3.143 \(\int \frac{x^5 (c+d x^2+e x^4+f x^6)}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=214 \[ \frac{\left (a+b x^2\right )^{5/2} \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac{a \left (a+b x^2\right )^{3/2} \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{a^2 \sqrt{a+b x^2} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}+\frac{\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac{\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac{f \left (a+b x^2\right )^{11/2}}{11 b^6} \]

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^6 - (a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)
*(a + b*x^2)^(3/2))/(3*b^6) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*(a + b*x^2)^(5/2))/(5*b^6) + ((b^2*d
- 4*a*b*e + 10*a^2*f)*(a + b*x^2)^(7/2))/(7*b^6) + ((b*e - 5*a*f)*(a + b*x^2)^(9/2))/(9*b^6) + (f*(a + b*x^2)
^(11/2))/(11*b^6)

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Rubi [A]  time = 0.251575, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1799, 1620} \[ \frac{\left (a+b x^2\right )^{5/2} \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac{a \left (a+b x^2\right )^{3/2} \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{a^2 \sqrt{a+b x^2} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}+\frac{\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac{\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac{f \left (a+b x^2\right )^{11/2}}{11 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^6 - (a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)
*(a + b*x^2)^(3/2))/(3*b^6) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*(a + b*x^2)^(5/2))/(5*b^6) + ((b^2*d
- 4*a*b*e + 10*a^2*f)*(a + b*x^2)^(7/2))/(7*b^6) + ((b*e - 5*a*f)*(a + b*x^2)^(9/2))/(9*b^6) + (f*(a + b*x^2)
^(11/2))/(11*b^6)

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{x^5 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 \left (c+d x+e x^2+f x^3\right )}{\sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 \sqrt{a+b x}}+\frac{a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) \sqrt{a+b x}}{b^5}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) (a+b x)^{3/2}}{b^5}+\frac{\left (b^2 d-4 a b e+10 a^2 f\right ) (a+b x)^{5/2}}{b^5}+\frac{(b e-5 a f) (a+b x)^{7/2}}{b^5}+\frac{f (a+b x)^{9/2}}{b^5}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt{a+b x^2}}{b^6}-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) \left (a+b x^2\right )^{3/2}}{3 b^6}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) \left (a+b x^2\right )^{5/2}}{5 b^6}+\frac{\left (b^2 d-4 a b e+10 a^2 f\right ) \left (a+b x^2\right )^{7/2}}{7 b^6}+\frac{(b e-5 a f) \left (a+b x^2\right )^{9/2}}{9 b^6}+\frac{f \left (a+b x^2\right )^{11/2}}{11 b^6}\\ \end{align*}

Mathematica [A]  time = 0.165804, size = 158, normalized size = 0.74 \[ \frac{\sqrt{a+b x^2} \left (8 a^2 b^3 \left (231 c+99 d x^2+66 e x^4+50 f x^6\right )-16 a^3 b^2 \left (99 d+44 e x^2+30 f x^4\right )+128 a^4 b \left (11 e+5 f x^2\right )-1280 a^5 f-2 a b^4 x^2 \left (462 c+297 d x^2+220 e x^4+175 f x^6\right )+b^5 x^4 \left (693 c+5 \left (99 d x^2+77 e x^4+63 f x^6\right )\right )\right )}{3465 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(-1280*a^5*f + 128*a^4*b*(11*e + 5*f*x^2) - 16*a^3*b^2*(99*d + 44*e*x^2 + 30*f*x^4) + 8*a^2*b
^3*(231*c + 99*d*x^2 + 66*e*x^4 + 50*f*x^6) - 2*a*b^4*x^2*(462*c + 297*d*x^2 + 220*e*x^4 + 175*f*x^6) + b^5*x^
4*(693*c + 5*(99*d*x^2 + 77*e*x^4 + 63*f*x^6))))/(3465*b^6)

