3.2.12 \(\int \frac {x^2 (c+d x^2+e x^4+f x^6)}{a+b x^2} \, dx\)

Optimal. Leaf size=136 \[ \frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{9/2}}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac {x^5 (b e-a f)}{5 b^2}+\frac {f x^7}{7 b} \]

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Rubi [A]  time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1802, 205} \begin {gather*} \frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{9/2}}+\frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac {x^5 (b e-a f)}{5 b^2}+\frac {f x^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^4 + ((b^2*d - a*b*e + a^2*f)*x^3)/(3*b^3) + ((b*e - a*f)*x^5)/(5*b^2
) + (f*x^7)/(7*b) - (Sqrt[a]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 128, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^{9/2}}+\frac {x \left (-105 a^3 f+35 a^2 b \left (3 e+f x^2\right )-7 a b^2 \left (15 d+5 e x^2+3 f x^4\right )+b^3 \left (105 c+35 d x^2+21 e x^4+15 f x^6\right )\right )}{105 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]

[Out]

(x*(-105*a^3*f + 35*a^2*b*(3*e + f*x^2) - 7*a*b^2*(15*d + 5*e*x^2 + 3*f*x^4) + b^3*(105*c + 35*d*x^2 + 21*e*x^
4 + 15*f*x^6)))/(105*b^4) + (Sqrt[a]*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*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^2 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]

[Out]

IntegrateAlgebraic[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2), x]

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fricas [A]  time = 0.79, size = 286, normalized size = 2.10 \begin {gather*} \left [\frac {30 \, b^{3} f x^{7} + 42 \, {\left (b^{3} e - a b^{2} f\right )} x^{5} + 70 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} - 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\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^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{210 \, b^{4}}, \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} - 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{105 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.42, size = 152, normalized size = 1.12 \begin {gather*} -\frac {{\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} f x^{7} - 21 \, a b^{5} f x^{5} + 21 \, b^{6} x^{5} e + 35 \, b^{6} d x^{3} + 35 \, a^{2} b^{4} f x^{3} - 35 \, a b^{5} x^{3} e + 105 \, b^{6} c x - 105 \, a b^{5} d x - 105 \, a^{3} b^{3} f x + 105 \, a^{2} b^{4} x e}{105 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 182, normalized size = 1.34 \begin {gather*} \frac {f \,x^{7}}{7 b}-\frac {a f \,x^{5}}{5 b^{2}}+\frac {e \,x^{5}}{5 b}+\frac {a^{2} f \,x^{3}}{3 b^{3}}-\frac {a e \,x^{3}}{3 b^{2}}+\frac {d \,x^{3}}{3 b}+\frac {a^{4} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {a^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {a^{3} f x}{b^{4}}+\frac {a^{2} e x}{b^{3}}-\frac {a d x}{b^{2}}+\frac {c x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x)

[Out]

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

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maxima [A]  time = 2.96, size = 133, normalized size = 0.98 \begin {gather*} -\frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{105 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.91, size = 193, normalized size = 1.42 \begin {gather*} x^5\,\left (\frac {e}{5\,b}-\frac {a\,f}{5\,b^2}\right )+x^3\,\left (\frac {d}{3\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{3\,b}\right )+x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )+\frac {f\,x^7}{7\,b}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^4-e\,a^3\,b+d\,a^2\,b^2-c\,a\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x)

[Out]

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

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sympy [A]  time = 1.13, size = 185, normalized size = 1.36 \begin {gather*} x^{5} \left (- \frac {a f}{5 b^{2}} + \frac {e}{5 b}\right ) + x^{3} \left (\frac {a^{2} f}{3 b^{3}} - \frac {a e}{3 b^{2}} + \frac {d}{3 b}\right ) + x \left (- \frac {a^{3} f}{b^{4}} + \frac {a^{2} e}{b^{3}} - \frac {a d}{b^{2}} + \frac {c}{b}\right ) - \frac {\sqrt {- \frac {a}{b^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- b^{4} \sqrt {- \frac {a}{b^{9}}} + x \right )}}{2} + \frac {\sqrt {- \frac {a}{b^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (b^{4} \sqrt {- \frac {a}{b^{9}}} + x \right )}}{2} + \frac {f x^{7}}{7 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a),x)

[Out]

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

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