\(\int \frac {(a-c x^4)^2}{(d+e x^2)^{9/2}} \, dx\) [338]

Optimal result

Integrand size = 22, antiderivative size = 210 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {\left (c d^2-a e^2\right )^2 x}{7 d e^4 \left (d+e x^2\right )^{7/2}}-\frac {2 \left (c d^2-a e^2\right ) \left (11 c d^2+3 a e^2\right ) x}{35 d^2 e^4 \left (d+e x^2\right )^{5/2}}+\frac {2 \left (61 c^2 d^4-3 a c d^2 e^2+12 a^2 e^4\right ) x}{105 d^3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {4 \left (44 c^2 d^4+3 a c d^2 e^2-12 a^2 e^4\right ) x}{105 d^4 e^4 \sqrt {d+e x^2}}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{9/2}} \] Output:

1/7*(-a*e^2+c*d^2)^2*x/d/e^4/(e*x^2+d)^(7/2)-2/35*(-a*e^2+c*d^2)*(3*a*e^2+ 
11*c*d^2)*x/d^2/e^4/(e*x^2+d)^(5/2)+2/105*(12*a^2*e^4-3*a*c*d^2*e^2+61*c^2 
*d^4)*x/d^3/e^4/(e*x^2+d)^(3/2)-4/105*(-12*a^2*e^4+3*a*c*d^2*e^2+44*c^2*d^ 
4)*x/d^4/e^4/(e*x^2+d)^(1/2)+c^2*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/e^(9/2 
)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=-\frac {x \left (6 a c d^2 e^4 x^4 \left (7 d+2 e x^2\right )-3 a^2 e^4 \left (35 d^3+70 d^2 e x^2+56 d e^2 x^4+16 e^3 x^6\right )+c^2 d^4 \left (105 d^3+350 d^2 e x^2+406 d e^2 x^4+176 e^3 x^6\right )\right )}{105 d^4 e^4 \left (d+e x^2\right )^{7/2}}-\frac {c^2 \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{e^{9/2}} \] Input:

Integrate[(a - c*x^4)^2/(d + e*x^2)^(9/2),x]
 

Output:

-1/105*(x*(6*a*c*d^2*e^4*x^4*(7*d + 2*e*x^2) - 3*a^2*e^4*(35*d^3 + 70*d^2* 
e*x^2 + 56*d*e^2*x^4 + 16*e^3*x^6) + c^2*d^4*(105*d^3 + 350*d^2*e*x^2 + 40 
6*d*e^2*x^4 + 176*e^3*x^6)))/(d^4*e^4*(d + e*x^2)^(7/2)) - (c^2*Log[-(Sqrt 
[e]*x) + Sqrt[d + e*x^2]])/e^(9/2)
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1472, 25, 2345, 25, 1471, 25, 25, 27, 298, 224, 219}

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.