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Maple [A]  time = 0.008, size = 193, normalized size = 0.9 \begin{align*} -{\frac{-315\,f{x}^{10}{b}^{5}+350\,a{b}^{4}f{x}^{8}-385\,{b}^{5}e{x}^{8}-400\,{a}^{2}{b}^{3}f{x}^{6}+440\,a{b}^{4}e{x}^{6}-495\,{b}^{5}d{x}^{6}+480\,{a}^{3}{b}^{2}f{x}^{4}-528\,{a}^{2}{b}^{3}e{x}^{4}+594\,a{b}^{4}d{x}^{4}-693\,{b}^{5}c{x}^{4}-640\,{a}^{4}bf{x}^{2}+704\,{a}^{3}{b}^{2}e{x}^{2}-792\,{a}^{2}{b}^{3}d{x}^{2}+924\,a{b}^{4}c{x}^{2}+1280\,{a}^{5}f-1408\,{a}^{4}be+1584\,{a}^{3}{b}^{2}d-1848\,{a}^{2}{b}^{3}c}{3465\,{b}^{6}}\sqrt{b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x)

[Out]

-1/3465*(b*x^2+a)^(1/2)*(-315*b^5*f*x^10+350*a*b^4*f*x^8-385*b^5*e*x^8-400*a^2*b^3*f*x^6+440*a*b^4*e*x^6-495*b
^5*d*x^6+480*a^3*b^2*f*x^4-528*a^2*b^3*e*x^4+594*a*b^4*d*x^4-693*b^5*c*x^4-640*a^4*b*f*x^2+704*a^3*b^2*e*x^2-7
92*a^2*b^3*d*x^2+924*a*b^4*c*x^2+1280*a^5*f-1408*a^4*b*e+1584*a^3*b^2*d-1848*a^2*b^3*c)/b^6

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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^5*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31254, size = 433, normalized size = 2.02 \begin{align*} \frac{{\left (315 \, b^{5} f x^{10} + 35 \,{\left (11 \, b^{5} e - 10 \, a b^{4} f\right )} x^{8} + 5 \,{\left (99 \, b^{5} d - 88 \, a b^{4} e + 80 \, a^{2} b^{3} f\right )} x^{6} + 1848 \, a^{2} b^{3} c - 1584 \, a^{3} b^{2} d + 1408 \, a^{4} b e - 1280 \, a^{5} f + 3 \,{\left (231 \, b^{5} c - 198 \, a b^{4} d + 176 \, a^{2} b^{3} e - 160 \, a^{3} b^{2} f\right )} x^{4} - 4 \,{\left (231 \, a b^{4} c - 198 \, a^{2} b^{3} d + 176 \, a^{3} b^{2} e - 160 \, a^{4} b f\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3465 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/3465*(315*b^5*f*x^10 + 35*(11*b^5*e - 10*a*b^4*f)*x^8 + 5*(99*b^5*d - 88*a*b^4*e + 80*a^2*b^3*f)*x^6 + 1848*
a^2*b^3*c - 1584*a^3*b^2*d + 1408*a^4*b*e - 1280*a^5*f + 3*(231*b^5*c - 198*a*b^4*d + 176*a^2*b^3*e - 160*a^3*
b^2*f)*x^4 - 4*(231*a*b^4*c - 198*a^2*b^3*d + 176*a^3*b^2*e - 160*a^4*b*f)*x^2)*sqrt(b*x^2 + a)/b^6

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Sympy [A]  time = 4.4592, size = 442, normalized size = 2.07 \begin{align*} \begin{cases} - \frac{256 a^{5} f \sqrt{a + b x^{2}}}{693 b^{6}} + \frac{128 a^{4} e \sqrt{a + b x^{2}}}{315 b^{5}} + \frac{128 a^{4} f x^{2} \sqrt{a + b x^{2}}}{693 b^{5}} - \frac{16 a^{3} d \sqrt{a + b x^{2}}}{35 b^{4}} - \frac{64 a^{3} e x^{2} \sqrt{a + b x^{2}}}{315 b^{4}} - \frac{32 a^{3} f x^{4} \sqrt{a + b x^{2}}}{231 b^{4}} + \frac{8 a^{2} c \sqrt{a + b x^{2}}}{15 b^{3}} + \frac{8 a^{2} d x^{2} \sqrt{a + b x^{2}}}{35 b^{3}} + \frac{16 a^{2} e x^{4} \sqrt{a + b x^{2}}}{105 b^{3}} + \frac{80 a^{2} f x^{6} \sqrt{a + b x^{2}}}{693 b^{3}} - \frac{4 a c x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} - \frac{6 a d x^{4} \sqrt{a + b x^{2}}}{35 b^{2}} - \frac{8 a e x^{6} \sqrt{a + b x^{2}}}{63 b^{2}} - \frac{10 a f x^{8} \sqrt{a + b x^{2}}}{99 b^{2}} + \frac{c x^{4} \sqrt{a + b x^{2}}}{5 b} + \frac{d x^{6} \sqrt{a + b x^{2}}}{7 b} + \frac{e x^{8} \sqrt{a + b x^{2}}}{9 b} + \frac{f x^{10} \sqrt{a + b x^{2}}}{11 b} & \text{for}\: b \neq 0 \\\frac{\frac{c x^{6}}{6} + \frac{d x^{8}}{8} + \frac{e x^{10}}{10} + \frac{f x^{12}}{12}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

