\(\int \frac {(a-c x^4)^2}{(d+e x^2)^{5/2}} \, dx\) [336]

Optimal result

Integrand size = 22, antiderivative size = 172 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\left (c d^2-a e^2\right )^2 x}{3 d e^4 \left (d+e x^2\right )^{3/2}}-\frac {2 \left (c d^2-a e^2\right ) \left (5 c d^2+a e^2\right ) x}{3 d^2 e^4 \sqrt {d+e x^2}}-\frac {11 c^2 d x \sqrt {d+e x^2}}{8 e^4}+\frac {c^2 x^3 \sqrt {d+e x^2}}{4 e^3}+\frac {c \left (35 c d^2-16 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 e^{9/2}} \] Output:

1/3*(-a*e^2+c*d^2)^2*x/d/e^4/(e*x^2+d)^(3/2)-2/3*(-a*e^2+c*d^2)*(a*e^2+5*c 
*d^2)*x/d^2/e^4/(e*x^2+d)^(1/2)-11/8*c^2*d*x*(e*x^2+d)^(1/2)/e^4+1/4*c^2*x 
^3*(e*x^2+d)^(1/2)/e^3+1/8*c*(-16*a*e^2+35*c*d^2)*arctanh(e^(1/2)*x/(e*x^2 
+d)^(1/2))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \left (8 a^2 e^4 \left (3 d+2 e x^2\right )+16 a c d^2 e^2 \left (3 d+4 e x^2\right )-c^2 d^2 \left (105 d^3+140 d^2 e x^2+21 d e^2 x^4-6 e^3 x^6\right )\right )}{24 d^2 e^4 \left (d+e x^2\right )^{3/2}}-\frac {c \left (35 c d^2-16 a e^2\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{8 e^{9/2}} \] Input:

Integrate[(a - c*x^4)^2/(d + e*x^2)^(5/2),x]
 

Output:

(x*(8*a^2*e^4*(3*d + 2*e*x^2) + 16*a*c*d^2*e^2*(3*d + 4*e*x^2) - c^2*d^2*( 
105*d^3 + 140*d^2*e*x^2 + 21*d*e^2*x^4 - 6*e^3*x^6)))/(24*d^2*e^4*(d + e*x 
^2)^(3/2)) - (c*(35*c*d^2 - 16*a*e^2)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]]) 
/(8*e^(9/2))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1472, 25, 2345, 27, 1473, 27, 299, 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.

\(\displaystyle \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1472

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}-\frac {\int -\frac {3 \left (\frac {c^2 d^2 x^4}{e^2}-\frac {2 c^2 d^3 x^2}{e^3}+\frac {c d^2 \left (3 c d^2-2 a e^2\right )}{e^4}\right )}{\sqrt {e x^2+d}}dx}{d}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {\frac {c^2 d^2 x^4}{e^2}-\frac {2 c^2 d^3 x^2}{e^3}+\frac {c d^2 \left (3 c d^2-2 a e^2\right )}{e^4}}{\sqrt {e x^2+d}}dx}{d}+\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1473

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {c d^2 \left (4 \left (3 c d^2-2 a e^2\right )-11 c d e x^2\right )}{e^3 \sqrt {e x^2+d}}dx}{4 e}+\frac {c^2 d^2 x^3 \sqrt {d+e x^2}}{4 e^3}\right )}{d}+\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {c d^2 \int \frac {4 \left (3 c d^2-2 a e^2\right )-11 c d e x^2}{\sqrt {e x^2+d}}dx}{4 e^4}+\frac {c^2 d^2 x^3 \sqrt {d+e x^2}}{4 e^3}\right )}{d}+\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {3 \left (\frac {c d^2 \left (\frac {1}{2} \left (35 c d^2-16 a e^2\right ) \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {11}{2} c d x \sqrt {d+e x^2}\right )}{4 e^4}+\frac {c^2 d^2 x^3 \sqrt {d+e x^2}}{4 e^3}\right )}{d}+\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {3 \left (\frac {c d^2 \left (\frac {1}{2} \left (35 c d^2-16 a e^2\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {11}{2} c d x \sqrt {d+e x^2}\right )}{4 e^4}+\frac {c^2 d^2 x^3 \sqrt {d+e x^2}}{4 e^3}\right )}{d}+\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {2 x \left (a^2+\frac {4 a c d^2}{e^2}-\frac {5 c^2 d^4}{e^4}\right )}{d \sqrt {d+e x^2}}+\frac {3 \left (\frac {c d^2 \left (\frac {\left (35 c d^2-16 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}-\frac {11}{2} c d x \sqrt {d+e x^2}\right )}{4 e^4}+\frac {c^2 d^2 x^3 \sqrt {d+e x^2}}{4 e^3}\right )}{d}}{3 d}+\frac {x \left (c d^2-a e^2\right )^2}{3 d e^4 \left (d+e x^2\right )^{3/2}}\)

