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/(e*x^2+d)^(9/2)+1/3*(8*a*e+b*d)*x^3/d^2/(e*x^2+d)^(9/2)+1/5*(c*d^2+2*e*(8*a*e+b*d))*x^5/d^3/(e*x^2+d)^(9
/2)+4/35*e*(c*d^2+2*e*(8*a*e+b*d))*x^7/d^4/(e*x^2+d)^(9/2)+8/315*e^2*(c*d^2+2*e*(8*a*e+b*d))*x^9/d^5/(e*x^2+d)
^(9/2)

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Rubi [A]  time = 0.21, 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 12

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

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]

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 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]

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*}

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Mathematica [A]  time = 0.12, size = 132, normalized size = 0.80 \[ \frac {a \left (315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right )+d x^3 \left (b \left (105 d^3+126 d^2 e x^2+72 d e^2 x^4+16 e^3 x^6\right )+c d x^2 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \]

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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fricas [A]  time = 0.73, size = 177, normalized size = 1.07 \[ \frac {{\left (8 \, {\left (c d^{2} e^{2} + 2 \, b d e^{3} + 16 \, a e^{4}\right )} x^{9} + 36 \, {\left (c d^{3} e + 2 \, b d^{2} e^{2} + 16 \, a d e^{3}\right )} x^{7} + 315 \, a d^{4} x + 63 \, {\left (c d^{4} + 2 \, b d^{3} e + 16 \, a d^{2} e^{2}\right )} x^{5} + 105 \, {\left (b d^{4} + 8 \, a d^{3} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{315 \, {\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\right )}} \]

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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giac [A]  time = 0.23, size = 148, normalized size = 0.90 \[ \frac {{\left ({\left ({\left (4 \, x^{2} {\left (\frac {2 \, {\left (c d^{2} e^{6} + 2 \, b d e^{7} + 16 \, a e^{8}\right )} x^{2} e^{\left (-4\right )}}{d^{5}} + \frac {9 \, {\left (c d^{3} e^{5} + 2 \, b d^{2} e^{6} + 16 \, a d e^{7}\right )} e^{\left (-4\right )}}{d^{5}}\right )} + \frac {63 \, {\left (c d^{4} e^{4} + 2 \, b d^{3} e^{5} + 16 \, a d^{2} e^{6}\right )} e^{\left (-4\right )}}{d^{5}}\right )} x^{2} + \frac {105 \, {\left (b d^{4} e^{4} + 8 \, a d^{3} e^{5}\right )} e^{\left (-4\right )}}{d^{5}}\right )} x^{2} + \frac {315 \, a}{d}\right )} x}{315 \, {\left (x^{2} e + d\right )}^{\frac {9}{2}}} \]

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)

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maple [A]  time = 0.01, size = 136, normalized size = 0.82 \[ \frac {\left (128 a \,e^{4} x^{8}+16 b d \,e^{3} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+72 b \,d^{2} e^{2} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+126 b \,d^{3} e \,x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+105 b \,d^{4} x^{2}+315 a \,d^{4}\right ) x}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}} \]

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 = 1.20, size = 281, normalized size = 1.70 \[ -\frac {c x^{3}}{6 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {128 \, a x}{315 \, \sqrt {e x^{2} + d} d^{5}} + \frac {64 \, a x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4}} + \frac {16 \, a x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, a x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2}} + \frac {a x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d} + \frac {c x}{126 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} + \frac {8 \, c x}{315 \, \sqrt {e x^{2} + d} d^{3} e^{2}} + \frac {4 \, c x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {c x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d e^{2}} - \frac {c d x}{18 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} - \frac {b x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {16 \, b x}{315 \, \sqrt {e x^{2} + d} d^{4} e} + \frac {8 \, b x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e} + \frac {2 \, b x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e} + \frac {b x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e} \]

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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mupad [B]  time = 4.75, size = 189, normalized size = 1.15 \[ \frac {x\,\left (\frac {a}{9\,d}-\frac {d\,\left (\frac {b}{9\,d}-\frac {c}{9\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{9/2}}-\frac {x\,\left (\frac {c}{7\,e^2}-\frac {-c\,d^2+b\,d\,e+8\,a\,e^2}{63\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (c\,d^2+2\,b\,d\,e+16\,a\,e^2\right )}{105\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (4\,c\,d^2+8\,b\,d\,e+64\,a\,e^2\right )}{315\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (8\,c\,d^2+16\,b\,d\,e+128\,a\,e^2\right )}{315\,d^5\,e^2\,\sqrt {e\,x^2+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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