\(\int \frac {a+b x^2+c x^4}{(d+e x^2)^{11/2}} \, dx\) [284]

Optimal result

Integrand size = 24, antiderivative size = 165 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {a x}{d \left (d+e x^2\right )^{9/2}}+\frac {(b d+8 a e) x^3}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac {\left (c d^2+2 e (b d+8 a e)\right ) x^5}{5 d^3 \left (d+e x^2\right )^{9/2}}+\frac {4 e \left (c d^2+2 e (b d+8 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac {8 e^2 \left (c d^2+2 e (b d+8 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{9/2}} \]

[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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1169, 1817, 12, 277, 270} \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {x^5 \left (\frac {2 e (8 a e+b d)}{d^2}+c\right )}{5 d \left (d+e x^2\right )^{9/2}}+\frac {8 e^2 x^9 \left (2 e (8 a e+b d)+c d^2\right )}{315 d^5 \left (d+e x^2\right )^{9/2}}+\frac {4 e x^7 \left (2 e (8 a e+b d)+c d^2\right )}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac {x^3 (8 a e+b d)}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac {a x}{d \left (d+e x^2\right )^{9/2}} \]

[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 270

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 277

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 1169

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 1817

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*} \text {integral}& = \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] (verified)

Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.80 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {a \left (315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right )+d x^3 \left (c d x^2 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )+b \left (105 d^3+126 d^2 e x^2+72 d e^2 x^4+16 e^3 x^6\right )\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \]

[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))

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {\left (\left (\frac {1}{5} c \,x^{4}+\frac {1}{3} b \,x^{2}+a \right ) d^{4}+\frac {8 e \left (\frac {3}{70} c \,x^{4}+\frac {3}{20} b \,x^{2}+a \right ) x^{2} d^{3}}{3}+\frac {16 e^{2} \left (\frac {1}{126} c \,x^{4}+\frac {1}{14} b \,x^{2}+a \right ) x^{4} d^{2}}{5}+\frac {64 \left (\frac {b \,x^{2}}{36}+a \right ) e^{3} x^{6} d}{35}+\frac {128 a \,e^{4} x^{8}}{315}\right ) x}{\left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(108\)
gosper	\(\frac {x \left (128 a \,e^{4} x^{8}+16 b d \,e^{3} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+72 b \,d^{2} e^{2} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+126 b \,d^{3} e \,x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+105 b \,d^{4} x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(136\)
trager	\(\frac {x \left (128 a \,e^{4} x^{8}+16 b d \,e^{3} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+72 b \,d^{2} e^{2} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+126 b \,d^{3} e \,x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+105 b \,d^{4} x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(136\)
default	\(a \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )+c \left (-\frac {x^{3}}{6 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )}{2 e}\right )+b \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )\)	\(358\)

[In]

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

[Out]

((1/5*c*x^4+1/3*b*x^2+a)*d^4+8/3*e*(3/70*c*x^4+3/20*b*x^2+a)*x^2*d^3+16/5*e^2*(1/126*c*x^4+1/14*b*x^2+a)*x^4*d
^2+64/35*(1/36*b*x^2+a)*e^3*x^6*d+128/315*a*e^4*x^8)/(e*x^2+d)^(9/2)*x/d^5

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left (8 \, {\left (c d^{2} e^{2} + 2 \, b d e^{3} + 16 \, a e^{4}\right )} x^{9} + 36 \, {\left (c d^{3} e + 2 \, b d^{2} e^{2} + 16 \, a d e^{3}\right )} x^{7} + 315 \, a d^{4} x + 63 \, {\left (c d^{4} + 2 \, b d^{3} e + 16 \, a d^{2} e^{2}\right )} x^{5} + 105 \, {\left (b d^{4} + 8 \, a d^{3} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{315 \, {\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5187 vs. \(2 (160) = 320\).

