Integrand size = 20, antiderivative size = 126 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {\left (a-\frac {c d^2}{e^2}\right ) x}{7 d \left (d+e x^2\right )^{7/2}}+\frac {2 \left (\frac {3 a}{d^2}+\frac {4 c}{e^2}\right ) x}{35 \left (d+e x^2\right )^{5/2}}+\frac {\left (\frac {8 a}{d^2}-\frac {c}{e^2}\right ) x}{35 d \left (d+e x^2\right )^{3/2}}-\frac {2 \left (c d^2-8 a e^2\right ) x}{35 d^4 e^2 \sqrt {d+e x^2}} \] Output:
1/7*(a-c*d^2/e^2)*x/d/(e*x^2+d)^(7/2)+2/35*(3*a/d^2+4*c/e^2)*x/(e*x^2+d)^( 5/2)+1/35*(8*a/d^2-c/e^2)*x/d/(e*x^2+d)^(3/2)-2/35*(-8*a*e^2+c*d^2)*x/d^4/ e^2/(e*x^2+d)^(1/2)
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.59 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {35 a d^3 x+70 a d^2 e x^3-7 c d^3 x^5+56 a d e^2 x^5-2 c d^2 e x^7+16 a e^3 x^7}{35 d^4 \left (d+e x^2\right )^{7/2}} \] Input:
Integrate[(a - c*x^4)/(d + e*x^2)^(9/2),x]
Output:
(35*a*d^3*x + 70*a*d^2*e*x^3 - 7*c*d^3*x^5 + 56*a*d*e^2*x^5 - 2*c*d^2*e*x^ 7 + 16*a*e^3*x^7)/(35*d^4*(d + e*x^2)^(7/2))
Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1470, 362, 245, 242}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1470 |
| \(\displaystyle \frac {\int \frac {x^2 \left (6 a e-c d x^2\right )}{\left (e x^2+d\right )^{9/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {6 a e}{d}+\frac {c d}{e}\right )}{7 \left (d+e x^2\right )^{7/2}}-\frac {3}{7} \left (\frac {c d}{e}-\frac {8 a e}{d}\right ) \int \frac {x^2}{\left (e x^2+d\right )^{7/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {6 a e}{d}+\frac {c d}{e}\right )}{7 \left (d+e x^2\right )^{7/2}}-\frac {3}{7} \left (\frac {c d}{e}-\frac {8 a e}{d}\right ) \left (\frac {2 e \int \frac {x^4}{\left (e x^2+d\right )^{7/2}}dx}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{5/2}}\right )}{d}+\frac {a x}{d \left (d+e x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {6 a e}{d}+\frac {c d}{e}\right )}{7 \left (d+e x^2\right )^{7/2}}-\frac {3}{7} \left (\frac {2 e x^5}{15 d^2 \left (d+e x^2\right )^{5/2}}+\frac {x^3}{3 d \left (d+e x^2\right )^{5/2}}\right ) \left (\frac {c d}{e}-\frac {8 a e}{d}\right )}{d}+\frac {a x}{d \left (d+e x^2\right )^{7/2}}\) |
Input:
Int[(a - c*x^4)/(d + e*x^2)^(9/2),x]
Output:
(a*x)/(d*(d + e*x^2)^(7/2)) + ((((c*d)/e + (6*a*e)/d)*x^3)/(7*(d + e*x^2)^ (7/2)) - (3*((c*d)/e - (8*a*e)/d)*(x^3/(3*d*(d + e*x^2)^(5/2)) + (2*e*x^5) /(15*d^2*(d + e*x^2)^(5/2))))/7)/d
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si mp[a^p*x*((d + e*x^2)^(q + 1)/d), x] + Simp[1/d Int[x^2*(d + e*x^2)^q*(d* PolynomialQuotient[(a + c*x^4)^p - a^p, x^2, x] - e*a^p*(2*q + 3)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.51
| method | result | size |
| pseudoelliptic | \(\frac {x \left (\left (-\frac {c \,x^{4}}{5}+a \right ) d^{3}+2 x^{2} e \left (-\frac {c \,x^{4}}{35}+a \right ) d^{2}+\frac {8 a d \,e^{2} x^{4}}{5}+\frac {16 a \,e^{3} x^{6}}{35}\right )}{\left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(64\) |
| gosper | \(\frac {x \left (16 a \,e^{3} x^{6}-2 c \,d^{2} e \,x^{6}+56 a d \,e^{2} x^{4}-7 c \,d^{3} x^{4}+70 a \,d^{2} e \,x^{2}+35 a \,d^{3}\right )}{35 \left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(71\) |