Input:

Int[(a - c*x^4)^2/(d + e*x^2)^(5/2),x]
 

Output:

((c*d^2 - a*e^2)^2*x)/(3*d*e^4*(d + e*x^2)^(3/2)) + ((2*(a^2 - (5*c^2*d^4) 
/e^4 + (4*a*c*d^2)/e^2)*x)/(d*Sqrt[d + e*x^2]) + (3*((c^2*d^2*x^3*Sqrt[d + 
 e*x^2])/(4*e^3) + (c*d^2*((-11*c*d*x*Sqrt[d + e*x^2])/2 + ((35*c*d^2 - 16 
*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e])))/(4*e^4)))/d)/( 
3*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 299

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

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 1473

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> Simp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))) 
, x] + Simp[1/(e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 
2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 
 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, 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 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 = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {-2 d^{2} \left (a \,e^{2}-\frac {35 c \,d^{2}}{16}\right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} c \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\frac {2 x \left (-\frac {105 c^{2} d^{5} \sqrt {e}}{16}+e^{\frac {3}{2}} \left (-\frac {35 c^{2} d^{4} x^{2}}{4}+3 c e \left (-\frac {7 c \,x^{4}}{16}+a \right ) d^{3}+4 x^{2} \left (\frac {3 c \,x^{4}}{32}+a \right ) c \,e^{2} d^{2}+\frac {3 e^{3} a^{2} d}{2}+e^{4} x^{2} a^{2}\right )\right )}{3}}{e^{\frac {9}{2}} \left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{2}}\)	\(147\)
default	\(a^{2} \left (\frac {x}{3 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {e \,x^{2}+d}}\right )+c^{2} \left (\frac {x^{7}}{4 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {7 d \left (\frac {x^{5}}{2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {5 d \left (-\frac {x^{3}}{3 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {-\frac {x}{e \sqrt {e \,x^{2}+d}}+\frac {\ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{e^{\frac {3}{2}}}}{e}\right )}{2 e}\right )}{4 e}\right )-2 a c \left (-\frac {x^{3}}{3 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {-\frac {x}{e \sqrt {e \,x^{2}+d}}+\frac {\ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{e^{\frac {3}{2}}}}{e}\right )\)	\(209\)
risch	\(-\frac {c^{2} x \left (-2 e \,x^{2}+11 d \right ) \sqrt {e \,x^{2}+d}}{8 e^{4}}-\frac {\frac {c \left (16 a \,e^{2}-35 c \,d^{2}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{\sqrt {e}}+\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{3 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{3 d \left (x -\frac {\sqrt {-d e}}{e}\right )}\right )}{e d}+\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{3 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{3 d \left (x +\frac {\sqrt {-d e}}{e}\right )}\right )}{e d}-\frac {2 \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}-7 c^{2} d^{4}\right ) \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{d^{2} e \left (x -\frac {\sqrt {-d e}}{e}\right )}-\frac {2 \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}-7 c^{2} d^{4}\right ) \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{d^{2} e \left (x +\frac {\sqrt {-d e}}{e}\right )}}{8 e^{4}}\)	\(559\)