\(\displaystyle \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 1472

\(\displaystyle \frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}-\frac {\int -\frac {\frac {7 c^2 d x^6}{e}-\frac {7 c^2 d^2 x^4}{e^2}+\frac {7 c d \left (c d^2-2 a e^2\right ) x^2}{e^3}+6 a^2+\frac {2 a c d^2}{e^2}-\frac {c^2 d^4}{e^4}}{\left (e x^2+d\right )^{7/2}}dx}{7 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {7 c^2 d x^6}{e}-\frac {7 c^2 d^2 x^4}{e^2}+\frac {7 c d \left (c d^2-2 a e^2\right ) x^2}{e^3}+6 a^2+\frac {2 a c d^2}{e^2}-\frac {c^2 d^4}{e^4}}{\left (e x^2+d\right )^{7/2}}dx}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}-\frac {\int -\frac {\frac {17 c^2 d^4}{e^4}-\frac {70 c^2 x^2 d^3}{e^3}+\frac {35 c^2 x^4 d^2}{e^2}-\frac {6 a c d^2}{e^2}+24 a^2}{\left (e x^2+d\right )^{5/2}}dx}{5 d}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\frac {17 c^2 d^4}{e^4}-\frac {70 c^2 x^2 d^3}{e^3}+\frac {35 c^2 x^4 d^2}{e^2}-\frac {6 a c d^2}{e^2}+24 a^2}{\left (e x^2+d\right )^{5/2}}dx}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {\frac {\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {105 c^2 x^2 d^3+\left (-\frac {71 c^2 d^4}{e^4}-\frac {12 a c d^2}{e^2}+48 a^2\right ) e^3}{e^3 \left (e x^2+d\right )^{3/2}}dx}{3 d}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int -\frac {\frac {71 c^2 d^4}{e}-105 c^2 x^2 d^3+12 a c e d^2-48 a^2 e^3}{e^3 \left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\int \frac {\frac {71 c^2 d^4}{e}-105 c^2 x^2 d^3+12 a c e d^2-48 a^2 e^3}{e^3 \left (e x^2+d\right )^{3/2}}dx}{3 d}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\int \frac {\frac {71 c^2 d^4}{e}-105 c^2 x^2 d^3+12 a c e d^2-48 a^2 e^3}{\left (e x^2+d\right )^{3/2}}dx}{3 d e^3}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\frac {4 x \left (-12 a^2 e^4+3 a c d^2 e^2+44 c^2 d^4\right )}{d e \sqrt {d+e x^2}}-\frac {105 c^2 d^3 \int \frac {1}{\sqrt {e x^2+d}}dx}{e}}{3 d e^3}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\frac {4 x \left (-12 a^2 e^4+3 a c d^2 e^2+44 c^2 d^4\right )}{d e \sqrt {d+e x^2}}-\frac {105 c^2 d^3 \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{e}}{3 d e^3}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {2 x \left (12 a^2-\frac {3 a c d^2}{e^2}+\frac {61 c^2 d^4}{e^4}\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {\frac {4 x \left (-12 a^2 e^4+3 a c d^2 e^2+44 c^2 d^4\right )}{d e \sqrt {d+e x^2}}-\frac {105 c^2 d^3 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}}{3 d e^3}}{5 d}+\frac {2 x \left (3 a^2+\frac {8 a c d^2}{e^2}-\frac {11 c^2 d^4}{e^4}\right )}{5 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {x \left (c d^2-a e^2\right )^2}{7 d e^4 \left (d+e x^2\right )^{7/2}}\)

Input:

Int[(a - c*x^4)^2/(d + e*x^2)^(9/2),x]
 

Output:

((c*d^2 - a*e^2)^2*x)/(7*d*e^4*(d + e*x^2)^(7/2)) + ((2*(3*a^2 - (11*c^2*d 
^4)/e^4 + (8*a*c*d^2)/e^2)*x)/(5*d*(d + e*x^2)^(5/2)) + ((2*(12*a^2 + (61* 
c^2*d^4)/e^4 - (3*a*c*d^2)/e^2)*x)/(3*d*(d + e*x^2)^(3/2)) - ((4*(44*c^2*d 
^4 + 3*a*c*d^2*e^2 - 12*a^2*e^4)*x)/(d*e*Sqrt[d + e*x^2]) - (105*c^2*d^3*A 
rcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(3/2))/(3*d*e^3))/(5*d))/(7*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 298

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

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], 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] && LtQ[q, -1]
 

rule 1472

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wi 
th[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x], R = Coeff[Polynom 
ialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, Simp[(-R)*x*((d + e*x^2) 
^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1))   Int[(d + e*x^2)^(q + 1 
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, c, d, 
e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {c^{2} d^{4} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) e^{4} \left (e \,x^{2}+d \right )^{\frac {7}{2}}+\frac {x \,e^{\frac {9}{2}} \left (48 a^{2} e^{7} x^{6}-12 a c \,d^{2} e^{5} x^{6}-176 c^{2} d^{4} e^{3} x^{6}+168 a^{2} d \,e^{6} x^{4}-42 a c \,d^{3} e^{4} x^{4}-406 c^{2} d^{5} e^{2} x^{4}+210 a^{2} d^{2} e^{5} x^{2}-350 c^{2} d^{6} e \,x^{2}+105 a^{2} d^{3} e^{4}-105 c^{2} d^{7}\right )}{105}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}} e^{\frac {17}{2}} d^{4}}\)	\(183\)
default	\(a^{2} \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )+c^{2} \left (-\frac {x^{7}}{7 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 e \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {-\frac {x}{e \sqrt {e \,x^{2}+d}}+\frac {\ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{e^{\frac {3}{2}}}}{e}}{e}}{e}\right )-2 a c \left (-\frac {x^{3}}{4 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {3 d \left (-\frac {x}{6 e \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {d \left (\frac {x}{7 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}}{d}\right )}{6 e}\right )}{4 e}\right )\)	\(308\)

