3.285 \(\int \frac{a+b x^2+c x^4}{(d+e x^2)^{13/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac{8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac{2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{a x}{d \left (d+e x^2\right )^{11/2}} \]

[Out]

(a*x)/(d*(d + e*x^2)^(11/2)) + ((b*d + 10*a*e)*x^3)/(3*d^2*(d + e*x^2)^(11/2)) + ((3*c*d^2 + 8*e*(b*d + 10*a*e
))*x^5)/(15*d^3*(d + e*x^2)^(11/2)) + (2*e*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^7)/(35*d^4*(d + e*x^2)^(11/2)) + (
8*e^2*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^9)/(315*d^5*(d + e*x^2)^(11/2)) + (16*e^3*(3*c*d^2 + 8*e*(b*d + 10*a*e)
)*x^11)/(3465*d^6*(d + e*x^2)^(11/2))

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Rubi [A]  time = 0.221537, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1155, 1803, 12, 271, 264} \[ \frac{16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac{8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac{2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{a x}{d \left (d+e x^2\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]

[Out]

(a*x)/(d*(d + e*x^2)^(11/2)) + ((b*d + 10*a*e)*x^3)/(3*d^2*(d + e*x^2)^(11/2)) + ((3*c*d^2 + 8*e*(b*d + 10*a*e
))*x^5)/(15*d^3*(d + e*x^2)^(11/2)) + (2*e*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^7)/(35*d^4*(d + e*x^2)^(11/2)) + (
8*e^2*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^9)/(315*d^5*(d + e*x^2)^(11/2)) + (16*e^3*(3*c*d^2 + 8*e*(b*d + 10*a*e)
)*x^11)/(3465*d^6*(d + e*x^2)^(11/2))

Rule 1155

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(a^p*x*(d + e*x^2
)^(q + 1))/d, x] + Dist[1/d, Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x]
- e*a^p*(2*q + 3)), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0
] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]

Rule 1803

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[(A*x^(m + 1)*(a + b*x^2)^(p + 1))/(a*(m + 1)), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{\int \frac{x^2 \left (10 a e+d \left (b+c x^2\right )\right )}{\left (d+e x^2\right )^{13/2}} \, dx}{d}\\ &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{\int \frac{\left (3 c d^2+8 e (b d+10 a e)\right ) x^4}{\left (d+e x^2\right )^{13/2}} \, dx}{3 d^2}\\ &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{1}{3} \left (3 c+\frac{8 e (b d+10 a e)}{d^2}\right ) \int \frac{x^4}{\left (d+e x^2\right )^{13/2}} \, dx\\ &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{\left (2 e \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac{x^6}{\left (d+e x^2\right )^{13/2}} \, dx}{5 d^3}\\ &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{\left (8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac{x^8}{\left (d+e x^2\right )^{13/2}} \, dx}{35 d^4}\\ &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac{\left (16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac{x^{10}}{\left (d+e x^2\right )^{13/2}} \, dx}{315 d^5}\\ &=\frac{a x}{d \left (d+e x^2\right )^{11/2}}+\frac{(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac{\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac{2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac{8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac{16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right ) x^{11}}{3465 d^6 \left (d+e x^2\right )^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.145272, size = 167, normalized size = 0.8 \[ \frac{5 a \left (3168 d^2 e^3 x^7+3696 d^3 e^2 x^5+2310 d^4 e x^3+693 d^5 x+1408 d e^4 x^9+256 e^5 x^{11}\right )+d x^3 \left (b \left (1584 d^2 e^2 x^4+1848 d^3 e x^2+1155 d^4+704 d e^3 x^6+128 e^4 x^8\right )+3 c d x^2 \left (198 d^2 e x^2+231 d^3+88 d e^2 x^4+16 e^3 x^6\right )\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]

[Out]

(5*a*(693*d^5*x + 2310*d^4*e*x^3 + 3696*d^3*e^2*x^5 + 3168*d^2*e^3*x^7 + 1408*d*e^4*x^9 + 256*e^5*x^11) + d*x^
3*(3*c*d*x^2*(231*d^3 + 198*d^2*e*x^2 + 88*d*e^2*x^4 + 16*e^3*x^6) + b*(1155*d^4 + 1848*d^3*e*x^2 + 1584*d^2*e
^2*x^4 + 704*d*e^3*x^6 + 128*e^4*x^8)))/(3465*d^6*(d + e*x^2)^(11/2))

