\frac {A \,x^{3}}{3 b^{2}}-\frac {2 B a \,x^{3}}{3 b^{3}}-\frac {A \,a^{2} x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 A \,a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {B \,a^{3} x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 B \,a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {2 A a x}{b^{3}}+\frac {3 B \,a^{2} x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

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

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maxima [A]  time = 2.23, size = 112, normalized size = 1.02 \[ \frac {{\left (B a^{3} - A a^{2} b\right )} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} - \frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, B b^{2} x^{5} - 5 \, {\left (2 \, B a b - A b^{2}\right )} x^{3} + 15 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} x}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 141, normalized size = 1.28 \[ x^3\,\left (\frac {A}{3\,b^2}-\frac {2\,B\,a}{3\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )+\frac {B\,x^5}{5\,b^2}+\frac {x\,\left (\frac {B\,a^3}{2}-\frac {A\,a^2\,b}{2}\right )}{b^5\,x^2+a\,b^4}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (5\,A\,b-7\,B\,a\right )}{7\,B\,a^3-5\,A\,a^2\,b}\right )\,\left (5\,A\,b-7\,B\,a\right )}{2\,b^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(A + B*x^2))/(a + b*x^2)^2,x)

[Out]

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

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sympy [B]  time = 0.72, size = 211, normalized size = 1.92 \[ \frac {B x^{5}}{5 b^{2}} + x^{3} \left (\frac {A}{3 b^{2}} - \frac {2 B a}{3 b^{3}}\right ) + x \left (- \frac {2 A a}{b^{3}} + \frac {3 B a^{2}}{b^{4}}\right ) + \frac {x \left (- A a^{2} b + B a^{3}\right )}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right ) \log {\left (- \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right )}{- 5 A a b + 7 B a^{2}} + x \right )}}{4} - \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right ) \log {\left (\frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right )}{- 5 A a b + 7 B a^{2}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

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

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