Input:

int((-c*x^4+a)^2/(e*x^2+d)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

(c^2*d^4*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))*e^4*(e*x^2+d)^(7/2)+1/105*x*e^ 
(9/2)*(48*a^2*e^7*x^6-12*a*c*d^2*e^5*x^6-176*c^2*d^4*e^3*x^6+168*a^2*d*e^6 
*x^4-42*a*c*d^3*e^4*x^4-406*c^2*d^5*e^2*x^4+210*a^2*d^2*e^5*x^2-350*c^2*d^ 
6*e*x^2+105*a^2*d^3*e^4-105*c^2*d^7))/(e*x^2+d)^(7/2)/e^(17/2)/d^4
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.65 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (c^{2} d^{4} e^{4} x^{8} + 4 \, c^{2} d^{5} e^{3} x^{6} + 6 \, c^{2} d^{6} e^{2} x^{4} + 4 \, c^{2} d^{7} e x^{2} + c^{2} d^{8}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, {\left (4 \, {\left (44 \, c^{2} d^{4} e^{4} + 3 \, a c d^{2} e^{6} - 12 \, a^{2} e^{8}\right )} x^{7} + 14 \, {\left (29 \, c^{2} d^{5} e^{3} + 3 \, a c d^{3} e^{5} - 12 \, a^{2} d e^{7}\right )} x^{5} + 70 \, {\left (5 \, c^{2} d^{6} e^{2} - 3 \, a^{2} d^{2} e^{6}\right )} x^{3} + 105 \, {\left (c^{2} d^{7} e - a^{2} d^{3} e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{210 \, {\left (d^{4} e^{9} x^{8} + 4 \, d^{5} e^{8} x^{6} + 6 \, d^{6} e^{7} x^{4} + 4 \, d^{7} e^{6} x^{2} + d^{8} e^{5}\right )}}, -\frac {105 \, {\left (c^{2} d^{4} e^{4} x^{8} + 4 \, c^{2} d^{5} e^{3} x^{6} + 6 \, c^{2} d^{6} e^{2} x^{4} + 4 \, c^{2} d^{7} e x^{2} + c^{2} d^{8}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (4 \, {\left (44 \, c^{2} d^{4} e^{4} + 3 \, a c d^{2} e^{6} - 12 \, a^{2} e^{8}\right )} x^{7} + 14 \, {\left (29 \, c^{2} d^{5} e^{3} + 3 \, a c d^{3} e^{5} - 12 \, a^{2} d e^{7}\right )} x^{5} + 70 \, {\left (5 \, c^{2} d^{6} e^{2} - 3 \, a^{2} d^{2} e^{6}\right )} x^{3} + 105 \, {\left (c^{2} d^{7} e - a^{2} d^{3} e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{105 \, {\left (d^{4} e^{9} x^{8} + 4 \, d^{5} e^{8} x^{6} + 6 \, d^{6} e^{7} x^{4} + 4 \, d^{7} e^{6} x^{2} + d^{8} e^{5}\right )}}\right ] \] Input:

integrate((-c*x^4+a)^2/(e*x^2+d)^(9/2),x, algorithm="fricas")
 

Output:

[1/210*(105*(c^2*d^4*e^4*x^8 + 4*c^2*d^5*e^3*x^6 + 6*c^2*d^6*e^2*x^4 + 4*c 
^2*d^7*e*x^2 + c^2*d^8)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x 
 - d) - 2*(4*(44*c^2*d^4*e^4 + 3*a*c*d^2*e^6 - 12*a^2*e^8)*x^7 + 14*(29*c^ 
2*d^5*e^3 + 3*a*c*d^3*e^5 - 12*a^2*d*e^7)*x^5 + 70*(5*c^2*d^6*e^2 - 3*a^2* 
d^2*e^6)*x^3 + 105*(c^2*d^7*e - a^2*d^3*e^5)*x)*sqrt(e*x^2 + d))/(d^4*e^9* 
x^8 + 4*d^5*e^8*x^6 + 6*d^6*e^7*x^4 + 4*d^7*e^6*x^2 + d^8*e^5), -1/105*(10 
5*(c^2*d^4*e^4*x^8 + 4*c^2*d^5*e^3*x^6 + 6*c^2*d^6*e^2*x^4 + 4*c^2*d^7*e*x 
^2 + c^2*d^8)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (4*(44*c^2*d^4 
*e^4 + 3*a*c*d^2*e^6 - 12*a^2*e^8)*x^7 + 14*(29*c^2*d^5*e^3 + 3*a*c*d^3*e^ 
5 - 12*a^2*d*e^7)*x^5 + 70*(5*c^2*d^6*e^2 - 3*a^2*d^2*e^6)*x^3 + 105*(c^2* 
d^7*e - a^2*d^3*e^5)*x)*sqrt(e*x^2 + d))/(d^4*e^9*x^8 + 4*d^5*e^8*x^6 + 6* 
d^6*e^7*x^4 + 4*d^7*e^6*x^2 + d^8*e^5)]
 