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Maple [A]  time = 0.006, size = 172, normalized size = 0.8 \begin{align*}{\frac{x \left ( 1280\,a{e}^{5}{x}^{10}+128\,bd{e}^{4}{x}^{10}+48\,c{d}^{2}{e}^{3}{x}^{10}+7040\,ad{e}^{4}{x}^{8}+704\,b{d}^{2}{e}^{3}{x}^{8}+264\,c{d}^{3}{e}^{2}{x}^{8}+15840\,a{d}^{2}{e}^{3}{x}^{6}+1584\,b{d}^{3}{e}^{2}{x}^{6}+594\,c{d}^{4}e{x}^{6}+18480\,a{d}^{3}{e}^{2}{x}^{4}+1848\,b{d}^{4}e{x}^{4}+693\,c{d}^{5}{x}^{4}+11550\,a{d}^{4}e{x}^{2}+1155\,b{d}^{5}{x}^{2}+3465\,a{d}^{5} \right ) }{3465\,{d}^{6}} \left ( e{x}^{2}+d \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x)

[Out]

1/3465*x*(1280*a*e^5*x^10+128*b*d*e^4*x^10+48*c*d^2*e^3*x^10+7040*a*d*e^4*x^8+704*b*d^2*e^3*x^8+264*c*d^3*e^2*
x^8+15840*a*d^2*e^3*x^6+1584*b*d^3*e^2*x^6+594*c*d^4*e*x^6+18480*a*d^3*e^2*x^4+1848*b*d^4*e*x^4+693*c*d^5*x^4+
11550*a*d^4*e*x^2+1155*b*d^5*x^2+3465*a*d^5)/(e*x^2+d)^(11/2)/d^6

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Maxima [A]  time = 0.989096, size = 452, normalized size = 2.15 \begin{align*} -\frac{c x^{3}}{8 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} e} + \frac{256 \, a x}{693 \, \sqrt{e x^{2} + d} d^{6}} + \frac{128 \, a x}{693 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{5}} + \frac{32 \, a x}{231 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{4}} + \frac{80 \, a x}{693 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d^{3}} + \frac{10 \, a x}{99 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} d^{2}} + \frac{a x}{11 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} d} + \frac{c x}{264 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e^{2}} + \frac{16 \, c x}{1155 \, \sqrt{e x^{2} + d} d^{4} e^{2}} + \frac{8 \, c x}{1155 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3} e^{2}} + \frac{2 \, c x}{385 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} e^{2}} + \frac{c x}{231 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d e^{2}} - \frac{3 \, c d x}{88 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} e^{2}} - \frac{b x}{11 \,{\left (e x^{2} + d\right )}^{\frac{11}{2}} e} + \frac{128 \, b x}{3465 \, \sqrt{e x^{2} + d} d^{5} e} + \frac{64 \, b x}{3465 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{4} e} + \frac{16 \, b x}{1155 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{3} e} + \frac{8 \, b x}{693 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d^{2} e} + \frac{b x}{99 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x, algorithm="maxima")

[Out]

-1/8*c*x^3/((e*x^2 + d)^(11/2)*e) + 256/693*a*x/(sqrt(e*x^2 + d)*d^6) + 128/693*a*x/((e*x^2 + d)^(3/2)*d^5) +
32/231*a*x/((e*x^2 + d)^(5/2)*d^4) + 80/693*a*x/((e*x^2 + d)^(7/2)*d^3) + 10/99*a*x/((e*x^2 + d)^(9/2)*d^2) +
1/11*a*x/((e*x^2 + d)^(11/2)*d) + 1/264*c*x/((e*x^2 + d)^(9/2)*e^2) + 16/1155*c*x/(sqrt(e*x^2 + d)*d^4*e^2) +
8/1155*c*x/((e*x^2 + d)^(3/2)*d^3*e^2) + 2/385*c*x/((e*x^2 + d)^(5/2)*d^2*e^2) + 1/231*c*x/((e*x^2 + d)^(7/2)*
d*e^2) - 3/88*c*d*x/((e*x^2 + d)^(11/2)*e^2) - 1/11*b*x/((e*x^2 + d)^(11/2)*e) + 128/3465*b*x/(sqrt(e*x^2 + d)
*d^5*e) + 64/3465*b*x/((e*x^2 + d)^(3/2)*d^4*e) + 16/1155*b*x/((e*x^2 + d)^(5/2)*d^3*e) + 8/693*b*x/((e*x^2 +
d)^(7/2)*d^2*e) + 1/99*b*x/((e*x^2 + d)^(9/2)*d*e)

