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

Optimal. Leaf size=98 \[ -\frac {a^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}}+\frac {a^2 x (A b-a B)}{b^4}-\frac {a x^3 (A b-a B)}{3 b^3}+\frac {x^5 (A b-a B)}{5 b^2}+\frac {B x^7}{7 b} \]

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Rubi [A]  time = 0.06, antiderivative size = 98, 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 = {459, 302, 205} \begin {gather*} \frac {a^2 x (A b-a B)}{b^4}-\frac {a^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}}+\frac {x^5 (A b-a B)}{5 b^2}-\frac {a x^3 (A b-a B)}{3 b^3}+\frac {B x^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^2))/(a + b*x^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 98, normalized size = 1.00 \begin {gather*} \frac {a^{5/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}}-\frac {a^2 x (a B-A b)}{b^4}+\frac {a x^3 (a B-A b)}{3 b^3}+\frac {x^5 (A b-a B)}{5 b^2}+\frac {B x^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 228, normalized size = 2.33 \begin {gather*} \left [\frac {30 \, B b^{3} x^{7} - 42 \, {\left (B a b^{2} - A b^{3}\right )} x^{5} + 70 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} - 105 \, {\left (B a^{3} - A a^{2} b\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 (B a^{3} - A a^{2} b\right )} x}{210 \, b^{4}}, \frac {15 \, B b^{3} x^{7} - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} + 105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 105 \, {\left (B a^{3} - A a^{2} b\right )} x}{105 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(30*B*b^3*x^7 - 42*(B*a*b^2 - A*b^3)*x^5 + 70*(B*a^2*b - A*a*b^2)*x^3 - 105*(B*a^3 - A*a^2*b)*sqrt(-a/b
)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 210*(B*a^3 - A*a^2*b)*x)/b^4, 1/105*(15*B*b^3*x^7 - 21*(B*
a*b^2 - A*b^3)*x^5 + 35*(B*a^2*b - A*a*b^2)*x^3 + 105*(B*a^3 - A*a^2*b)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 10
5*(B*a^3 - A*a^2*b)*x)/b^4]

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giac [A]  time = 0.33, size = 108, normalized size = 1.10 \begin {gather*} \frac {{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, B b^{6} x^{7} - 21 \, B a b^{5} x^{5} + 21 \, A b^{6} x^{5} + 35 \, B a^{2} b^{4} x^{3} - 35 \, A a b^{5} x^{3} - 105 \, B a^{3} b^{3} x + 105 \, A a^{2} b^{4} x}{105 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 116, normalized size = 1.18 \begin {gather*} \frac {B \,x^{7}}{7 b}+\frac {A \,x^{5}}{5 b}-\frac {B a \,x^{5}}{5 b^{2}}-\frac {A a \,x^{3}}{3 b^{2}}+\frac {B \,a^{2} x^{3}}{3 b^{3}}-\frac {A \,a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {B \,a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}+\frac {A \,a^{2} x}{b^{3}}-\frac {B \,a^{3} x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.41, size = 100, normalized size = 1.02 \begin {gather*} \frac {{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, B b^{3} x^{7} - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} - 105 \, {\left (B a^{3} - A a^{2} b\right )} x}{105 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 118, normalized size = 1.20 \begin {gather*} x^5\,\left (\frac {A}{5\,b}-\frac {B\,a}{5\,b^2}\right )+\frac {B\,x^7}{7\,b}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (A\,b-B\,a\right )}{B\,a^4-A\,a^3\,b}\right )\,\left (A\,b-B\,a\right )}{b^{9/2}}-\frac {a\,x^3\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{3\,b}+\frac {a^2\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(A + B*x^2))/(a + b*x^2),x)

[Out]

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

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sympy [B]  time = 0.41, size = 180, normalized size = 1.84 \begin {gather*} \frac {B x^{7}}{7 b} + x^{5} \left (\frac {A}{5 b} - \frac {B a}{5 b^{2}}\right ) + x^{3} \left (- \frac {A a}{3 b^{2}} + \frac {B a^{2}}{3 b^{3}}\right ) + x \left (\frac {A a^{2}}{b^{3}} - \frac {B a^{3}}{b^{4}}\right ) - \frac {\sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right ) \log {\left (- \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right ) \log {\left (\frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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