3.284 \(\int \frac{a+b x^2+c x^4}{(d+e x^2)^{11/2}} \, dx\)

Optimal. Leaf size=165 \[ \frac{8 e^2 x^9 \left (2 e (8 a e+b d)+c d^2\right )}{315 d^5 \left (d+e x^2\right )^{9/2}}+\frac{4 e x^7 \left (2 e (8 a e+b d)+c d^2\right )}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac{x^5 \left (2 e (8 a e+b d)+c d^2\right )}{5 d^3 \left (d+e x^2\right )^{9/2}}+\frac{x^3 (8 a e+b d)}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac{a x}{d \left (d+e x^2\right )^{9/2}} \]

[Out]

(a*x)/(d*(d + e*x^2)^(9/2)) + ((b*d + 8*a*e)*x^3)/(3*d^2*(d + e*x^2)^(9/2)) + ((c*d^2 + 2*e*(b*d + 8*a*e))*x^5
)/(5*d^3*(d + e*x^2)^(9/2)) + (4*e*(c*d^2 + 2*e*(b*d + 8*a*e))*x^7)/(35*d^4*(d + e*x^2)^(9/2)) + (8*e^2*(c*d^2
+ 2*e*(b*d + 8*a*e))*x^9)/(315*d^5*(d + e*x^2)^(9/2))

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Rubi [A]  time = 0.210036, antiderivative size = 164, normalized size of antiderivative = 0.99, number of steps used = 6, 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} \[ \frac{8 e^2 x^9 \left (2 e (8 a e+b d)+c d^2\right )}{315 d^5 \left (d+e x^2\right )^{9/2}}+\frac{4 e x^7 \left (2 e (8 a e+b d)+c d^2\right )}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac{x^5 \left (\frac{2 e (8 a e+b d)}{d^2}+c\right )}{5 d \left (d+e x^2\right )^{9/2}}+\frac{x^3 (8 a e+b d)}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac{a x}{d \left (d+e x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(11/2),x]

[Out]

(a*x)/(d*(d + e*x^2)^(9/2)) + ((b*d + 8*a*e)*x^3)/(3*d^2*(d + e*x^2)^(9/2)) + ((c + (2*e*(b*d + 8*a*e))/d^2)*x
^5)/(5*d*(d + e*x^2)^(9/2)) + (4*e*(c*d^2 + 2*e*(b*d + 8*a*e))*x^7)/(35*d^4*(d + e*x^2)^(9/2)) + (8*e^2*(c*d^2
+ 2*e*(b*d + 8*a*e))*x^9)/(315*d^5*(d + e*x^2)^(9/2))

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

Mathematica [A]  time = 0.118709, size = 132, normalized size = 0.8 \[ \frac{a \left (1008 d^2 e^2 x^5+840 d^3 e x^3+315 d^4 x+576 d e^3 x^7+128 e^4 x^9\right )+d x^3 \left (b \left (126 d^2 e x^2+105 d^3+72 d e^2 x^4+16 e^3 x^6\right )+c d x^2 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(11/2),x]

[Out]

(a*(315*d^4*x + 840*d^3*e*x^3 + 1008*d^2*e^2*x^5 + 576*d*e^3*x^7 + 128*e^4*x^9) + d*x^3*(c*d*x^2*(63*d^2 + 36*
d*e*x^2 + 8*e^2*x^4) + b*(105*d^3 + 126*d^2*e*x^2 + 72*d*e^2*x^4 + 16*e^3*x^6)))/(315*d^5*(d + e*x^2)^(9/2))

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Maple [A]  time = 0.005, size = 136, normalized size = 0.8 \begin{align*}{\frac{x \left ( 128\,a{e}^{4}{x}^{8}+16\,bd{e}^{3}{x}^{8}+8\,c{d}^{2}{e}^{2}{x}^{8}+576\,ad{e}^{3}{x}^{6}+72\,b{d}^{2}{e}^{2}{x}^{6}+36\,c{d}^{3}e{x}^{6}+1008\,a{d}^{2}{e}^{2}{x}^{4}+126\,b{d}^{3}e{x}^{4}+63\,c{d}^{4}{x}^{4}+840\,a{d}^{3}e{x}^{2}+105\,b{d}^{4}{x}^{2}+315\,a{d}^{4} \right ) }{315\,{d}^{5}} \left ( e{x}^{2}+d \right ) ^{-{\frac{9}{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)^(11/2),x)

[Out]

1/315*x*(128*a*e^4*x^8+16*b*d*e^3*x^8+8*c*d^2*e^2*x^8+576*a*d*e^3*x^6+72*b*d^2*e^2*x^6+36*c*d^3*e*x^6+1008*a*d
^2*e^2*x^4+126*b*d^3*e*x^4+63*c*d^4*x^4+840*a*d^3*e*x^2+105*b*d^4*x^2+315*a*d^4)/(e*x^2+d)^(9/2)/d^5

