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

Optimal. Leaf size=210 \[ \frac{x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}-\frac{a x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{13/2}}+\frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{11}}{11 b} \]

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^6 - (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^5) + ((b^3*
c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^9)/(9*b^2
) + (f*x^11)/(11*b) - (a^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(13/2)

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Rubi [A]  time = 0.161395, antiderivative size = 210, normalized size of antiderivative = 1., 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} \[ \frac{x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}-\frac{a x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{13/2}}+\frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{11}}{11 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^6 - (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^5) + ((b^3*
c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^9)/(9*b^2
) + (f*x^11)/(11*b) - (a^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(13/2)

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]

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]

Rubi steps

\begin{align*} \int \frac{x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx &=\int \left (\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^6}{b^3}+\frac{(b e-a f) x^8}{b^2}+\frac{f x^{10}}{b}+\frac{-a^3 b^3 c+a^4 b^2 d-a^5 b e+a^6 f}{b^6 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^9}{9 b^2}+\frac{f x^{11}}{11 b}-\frac{\left (a^3 \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^6}\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac{(b e-a f) x^9}{9 b^2}+\frac{f x^{11}}{11 b}-\frac{a^{5/2} \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^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.14573, size = 210, normalized size = 1. \[ \frac{x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}+\frac{a x^3 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 b^5}-\frac{a^2 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^6}+\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^{13/2}}+\frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{11}}{11 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^6) + (a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x^3)/(3*b^5)
+ ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^
9)/(9*b^2) + (f*x^11)/(11*b) + (a^(5/2)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^
(13/2)

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Maple [A]  time = 0.006, size = 278, normalized size = 1.3 \begin{align*}{\frac{f{x}^{11}}{11\,b}}-{\frac{{x}^{9}af}{9\,{b}^{2}}}+{\frac{{x}^{9}e}{9\,b}}+{\frac{{x}^{7}{a}^{2}f}{7\,{b}^{3}}}-{\frac{{x}^{7}ae}{7\,{b}^{2}}}+{\frac{{x}^{7}d}{7\,b}}-{\frac{{x}^{5}{a}^{3}f}{5\,{b}^{4}}}+{\frac{{x}^{5}{a}^{2}e}{5\,{b}^{3}}}-{\frac{{x}^{5}ad}{5\,{b}^{2}}}+{\frac{{x}^{5}c}{5\,b}}+{\frac{{x}^{3}{a}^{4}f}{3\,{b}^{5}}}-{\frac{{x}^{3}{a}^{3}e}{3\,{b}^{4}}}+{\frac{{x}^{3}{a}^{2}d}{3\,{b}^{3}}}-{\frac{a{x}^{3}c}{3\,{b}^{2}}}-{\frac{{a}^{5}fx}{{b}^{6}}}+{\frac{{a}^{4}ex}{{b}^{5}}}-{\frac{{a}^{3}dx}{{b}^{4}}}+{\frac{{a}^{2}cx}{{b}^{3}}}+{\frac{{a}^{6}f}{{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{a}^{5}e}{{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{a}^{4}d}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{a}^{3}c}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*f*x^11/b-1/9/b^2*x^9*a*f+1/9/b*x^9*e+1/7/b^3*x^7*a^2*f-1/7/b^2*x^7*a*e+1/7/b*x^7*d-1/5/b^4*x^5*a^3*f+1/5/
b^3*x^5*a^2*e-1/5/b^2*x^5*a*d+1/5/b*x^5*c+1/3/b^5*x^3*a^4*f-1/3/b^4*x^3*a^3*e+1/3/b^3*x^3*a^2*d-1/3/b^2*x^3*a*
c-1/b^6*a^5*f*x+1/b^5*a^4*e*x-1/b^4*a^3*d*x+1/b^3*a^2*c*x+a^6/b^6/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f-a^5/b^
5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e+a^4/b^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d-a^3/b^3/(a*b)^(1/2)*arct
an(b*x/(a*b)^(1/2))*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51908, size = 953, normalized size = 4.54 \begin{align*} \left [\frac{630 \, b^{5} f x^{11} + 770 \,{\left (b^{5} e - a b^{4} f\right )} x^{9} + 990 \,{\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 1386 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 2310 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} 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 ) + 6930 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{6930 \, b^{6}}, \frac{315 \, b^{5} f x^{11} + 385 \,{\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \,{\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 3465 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6930*(630*b^5*f*x^11 + 770*(b^5*e - a*b^4*f)*x^9 + 990*(b^5*d - a*b^4*e + a^2*b^3*f)*x^7 + 1386*(b^5*c - a*
b^4*d + a^2*b^3*e - a^3*b^2*f)*x^5 - 2310*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 - 3465*(a^2*b^3*c -
a^3*b^2*d + a^4*b*e - a^5*f)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 6930*(a^2*b^3*c - a^
3*b^2*d + a^4*b*e - a^5*f)*x)/b^6, 1/3465*(315*b^5*f*x^11 + 385*(b^5*e - a*b^4*f)*x^9 + 495*(b^5*d - a*b^4*e +
a^2*b^3*f)*x^7 + 693*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^5 - 1155*(a*b^4*c - a^2*b^3*d + a^3*b^2*e -
a^4*b*f)*x^3 - 3465*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 3465*(a^2*b^
3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x)/b^6]