Input:

int((-c*x^4+a)^2/(e*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/3/e^(9/2)*(-3*d^2*(a*e^2-35/16*c*d^2)*(e*x^2+d)^(3/2)*c*arctanh((e*x^2+d 
)^(1/2)/x/e^(1/2))+x*(-105/16*c^2*d^5*e^(1/2)+e^(3/2)*(-35/4*c^2*d^4*x^2+3 
*c*e*(-7/16*c*x^4+a)*d^3+4*x^2*(3/32*c*x^4+a)*c*e^2*d^2+3/2*e^3*a^2*d+e^4* 
x^2*a^2)))/(e*x^2+d)^(3/2)/d^2
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.78 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (35 \, c^{2} d^{6} - 16 \, a c d^{4} e^{2} + {\left (35 \, c^{2} d^{4} e^{2} - 16 \, a c d^{2} e^{4}\right )} x^{4} + 2 \, {\left (35 \, c^{2} d^{5} e - 16 \, a c d^{3} e^{3}\right )} x^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, {\left (6 \, c^{2} d^{2} e^{4} x^{7} - 21 \, c^{2} d^{3} e^{3} x^{5} - 4 \, {\left (35 \, c^{2} d^{4} e^{2} - 16 \, a c d^{2} e^{4} - 4 \, a^{2} e^{6}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{5} e - 16 \, a c d^{3} e^{3} - 8 \, a^{2} d e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{48 \, {\left (d^{2} e^{7} x^{4} + 2 \, d^{3} e^{6} x^{2} + d^{4} e^{5}\right )}}, -\frac {3 \, {\left (35 \, c^{2} d^{6} - 16 \, a c d^{4} e^{2} + {\left (35 \, c^{2} d^{4} e^{2} - 16 \, a c d^{2} e^{4}\right )} x^{4} + 2 \, {\left (35 \, c^{2} d^{5} e - 16 \, a c d^{3} e^{3}\right )} x^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (6 \, c^{2} d^{2} e^{4} x^{7} - 21 \, c^{2} d^{3} e^{3} x^{5} - 4 \, {\left (35 \, c^{2} d^{4} e^{2} - 16 \, a c d^{2} e^{4} - 4 \, a^{2} e^{6}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{5} e - 16 \, a c d^{3} e^{3} - 8 \, a^{2} d e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{24 \, {\left (d^{2} e^{7} x^{4} + 2 \, d^{3} e^{6} x^{2} + d^{4} e^{5}\right )}}\right ] \] Input:

integrate((-c*x^4+a)^2/(e*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

[-1/48*(3*(35*c^2*d^6 - 16*a*c*d^4*e^2 + (35*c^2*d^4*e^2 - 16*a*c*d^2*e^4) 
*x^4 + 2*(35*c^2*d^5*e - 16*a*c*d^3*e^3)*x^2)*sqrt(e)*log(-2*e*x^2 + 2*sqr 
t(e*x^2 + d)*sqrt(e)*x - d) - 2*(6*c^2*d^2*e^4*x^7 - 21*c^2*d^3*e^3*x^5 - 
4*(35*c^2*d^4*e^2 - 16*a*c*d^2*e^4 - 4*a^2*e^6)*x^3 - 3*(35*c^2*d^5*e - 16 
*a*c*d^3*e^3 - 8*a^2*d*e^5)*x)*sqrt(e*x^2 + d))/(d^2*e^7*x^4 + 2*d^3*e^6*x 
^2 + d^4*e^5), -1/24*(3*(35*c^2*d^6 - 16*a*c*d^4*e^2 + (35*c^2*d^4*e^2 - 1 
6*a*c*d^2*e^4)*x^4 + 2*(35*c^2*d^5*e - 16*a*c*d^3*e^3)*x^2)*sqrt(-e)*arcta 
n(sqrt(-e)*x/sqrt(e*x^2 + d)) - (6*c^2*d^2*e^4*x^7 - 21*c^2*d^3*e^3*x^5 - 
4*(35*c^2*d^4*e^2 - 16*a*c*d^2*e^4 - 4*a^2*e^6)*x^3 - 3*(35*c^2*d^5*e - 16 
*a*c*d^3*e^3 - 8*a^2*d*e^5)*x)*sqrt(e*x^2 + d))/(d^2*e^7*x^4 + 2*d^3*e^6*x 
^2 + d^4*e^5)]
 