| trager | \(\frac {x \left (16 a \,e^{3} x^{6}-2 c \,d^{2} e \,x^{6}+56 a d \,e^{2} x^{4}-7 c \,d^{3} x^{4}+70 a \,d^{2} e \,x^{2}+35 a \,d^{3}\right )}{35 \left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(71\) |
| orering | \(\frac {x \left (16 a \,e^{3} x^{6}-2 c \,d^{2} e \,x^{6}+56 a d \,e^{2} x^{4}-7 c \,d^{3} x^{4}+70 a \,d^{2} e \,x^{2}+35 a \,d^{3}\right )}{35 \left (e \,x^{2}+d \right )^{\frac {7}{2}} d^{4}}\) | \(71\) |
| default | \(a \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )-c \left (-\frac {x^{3}}{4 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {3 d \left (-\frac {x}{6 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {d \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )}{6 e}\right )}{4 e}\right )\) | \(199\) |
Input:
int((-c*x^4+a)/(e*x^2+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
x/(e*x^2+d)^(7/2)*((-1/5*c*x^4+a)*d^3+2*x^2*e*(-1/35*c*x^4+a)*d^2+8/5*a*d* e^2*x^4+16/35*a*e^3*x^6)/d^4
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=-\frac {{\left (2 \, {\left (c d^{2} e - 8 \, a e^{3}\right )} x^{7} - 70 \, a d^{2} e x^{3} + 7 \, {\left (c d^{3} - 8 \, a d e^{2}\right )} x^{5} - 35 \, a d^{3} x\right )} \sqrt {e x^{2} + d}}{35 \, {\left (d^{4} e^{4} x^{8} + 4 \, d^{5} e^{3} x^{6} + 6 \, d^{6} e^{2} x^{4} + 4 \, d^{7} e x^{2} + d^{8}\right )}} \] Input:
integrate((-c*x^4+a)/(e*x^2+d)^(9/2),x, algorithm="fricas")
Output:
-1/35*(2*(c*d^2*e - 8*a*e^3)*x^7 - 70*a*d^2*e*x^3 + 7*(c*d^3 - 8*a*d*e^2)* x^5 - 35*a*d^3*x)*sqrt(e*x^2 + d)/(d^4*e^4*x^8 + 4*d^5*e^3*x^6 + 6*d^6*e^2 *x^4 + 4*d^7*e*x^2 + d^8)
Leaf count of result is larger than twice the leaf count of optimal. 1469 vs. \(2 (119) = 238\).
Time = 22.09 (sec) , antiderivative size = 1469, normalized size of antiderivative = 11.66 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((-c*x**4+a)/(e*x**2+d)**(9/2),x)
Output:
a*(35*d**14*x/(35*d**(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt (1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 700*d**(31/2 )*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d ) + 210*d**(27/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12* sqrt(1 + e*x**2/d)) + 175*d**13*e*x**3/(35*d**(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e* *4*x**8*sqrt(1 + e*x**2/d) + 210*d**(27/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 371*d**12*e**2*x**5/(35*d** (37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525* d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 210*d**(27/2)*e** 5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6*x**12*sqrt(1 + e*x**2/d)) + 429*d**11*e**3*x**7/(35*d**(37/2)*sqrt(1 + e*x**2/d) + 210*d**(35/2)*e*x* *2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 700*d **(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525*d**(29/2)*e**4*x**8*sqrt(1 + e *x**2/d) + 210*d**(27/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 35*d**(25/2)*e**6 *x**12*sqrt(1 + e*x**2/d)) + 286*d**10*e**4*x**9/(35*d**(37/2)*sqrt(1 + e* x**2/d) + 210*d**(35/2)*e*x**2*sqrt(1 + e*x**2/d) + 525*d**(33/2)*e**2*x** 4*sqrt(1 + e*x**2/d) + 700*d**(31/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 525...