Sympy [F]

\[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (- a + c x^{4}\right )^{2}}{\left (d + e x^{2}\right )^{\frac {9}{2}}}\, dx \] Input:

integrate((-c*x**4+a)**2/(e*x**2+d)**(9/2),x)
 

Output:

Integral((-a + c*x**4)**2/(d + e*x**2)**(9/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c*x^4+a)^2/(e*x^2+d)^(9/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=-\frac {{\left (2 \, {\left (x^{2} {\left (\frac {2 \, {\left (44 \, c^{2} d^{4} e^{6} + 3 \, a c d^{2} e^{8} - 12 \, a^{2} e^{10}\right )} x^{2}}{d^{4} e^{7}} + \frac {7 \, {\left (29 \, c^{2} d^{5} e^{5} + 3 \, a c d^{3} e^{7} - 12 \, a^{2} d e^{9}\right )}}{d^{4} e^{7}}\right )} + \frac {35 \, {\left (5 \, c^{2} d^{6} e^{4} - 3 \, a^{2} d^{2} e^{8}\right )}}{d^{4} e^{7}}\right )} x^{2} + \frac {105 \, {\left (c^{2} d^{7} e^{3} - a^{2} d^{3} e^{7}\right )}}{d^{4} e^{7}}\right )} x}{105 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}}} - \frac {c^{2} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{e^{\frac {9}{2}}} \] Input:

integrate((-c*x^4+a)^2/(e*x^2+d)^(9/2),x, algorithm="giac")
 

Output:

-1/105*(2*(x^2*(2*(44*c^2*d^4*e^6 + 3*a*c*d^2*e^8 - 12*a^2*e^10)*x^2/(d^4* 
e^7) + 7*(29*c^2*d^5*e^5 + 3*a*c*d^3*e^7 - 12*a^2*d*e^9)/(d^4*e^7)) + 35*( 
5*c^2*d^6*e^4 - 3*a^2*d^2*e^8)/(d^4*e^7))*x^2 + 105*(c^2*d^7*e^3 - a^2*d^3 
*e^7)/(d^4*e^7))*x/(e*x^2 + d)^(7/2) - c^2*log(abs(-sqrt(e)*x + sqrt(e*x^2 
 + d)))/e^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^2}{{\left (e\,x^2+d\right )}^{9/2}} \,d x \] Input:

int((a - c*x^4)^2/(d + e*x^2)^(9/2),x)
 

Output:

int((a - c*x^4)^2/(d + e*x^2)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 641, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{9/2}} \, dx=\frac {56 \sqrt {e}\, c^{2} d^{8}-42 \sqrt {e \,x^{2}+d}\, a c \,d^{3} e^{5} x^{5}-12 \sqrt {e \,x^{2}+d}\, a c \,d^{2} e^{6} x^{7}+420 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{7} e \,x^{2}+630 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{6} e^{2} x^{4}+420 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{5} e^{3} x^{6}+105 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{4} e^{4} x^{8}+48 \sqrt {e}\, a c \,d^{5} e^{3} x^{2}+72 \sqrt {e}\, a c \,d^{4} e^{4} x^{4}+48 \sqrt {e}\, a c \,d^{3} e^{5} x^{6}+12 \sqrt {e}\, a c \,d^{2} e^{6} x^{8}+48 \sqrt {e \,x^{2}+d}\, a^{2} e^{8} x^{7}+105 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{8}-48 \sqrt {e}\, a^{2} d^{4} e^{4}-48 \sqrt {e}\, a^{2} e^{8} x^{8}+105 \sqrt {e \,x^{2}+d}\, a^{2} d^{3} e^{5} x +210 \sqrt {e \,x^{2}+d}\, a^{2} d^{2} e^{6} x^{3}+168 \sqrt {e \,x^{2}+d}\, a^{2} d \,e^{7} x^{5}-105 \sqrt {e \,x^{2}+d}\, c^{2} d^{7} e x -350 \sqrt {e \,x^{2}+d}\, c^{2} d^{6} e^{2} x^{3}-406 \sqrt {e \,x^{2}+d}\, c^{2} d^{5} e^{3} x^{5}-176 \sqrt {e \,x^{2}+d}\, c^{2} d^{4} e^{4} x^{7}-192 \sqrt {e}\, a^{2} d^{3} e^{5} x^{2}-288 \sqrt {e}\, a^{2} d^{2} e^{6} x^{4}-192 \sqrt {e}\, a^{2} d \,e^{7} x^{6}+12 \sqrt {e}\, a c \,d^{6} e^{2}+224 \sqrt {e}\, c^{2} d^{7} e \,x^{2}+336 \sqrt {e}\, c^{2} d^{6} e^{2} x^{4}+224 \sqrt {e}\, c^{2} d^{5} e^{3} x^{6}+56 \sqrt {e}\, c^{2} d^{4} e^{4} x^{8}}{105 d^{4} e^{5} \left (e^{4} x^{8}+4 d \,e^{3} x^{6}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{2}+d^{4}\right )} \] Input:

int((-c*x^4+a)^2/(e*x^2+d)^(9/2),x)
 

Output:

(105*sqrt(d + e*x**2)*a**2*d**3*e**5*x + 210*sqrt(d + e*x**2)*a**2*d**2*e* 
*6*x**3 + 168*sqrt(d + e*x**2)*a**2*d*e**7*x**5 + 48*sqrt(d + e*x**2)*a**2 
*e**8*x**7 - 42*sqrt(d + e*x**2)*a*c*d**3*e**5*x**5 - 12*sqrt(d + e*x**2)* 
a*c*d**2*e**6*x**7 - 105*sqrt(d + e*x**2)*c**2*d**7*e*x - 350*sqrt(d + e*x 
**2)*c**2*d**6*e**2*x**3 - 406*sqrt(d + e*x**2)*c**2*d**5*e**3*x**5 - 176* 
sqrt(d + e*x**2)*c**2*d**4*e**4*x**7 + 105*sqrt(e)*log((sqrt(d + e*x**2) + 
 sqrt(e)*x)/sqrt(d))*c**2*d**8 + 420*sqrt(e)*log((sqrt(d + e*x**2) + sqrt( 
e)*x)/sqrt(d))*c**2*d**7*e*x**2 + 630*sqrt(e)*log((sqrt(d + e*x**2) + sqrt 
(e)*x)/sqrt(d))*c**2*d**6*e**2*x**4 + 420*sqrt(e)*log((sqrt(d + e*x**2) + 
sqrt(e)*x)/sqrt(d))*c**2*d**5*e**3*x**6 + 105*sqrt(e)*log((sqrt(d + e*x**2 
) + sqrt(e)*x)/sqrt(d))*c**2*d**4*e**4*x**8 - 48*sqrt(e)*a**2*d**4*e**4 - 
192*sqrt(e)*a**2*d**3*e**5*x**2 - 288*sqrt(e)*a**2*d**2*e**6*x**4 - 192*sq 
rt(e)*a**2*d*e**7*x**6 - 48*sqrt(e)*a**2*e**8*x**8 + 12*sqrt(e)*a*c*d**6*e 
**2 + 48*sqrt(e)*a*c*d**5*e**3*x**2 + 72*sqrt(e)*a*c*d**4*e**4*x**4 + 48*s 
qrt(e)*a*c*d**3*e**5*x**6 + 12*sqrt(e)*a*c*d**2*e**6*x**8 + 56*sqrt(e)*c** 
2*d**8 + 224*sqrt(e)*c**2*d**7*e*x**2 + 336*sqrt(e)*c**2*d**6*e**2*x**4 + 
224*sqrt(e)*c**2*d**5*e**3*x**6 + 56*sqrt(e)*c**2*d**4*e**4*x**8)/(105*d** 
4*e**5*(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))