Time = 80.52 (sec) , antiderivative size = 5187, normalized size of antiderivative = 31.44 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

a*(315*d**30*x/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*
e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1
+ e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) +
37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2
)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d)) + 2730*d**29*e*x**3/(315*d**(7
1/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2
/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(
61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**
14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x
**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d)) + 10773*d**28*e**2*x**5/(315*d**(71/2)*sqrt(1 + e*x**2/
d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)
*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt
(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d)
+ 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2
)*e**10*x**20*sqrt(1 + e*x**2/d)) + 25524*d**27*e**3*x**7/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e
*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e
*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 6615
0*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e*
*8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1
+ e*x**2/d)) + 40229*d**26*e**4*x**9/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2
/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(
63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**1
2*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x
**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d)) + 44058*
d**25*e**5*x**11/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2
)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt
(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d)
+ 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53
/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d)) + 33915*d**24*e**6*x**13/(31
5*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 +
e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 793
80*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e
**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(
1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d)) + 18088*d**23*e**7*x**15/(315*d**(71/2)*sqrt(1 +
e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d
**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x*
*10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e
*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*
d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d)) + 6384*d**22*e**8*x**17/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**
(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sq
rt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d
) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(
55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20
*sqrt(1 + e*x**2/d)) + 1344*d**21*e**9*x**19/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1
+ e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66
150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e
**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt
(1 + e*x**2/d) + 3150*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d))
+ 128*d**20*e**10*x**21/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d
**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x*
*8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*
x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150
*d**(53/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d))) + b*(105*d**14*x**3/
(315*d**(39/2)*sqrt(1 + e*x**2/d) + 2205*d**(37/2)*e*x**2*sqrt(1 + e*x**2/d) + 6615*d**(35/2)*e**2*x**4*sqrt(1
+ e*x**2/d) + 11025*d**(33/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 11025*d**(31/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 6
615*d**(29/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 2205*d**(27/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 315*d**(25/2)*e**
7*x**14*sqrt(1 + e*x**2/d)) + 441*d**13*e*x**5/(315*d**(39/2)*sqrt(1 + e*x**2/d) + 2205*d**(37/2)*e*x**2*sqrt(
1 + e*x**2/d) + 6615*d**(35/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 11025*d**(33/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1
1025*d**(31/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 6615*d**(29/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 2205*d**(27/2)*e*
*6*x**12*sqrt(1 + e*x**2/d) + 315*d**(25/2)*e**7*x**14*sqrt(1 + e*x**2/d)) + 765*d**12*e**2*x**7/(315*d**(39/2
)*sqrt(1 + e*x**2/d) + 2205*d**(37/2)*e*x**2*sqrt(1 + e*x**2/d) + 6615*d**(35/2)*e**2*x**4*sqrt(1 + e*x**2/d)
+ 11025*d**(33/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 11025*d**(31/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 6615*d**(29/2)
*e**5*x**10*sqrt(1 + e*x**2/d) + 2205*d**(27/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 315*d**(25/2)*e**7*x**14*sqrt(
1 + e*x**2/d)) + 715*d**11*e**3*x**9/(315*d**(39/2)*sqrt(1 + e*x**2/d) + 2205*d**(37/2)*e*x**2*sqrt(1 + e*x**2
/d) + 6615*d**(35/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 11025*d**(33/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 11025*d**(3
