3.19.43 \(\int \frac {x^6}{a+\frac {b}{x^2}} \, dx\) [1843]

Optimal. Leaf size=68 \[ -\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 308, 211} \begin {gather*} \frac {b^{7/2} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}}-\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b/x^2),x]

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 269

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

Rule 308

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 68, normalized size = 1.00 \begin {gather*} -\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b/x^2),x]

[Out]

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

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Maple [A]
time = 0.08, size = 60, normalized size = 0.88

method	result	size
default	\(\frac {\frac {1}{7} a^{3} x^{7}-\frac {1}{5} a^{2} b \,x^{5}+\frac {1}{3} a \,b^{2} x^{3}-b^{3} x}{a^{4}}+\frac {b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}\)	\(60\)
risch	\(\frac {x^{7}}{7 a}-\frac {b \,x^{5}}{5 a^{2}}+\frac {b^{2} x^{3}}{3 a^{3}}-\frac {x \,b^{3}}{a^{4}}+\frac {\sqrt {-a b}\, b^{3} \ln \left (-\sqrt {-a b}\, x +b \right )}{2 a^{5}}-\frac {\sqrt {-a b}\, b^{3} \ln \left (\sqrt {-a b}\, x +b \right )}{2 a^{5}}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b/x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 60, normalized size = 0.88 \begin {gather*} \frac {b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {15 \, a^{3} x^{7} - 21 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} - 105 \, b^{3} x}{105 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 148, normalized size = 2.18 \begin {gather*} \left [\frac {30 \, a^{3} x^{7} - 42 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) - 210 \, b^{3} x}{210 \, a^{4}}, \frac {15 \, a^{3} x^{7} - 21 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) - 105 \, b^{3} x}{105 \, a^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.08, size = 107, normalized size = 1.57 \begin {gather*} - \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (- \frac {a^{4} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (\frac {a^{4} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac {x^{7}}{7 a} - \frac {b x^{5}}{5 a^{2}} + \frac {b^{2} x^{3}}{3 a^{3}} - \frac {b^{3} x}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.36, size = 65, normalized size = 0.96 \begin {gather*} \frac {b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {15 \, a^{6} x^{7} - 21 \, a^{5} b x^{5} + 35 \, a^{4} b^{2} x^{3} - 105 \, a^{3} b^{3} x}{105 \, a^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.10, size = 54, normalized size = 0.79 \begin {gather*} \frac {x^7}{7\,a}-\frac {b\,x^5}{5\,a^2}-\frac {b^3\,x}{a^4}+\frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{a^{9/2}}+\frac {b^2\,x^3}{3\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b/x^2),x)

[Out]

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

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