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Maxima [A]  time = 0.989696, size = 379, normalized size = 2.3 \begin{align*} -\frac{c x^{3}}{6 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e} + \frac{128 \, a x}{315 \, \sqrt{e x^{2} + d} d^{5}} + \frac{64 \, a x}{315 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{4}} + \frac{16 \, a x}{105 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{3}} + \frac{8 \, a x}{63 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} d^{2}} + \frac{a x}{9 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} d} + \frac{c x}{126 \,{\left (e x^{2} + d\right )}^{\frac{7}{2}} e^{2}} + \frac{8 \, c x}{315 \, \sqrt{e x^{2} + d} d^{3} e^{2}} + \frac{4 \, c x}{315 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2} e^{2}} + \frac{c x}{105 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d e^{2}} - \frac{c d x}{18 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e^{2}} - \frac{b x}{9 \,{\left (e x^{2} + d\right )}^{\frac{9}{2}} e} + \frac{16 \, b x}{315 \, \sqrt{e x^{2} + d} d^{4} e} + \frac{8 \, b x}{315 \,{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{3} e} + \frac{2 \, b x}{105 \,{\left (e x^{2} + d\right )}^{\frac{5}{2}} d^{2} e} + \frac{b x}{63 \,{\left (e x^{2} + d\right )}^{\frac{7}{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)^(11/2),x, algorithm="maxima")

[Out]

-1/6*c*x^3/((e*x^2 + d)^(9/2)*e) + 128/315*a*x/(sqrt(e*x^2 + d)*d^5) + 64/315*a*x/((e*x^2 + d)^(3/2)*d^4) + 16
/105*a*x/((e*x^2 + d)^(5/2)*d^3) + 8/63*a*x/((e*x^2 + d)^(7/2)*d^2) + 1/9*a*x/((e*x^2 + d)^(9/2)*d) + 1/126*c*
x/((e*x^2 + d)^(7/2)*e^2) + 8/315*c*x/(sqrt(e*x^2 + d)*d^3*e^2) + 4/315*c*x/((e*x^2 + d)^(3/2)*d^2*e^2) + 1/10
5*c*x/((e*x^2 + d)^(5/2)*d*e^2) - 1/18*c*d*x/((e*x^2 + d)^(9/2)*e^2) - 1/9*b*x/((e*x^2 + d)^(9/2)*e) + 16/315*
b*x/(sqrt(e*x^2 + d)*d^4*e) + 8/315*b*x/((e*x^2 + d)^(3/2)*d^3*e) + 2/105*b*x/((e*x^2 + d)^(5/2)*d^2*e) + 1/63
*b*x/((e*x^2 + d)^(7/2)*d*e)

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Fricas [A]  time = 6.21639, size = 386, normalized size = 2.34 \begin{align*} \frac{{\left (8 \,{\left (c d^{2} e^{2} + 2 \, b d e^{3} + 16 \, a e^{4}\right )} x^{9} + 36 \,{\left (c d^{3} e + 2 \, b d^{2} e^{2} + 16 \, a d e^{3}\right )} x^{7} + 315 \, a d^{4} x + 63 \,{\left (c d^{4} + 2 \, b d^{3} e + 16 \, a d^{2} e^{2}\right )} x^{5} + 105 \,{\left (b d^{4} + 8 \, a d^{3} e\right )} x^{3}\right )} \sqrt{e x^{2} + d}}{315 \,{\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\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)^(11/2),x, algorithm="fricas")

[Out]

1/315*(8*(c*d^2*e^2 + 2*b*d*e^3 + 16*a*e^4)*x^9 + 36*(c*d^3*e + 2*b*d^2*e^2 + 16*a*d*e^3)*x^7 + 315*a*d^4*x +
63*(c*d^4 + 2*b*d^3*e + 16*a*d^2*e^2)*x^5 + 105*(b*d^4 + 8*a*d^3*e)*x^3)*sqrt(e*x^2 + d)/(d^5*e^5*x^10 + 5*d^6
*e^4*x^8 + 10*d^7*e^3*x^6 + 10*d^8*e^2*x^4 + 5*d^9*e*x^2 + d^10)

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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)**(11/2),x)

[Out]

Timed out

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Giac [A]  time = 1.14314, size = 200, normalized size = 1.21 \begin{align*} \frac{{\left ({\left ({\left (4 \, x^{2}{\left (\frac{2 \,{\left (c d^{2} e^{6} + 2 \, b d e^{7} + 16 \, a e^{8}\right )} x^{2} e^{\left (-4\right )}}{d^{5}} + \frac{9 \,{\left (c d^{3} e^{5} + 2 \, b d^{2} e^{6} + 16 \, a d e^{7}\right )} e^{\left (-4\right )}}{d^{5}}\right )} + \frac{63 \,{\left (c d^{4} e^{4} + 2 \, b d^{3} e^{5} + 16 \, a d^{2} e^{6}\right )} e^{\left (-4\right )}}{d^{5}}\right )} x^{2} + \frac{105 \,{\left (b d^{4} e^{4} + 8 \, a d^{3} e^{5}\right )} e^{\left (-4\right )}}{d^{5}}\right )} x^{2} + \frac{315 \, a}{d}\right )} x}{315 \,{\left (x^{2} e + d\right )}^{\frac{9}{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)^(11/2),x, algorithm="giac")

[Out]

1/315*(((4*x^2*(2*(c*d^2*e^6 + 2*b*d*e^7 + 16*a*e^8)*x^2*e^(-4)/d^5 + 9*(c*d^3*e^5 + 2*b*d^2*e^6 + 16*a*d*e^7)
*e^(-4)/d^5) + 63*(c*d^4*e^4 + 2*b*d^3*e^5 + 16*a*d^2*e^6)*e^(-4)/d^5)*x^2 + 105*(b*d^4*e^4 + 8*a*d^3*e^5)*e^(
-4)/d^5)*x^2 + 315*a/d)*x/(x^2*e + d)^(9/2)