1/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 6615*d**(29/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 2205*d**(27/2)*e**6*x**12*s
qrt(1 + e*x**2/d) + 315*d**(25/2)*e**7*x**14*sqrt(1 + e*x**2/d)) + 390*d**10*e**4*x**11/(315*d**(39/2)*sqrt(1
+ e*x**2/d) + 2205*d**(37/2)*e*x**2*sqrt(1 + e*x**2/d) + 6615*d**(35/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 11025*d
**(33/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 11025*d**(31/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 6615*d**(29/2)*e**5*x**
10*sqrt(1 + e*x**2/d) + 2205*d**(27/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 315*d**(25/2)*e**7*x**14*sqrt(1 + e*x**
2/d)) + 120*d**9*e**5*x**13/(315*d**(39/2)*sqrt(1 + e*x**2/d) + 2205*d**(37/2)*e*x**2*sqrt(1 + e*x**2/d) + 661
5*d**(35/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 11025*d**(33/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 11025*d**(31/2)*e**4
*x**8*sqrt(1 + e*x**2/d) + 6615*d**(29/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 2205*d**(27/2)*e**6*x**12*sqrt(1 + e
*x**2/d) + 315*d**(25/2)*e**7*x**14*sqrt(1 + e*x**2/d)) + 16*d**8*e**6*x**15/(315*d**(39/2)*sqrt(1 + e*x**2/d)
+ 2205*d**(37/2)*e*x**2*sqrt(1 + e*x**2/d) + 6615*d**(35/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 11025*d**(33/2)*e*
*3*x**6*sqrt(1 + e*x**2/d) + 11025*d**(31/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 6615*d**(29/2)*e**5*x**10*sqrt(1 +
e*x**2/d) + 2205*d**(27/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 315*d**(25/2)*e**7*x**14*sqrt(1 + e*x**2/d))) + c*
(63*d**5*x**5/(315*d**(21/2)*sqrt(1 + e*x**2/d) + 1575*d**(19/2)*e*x**2*sqrt(1 + e*x**2/d) + 3150*d**(17/2)*e*
*2*x**4*sqrt(1 + e*x**2/d) + 3150*d**(15/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1575*d**(13/2)*e**4*x**8*sqrt(1 + e
*x**2/d) + 315*d**(11/2)*e**5*x**10*sqrt(1 + e*x**2/d)) + 99*d**4*e*x**7/(315*d**(21/2)*sqrt(1 + e*x**2/d) + 1
575*d**(19/2)*e*x**2*sqrt(1 + e*x**2/d) + 3150*d**(17/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 3150*d**(15/2)*e**3*x*
*6*sqrt(1 + e*x**2/d) + 1575*d**(13/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 315*d**(11/2)*e**5*x**10*sqrt(1 + e*x**2
/d)) + 44*d**3*e**2*x**9/(315*d**(21/2)*sqrt(1 + e*x**2/d) + 1575*d**(19/2)*e*x**2*sqrt(1 + e*x**2/d) + 3150*d
**(17/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 3150*d**(15/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1575*d**(13/2)*e**4*x**8
*sqrt(1 + e*x**2/d) + 315*d**(11/2)*e**5*x**10*sqrt(1 + e*x**2/d)) + 8*d**2*e**3*x**11/(315*d**(21/2)*sqrt(1 +
e*x**2/d) + 1575*d**(19/2)*e*x**2*sqrt(1 + e*x**2/d) + 3150*d**(17/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 3150*d**
(15/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1575*d**(13/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 315*d**(11/2)*e**5*x**10*s
qrt(1 + e*x**2/d)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.70 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=-\frac {c x^{3}}{6 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {128 \, a x}{315 \, \sqrt {e x^{2} + d} d^{5}} + \frac {64 \, a x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4}} + \frac {16 \, a x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, a x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2}} + \frac {a x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d} + \frac {c x}{126 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} + \frac {8 \, c x}{315 \, \sqrt {e x^{2} + d} d^{3} e^{2}} + \frac {4 \, c x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {c x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d e^{2}} - \frac {c d x}{18 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} - \frac {b x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {16 \, b x}{315 \, \sqrt {e x^{2} + d} d^{4} e} + \frac {8 \, b x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e} + \frac {2 \, b x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e} + \frac {b x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left ({\left ({\left (4 \, x^{2} {\left (\frac {2 \, {\left (c d^{2} e^{6} + 2 \, b d e^{7} + 16 \, a e^{8}\right )} x^{2}}{d^{5} e^{4}} + \frac {9 \, {\left (c d^{3} e^{5} + 2 \, b d^{2} e^{6} + 16 \, a d e^{7}\right )}}{d^{5} e^{4}}\right )} + \frac {63 \, {\left (c d^{4} e^{4} + 2 \, b d^{3} e^{5} + 16 \, a d^{2} e^{6}\right )}}{d^{5} e^{4}}\right )} x^{2} + \frac {105 \, {\left (b d^{4} e^{4} + 8 \, a d^{3} e^{5}\right )}}{d^{5} e^{4}}\right )} x^{2} + \frac {315 \, a}{d}\right )} x}{315 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}}} \]

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

Mupad [B] (verification not implemented)

Time = 7.87 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {x\,\left (\frac {a}{9\,d}-\frac {d\,\left (\frac {b}{9\,d}-\frac {c}{9\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{9/2}}-\frac {x\,\left (\frac {c}{7\,e^2}-\frac {-c\,d^2+b\,d\,e+8\,a\,e^2}{63\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (c\,d^2+2\,b\,d\,e+16\,a\,e^2\right )}{105\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (4\,c\,d^2+8\,b\,d\,e+64\,a\,e^2\right )}{315\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (8\,c\,d^2+16\,b\,d\,e+128\,a\,e^2\right )}{315\,d^5\,e^2\,\sqrt {e\,x^2+d}} \]

[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))