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.22 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {c x^{3}}{4 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e} + \frac {16 \, a x}{35 \, \sqrt {e x^{2} + d} d^{4}} + \frac {8 \, a x}{35 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3}} + \frac {6 \, a x}{35 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2}} + \frac {a x}{7 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d} - \frac {3 \, c x}{140 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, c x}{35 \, \sqrt {e x^{2} + d} d^{2} e^{2}} - \frac {c x}{35 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d e^{2}} + \frac {3 \, c d x}{28 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} \] Input:
integrate((-c*x^4+a)/(e*x^2+d)^(9/2),x, algorithm="maxima")
Output:
1/4*c*x^3/((e*x^2 + d)^(7/2)*e) + 16/35*a*x/(sqrt(e*x^2 + d)*d^4) + 8/35*a *x/((e*x^2 + d)^(3/2)*d^3) + 6/35*a*x/((e*x^2 + d)^(5/2)*d^2) + 1/7*a*x/(( e*x^2 + d)^(7/2)*d) - 3/140*c*x/((e*x^2 + d)^(5/2)*e^2) - 2/35*c*x/(sqrt(e *x^2 + d)*d^2*e^2) - 1/35*c*x/((e*x^2 + d)^(3/2)*d*e^2) + 3/28*c*d*x/((e*x ^2 + d)^(7/2)*e^2)
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.68 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {2 \, {\left (c d^{2} e^{4} - 8 \, a e^{6}\right )} x^{2}}{d^{4} e^{3}} + \frac {7 \, {\left (c d^{3} e^{3} - 8 \, a d e^{5}\right )}}{d^{4} e^{3}}\right )} - \frac {70 \, a e}{d^{2}}\right )} x^{2} - \frac {35 \, a}{d}\right )} x}{35 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \] Input:
integrate((-c*x^4+a)/(e*x^2+d)^(9/2),x, algorithm="giac")
Output:
-1/35*((x^2*(2*(c*d^2*e^4 - 8*a*e^6)*x^2/(d^4*e^3) + 7*(c*d^3*e^3 - 8*a*d* e^5)/(d^4*e^3)) - 70*a*e/d^2)*x^2 - 35*a/d)*x/(e*x^2 + d)^(7/2)
Time = 17.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {x\,\left (\frac {a}{7\,d}-\frac {c\,d}{7\,e^2}\right )}{{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (\frac {c}{5\,e^2}+\frac {c\,d^2+6\,a\,e^2}{35\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (8\,a\,e^2-c\,d^2\right )}{35\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (16\,a\,e^2-2\,c\,d^2\right )}{35\,d^4\,e^2\,\sqrt {e\,x^2+d}} \] Input:
int((a - c*x^4)/(d + e*x^2)^(9/2),x)
Output:
(x*(a/(7*d) - (c*d)/(7*e^2)))/(d + e*x^2)^(7/2) + (x*(c/(5*e^2) + (6*a*e^2 + c*d^2)/(35*d^2*e^2)))/(d + e*x^2)^(5/2) + (x*(8*a*e^2 - c*d^2))/(35*d^3 *e^2*(d + e*x^2)^(3/2)) + (x*(16*a*e^2 - 2*c*d^2))/(35*d^4*e^2*(d + e*x^2) ^(1/2))
Time = 0.24 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.29 \[ \int \frac {a-c x^4}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {35 \sqrt {e \,x^{2}+d}\, a \,d^{3} e^{3} x +70 \sqrt {e \,x^{2}+d}\, a \,d^{2} e^{4} x^{3}+56 \sqrt {e \,x^{2}+d}\, a d \,e^{5} x^{5}+16 \sqrt {e \,x^{2}+d}\, a \,e^{6} x^{7}-7 \sqrt {e \,x^{2}+d}\, c \,d^{3} e^{3} x^{5}-2 \sqrt {e \,x^{2}+d}\, c \,d^{2} e^{4} x^{7}-16 \sqrt {e}\, a \,d^{4} e^{2}-64 \sqrt {e}\, a \,d^{3} e^{3} x^{2}-96 \sqrt {e}\, a \,d^{2} e^{4} x^{4}-64 \sqrt {e}\, a d \,e^{5} x^{6}-16 \sqrt {e}\, a \,e^{6} x^{8}+2 \sqrt {e}\, c \,d^{6}+8 \sqrt {e}\, c \,d^{5} e \,x^{2}+12 \sqrt {e}\, c \,d^{4} e^{2} x^{4}+8 \sqrt {e}\, c \,d^{3} e^{3} x^{6}+2 \sqrt {e}\, c \,d^{2} e^{4} x^{8}}{35 d^{4} e^{3} \left (e^{4} x^{8}+4 d \,e^{3} x^{6}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{2}+d^{4}\right )} \] Input:
int((-c*x^4+a)/(e*x^2+d)^(9/2),x)
Output:
(35*sqrt(d + e*x**2)*a*d**3*e**3*x + 70*sqrt(d + e*x**2)*a*d**2*e**4*x**3 + 56*sqrt(d + e*x**2)*a*d*e**5*x**5 + 16*sqrt(d + e*x**2)*a*e**6*x**7 - 7* sqrt(d + e*x**2)*c*d**3*e**3*x**5 - 2*sqrt(d + e*x**2)*c*d**2*e**4*x**7 - 16*sqrt(e)*a*d**4*e**2 - 64*sqrt(e)*a*d**3*e**3*x**2 - 96*sqrt(e)*a*d**2*e **4*x**4 - 64*sqrt(e)*a*d*e**5*x**6 - 16*sqrt(e)*a*e**6*x**8 + 2*sqrt(e)*c *d**6 + 8*sqrt(e)*c*d**5*e*x**2 + 12*sqrt(e)*c*d**4*e**2*x**4 + 8*sqrt(e)* c*d**3*e**3*x**6 + 2*sqrt(e)*c*d**2*e**4*x**8)/(35*d**4*e**3*(d**4 + 4*d** 3*e*x**2 + 6*d**2*e**2*x**4 + 4*d*e**3*x**6 + e**4*x**8))