Piecewise((-256*a**5*f*sqrt(a + b*x**2)/(693*b**6) + 128*a**4*e*sqrt(a + b*x**2)/(315*b**5) + 128*a**4*f*x**2*
sqrt(a + b*x**2)/(693*b**5) - 16*a**3*d*sqrt(a + b*x**2)/(35*b**4) - 64*a**3*e*x**2*sqrt(a + b*x**2)/(315*b**4
) - 32*a**3*f*x**4*sqrt(a + b*x**2)/(231*b**4) + 8*a**2*c*sqrt(a + b*x**2)/(15*b**3) + 8*a**2*d*x**2*sqrt(a +
b*x**2)/(35*b**3) + 16*a**2*e*x**4*sqrt(a + b*x**2)/(105*b**3) + 80*a**2*f*x**6*sqrt(a + b*x**2)/(693*b**3) -
4*a*c*x**2*sqrt(a + b*x**2)/(15*b**2) - 6*a*d*x**4*sqrt(a + b*x**2)/(35*b**2) - 8*a*e*x**6*sqrt(a + b*x**2)/(6
3*b**2) - 10*a*f*x**8*sqrt(a + b*x**2)/(99*b**2) + c*x**4*sqrt(a + b*x**2)/(5*b) + d*x**6*sqrt(a + b*x**2)/(7*
b) + e*x**8*sqrt(a + b*x**2)/(9*b) + f*x**10*sqrt(a + b*x**2)/(11*b), Ne(b, 0)), ((c*x**6/6 + d*x**8/8 + e*x**
10/10 + f*x**12/12)/sqrt(a), True))

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Giac [A]  time = 1.21487, size = 387, normalized size = 1.81 \begin{align*} \frac{693 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{3} c - 2310 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{3} c + 3465 \, \sqrt{b x^{2} + a} a^{2} b^{3} c + 495 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2} d - 2079 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b^{2} d + 3465 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b^{2} d - 3465 \, \sqrt{b x^{2} + a} a^{3} b^{2} d + 315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} f - 1925 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a f + 4950 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} f - 6930 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} f + 5775 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4} f - 3465 \, \sqrt{b x^{2} + a} a^{5} f + 385 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} b e - 1980 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a b e + 4158 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} b e - 4620 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} b e + 3465 \, \sqrt{b x^{2} + a} a^{4} b e}{3465 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/3465*(693*(b*x^2 + a)^(5/2)*b^3*c - 2310*(b*x^2 + a)^(3/2)*a*b^3*c + 3465*sqrt(b*x^2 + a)*a^2*b^3*c + 495*(b
*x^2 + a)^(7/2)*b^2*d - 2079*(b*x^2 + a)^(5/2)*a*b^2*d + 3465*(b*x^2 + a)^(3/2)*a^2*b^2*d - 3465*sqrt(b*x^2 +
a)*a^3*b^2*d + 315*(b*x^2 + a)^(11/2)*f - 1925*(b*x^2 + a)^(9/2)*a*f + 4950*(b*x^2 + a)^(7/2)*a^2*f - 6930*(b*
x^2 + a)^(5/2)*a^3*f + 5775*(b*x^2 + a)^(3/2)*a^4*f - 3465*sqrt(b*x^2 + a)*a^5*f + 385*(b*x^2 + a)^(9/2)*b*e -
1980*(b*x^2 + a)^(7/2)*a*b*e + 4158*(b*x^2 + a)^(5/2)*a^2*b*e - 4620*(b*x^2 + a)^(3/2)*a^3*b*e + 3465*sqrt(b*
x^2 + a)*a^4*b*e)/b^6