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Sympy [A]  time = 0.900026, size = 366, normalized size = 1.74 \begin{align*} - \frac{\sqrt{- \frac{a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- \frac{b^{6} \sqrt{- \frac{a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac{\sqrt{- \frac{a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{b^{6} \sqrt{- \frac{a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac{f x^{11}}{11 b} - \frac{x^{9} \left (a f - b e\right )}{9 b^{2}} + \frac{x^{7} \left (a^{2} f - a b e + b^{2} d\right )}{7 b^{3}} - \frac{x^{5} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{5 b^{4}} + \frac{x^{3} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{3 b^{5}} - \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a**5/b**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-b**6*sqrt(-a**5/b**13)*(a**3*f - a**2*b*e + a*
b**2*d - b**3*c)/(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c) + x)/2 + sqrt(-a**5/b**13)*(a**3*f - a**2*b*e
+ a*b**2*d - b**3*c)*log(b**6*sqrt(-a**5/b**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**5*f - a**4*b*e +
a**3*b**2*d - a**2*b**3*c) + x)/2 + f*x**11/(11*b) - x**9*(a*f - b*e)/(9*b**2) + x**7*(a**2*f - a*b*e + b**2*d
)/(7*b**3) - x**5*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(5*b**4) + x**3*(a**4*f - a**3*b*e + a**2*b**2*d - a
*b**3*c)/(3*b**5) - x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/b**6

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Giac [A]  time = 1.19992, size = 338, normalized size = 1.61 \begin{align*} -\frac{{\left (a^{3} b^{3} c - a^{4} b^{2} d - a^{6} f + a^{5} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{6}} + \frac{315 \, b^{10} f x^{11} - 385 \, a b^{9} f x^{9} + 385 \, b^{10} x^{9} e + 495 \, b^{10} d x^{7} + 495 \, a^{2} b^{8} f x^{7} - 495 \, a b^{9} x^{7} e + 693 \, b^{10} c x^{5} - 693 \, a b^{9} d x^{5} - 693 \, a^{3} b^{7} f x^{5} + 693 \, a^{2} b^{8} x^{5} e - 1155 \, a b^{9} c x^{3} + 1155 \, a^{2} b^{8} d x^{3} + 1155 \, a^{4} b^{6} f x^{3} - 1155 \, a^{3} b^{7} x^{3} e + 3465 \, a^{2} b^{8} c x - 3465 \, a^{3} b^{7} d x - 3465 \, a^{5} b^{5} f x + 3465 \, a^{4} b^{6} x e}{3465 \, b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*b^3*c - a^4*b^2*d - a^6*f + a^5*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/3465*(315*b^10*f*x^11 - 3
85*a*b^9*f*x^9 + 385*b^10*x^9*e + 495*b^10*d*x^7 + 495*a^2*b^8*f*x^7 - 495*a*b^9*x^7*e + 693*b^10*c*x^5 - 693*
a*b^9*d*x^5 - 693*a^3*b^7*f*x^5 + 693*a^2*b^8*x^5*e - 1155*a*b^9*c*x^3 + 1155*a^2*b^8*d*x^3 + 1155*a^4*b^6*f*x
^3 - 1155*a^3*b^7*x^3*e + 3465*a^2*b^8*c*x - 3465*a^3*b^7*d*x - 3465*a^5*b^5*f*x + 3465*a^4*b^6*x*e)/b^11