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Fricas [A]  time = 7.45462, size = 502, normalized size = 2.39 \begin{align*} \frac{{\left (16 \,{\left (3 \, c d^{2} e^{3} + 8 \, b d e^{4} + 80 \, a e^{5}\right )} x^{11} + 88 \,{\left (3 \, c d^{3} e^{2} + 8 \, b d^{2} e^{3} + 80 \, a d e^{4}\right )} x^{9} + 198 \,{\left (3 \, c d^{4} e + 8 \, b d^{3} e^{2} + 80 \, a d^{2} e^{3}\right )} x^{7} + 3465 \, a d^{5} x + 231 \,{\left (3 \, c d^{5} + 8 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} x^{5} + 1155 \,{\left (b d^{5} + 10 \, a d^{4} e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{3465 \,{\left (d^{6} e^{6} x^{12} + 6 \, d^{7} e^{5} x^{10} + 15 \, d^{8} e^{4} x^{8} + 20 \, d^{9} e^{3} x^{6} + 15 \, d^{10} e^{2} x^{4} + 6 \, d^{11} e x^{2} + d^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x, algorithm="fricas")

[Out]

1/3465*(16*(3*c*d^2*e^3 + 8*b*d*e^4 + 80*a*e^5)*x^11 + 88*(3*c*d^3*e^2 + 8*b*d^2*e^3 + 80*a*d*e^4)*x^9 + 198*(
3*c*d^4*e + 8*b*d^3*e^2 + 80*a*d^2*e^3)*x^7 + 3465*a*d^5*x + 231*(3*c*d^5 + 8*b*d^4*e + 80*a*d^3*e^2)*x^5 + 11
55*(b*d^5 + 10*a*d^4*e)*x^3)*sqrt(e*x^2 + d)/(d^6*e^6*x^12 + 6*d^7*e^5*x^10 + 15*d^8*e^4*x^8 + 20*d^9*e^3*x^6
+ 15*d^10*e^2*x^4 + 6*d^11*e*x^2 + d^12)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(13/2),x)

[Out]

Timed out

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Giac [A]  time = 1.19784, size = 255, normalized size = 1.21 \begin{align*} \frac{{\left ({\left ({\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \,{\left (3 \, c d^{2} e^{8} + 8 \, b d e^{9} + 80 \, a e^{10}\right )} x^{2} e^{\left (-5\right )}}{d^{6}} + \frac{11 \,{\left (3 \, c d^{3} e^{7} + 8 \, b d^{2} e^{8} + 80 \, a d e^{9}\right )} e^{\left (-5\right )}}{d^{6}}\right )} + \frac{99 \,{\left (3 \, c d^{4} e^{6} + 8 \, b d^{3} e^{7} + 80 \, a d^{2} e^{8}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac{231 \,{\left (3 \, c d^{5} e^{5} + 8 \, b d^{4} e^{6} + 80 \, a d^{3} e^{7}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac{1155 \,{\left (b d^{5} e^{5} + 10 \, a d^{4} e^{6}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac{3465 \, a}{d}\right )} x}{3465 \,{\left (x^{2} e + d\right )}^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x, algorithm="giac")

[Out]

1/3465*(((2*(4*x^2*(2*(3*c*d^2*e^8 + 8*b*d*e^9 + 80*a*e^10)*x^2*e^(-5)/d^6 + 11*(3*c*d^3*e^7 + 8*b*d^2*e^8 + 8
0*a*d*e^9)*e^(-5)/d^6) + 99*(3*c*d^4*e^6 + 8*b*d^3*e^7 + 80*a*d^2*e^8)*e^(-5)/d^6)*x^2 + 231*(3*c*d^5*e^5 + 8*
b*d^4*e^6 + 80*a*d^3*e^7)*e^(-5)/d^6)*x^2 + 1155*(b*d^5*e^5 + 10*a*d^4*e^6)*e^(-5)/d^6)*x^2 + 3465*a/d)*x/(x^2
*e + d)^(11/2)