Sympy [F]

\[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {\left (- a + c x^{4}\right )^{2}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((-c*x**4+a)**2/(e*x**2+d)**(5/2),x)
 

Output:

Integral((-a + c*x**4)**2/(d + e*x**2)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c*x^4+a)^2/(e*x^2+d)^(5/2),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 = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, c^{2} x^{2}}{e} - \frac {7 \, c^{2} d}{e^{2}}\right )} x^{2} - \frac {4 \, {\left (35 \, c^{2} d^{4} e^{4} - 16 \, a c d^{2} e^{6} - 4 \, a^{2} e^{8}\right )}}{d^{2} e^{7}}\right )} x^{2} - \frac {3 \, {\left (35 \, c^{2} d^{5} e^{3} - 16 \, a c d^{3} e^{5} - 8 \, a^{2} d e^{7}\right )}}{d^{2} e^{7}}\right )} x}{24 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, c^{2} d^{2} - 16 \, a c e^{2}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{8 \, e^{\frac {9}{2}}} \] Input:

integrate((-c*x^4+a)^2/(e*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

1/24*((3*(2*c^2*x^2/e - 7*c^2*d/e^2)*x^2 - 4*(35*c^2*d^4*e^4 - 16*a*c*d^2* 
e^6 - 4*a^2*e^8)/(d^2*e^7))*x^2 - 3*(35*c^2*d^5*e^3 - 16*a*c*d^3*e^5 - 8*a 
^2*d*e^7)/(d^2*e^7))*x/(e*x^2 + d)^(3/2) - 1/8*(35*c^2*d^2 - 16*a*c*e^2)*l 
og(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/e^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^2}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((a - c*x^4)^2/(d + e*x^2)^(5/2),x)
 

Output:

int((a - c*x^4)^2/(d + e*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a-c x^4\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {192 \sqrt {e \,x^{2}+d}\, a^{2} d \,e^{5} x +128 \sqrt {e \,x^{2}+d}\, a^{2} e^{6} x^{3}+384 \sqrt {e \,x^{2}+d}\, a c \,d^{3} e^{3} x +512 \sqrt {e \,x^{2}+d}\, a c \,d^{2} e^{4} x^{3}-840 \sqrt {e \,x^{2}+d}\, c^{2} d^{5} e x -1120 \sqrt {e \,x^{2}+d}\, c^{2} d^{4} e^{2} x^{3}-168 \sqrt {e \,x^{2}+d}\, c^{2} d^{3} e^{3} x^{5}+48 \sqrt {e \,x^{2}+d}\, c^{2} d^{2} e^{4} x^{7}-384 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a c \,d^{4} e^{2}-768 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a c \,d^{3} e^{3} x^{2}-384 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a c \,d^{2} e^{4} x^{4}+840 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{6}+1680 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{5} e \,x^{2}+840 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{4} e^{2} x^{4}-128 \sqrt {e}\, a^{2} d^{2} e^{4}-256 \sqrt {e}\, a^{2} d \,e^{5} x^{2}-128 \sqrt {e}\, a^{2} e^{6} x^{4}+175 \sqrt {e}\, c^{2} d^{6}+350 \sqrt {e}\, c^{2} d^{5} e \,x^{2}+175 \sqrt {e}\, c^{2} d^{4} e^{2} x^{4}}{192 d^{2} e^{5} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((-c*x^4+a)^2/(e*x^2+d)^(5/2),x)
 

Output:

(192*sqrt(d + e*x**2)*a**2*d*e**5*x + 128*sqrt(d + e*x**2)*a**2*e**6*x**3 
+ 384*sqrt(d + e*x**2)*a*c*d**3*e**3*x + 512*sqrt(d + e*x**2)*a*c*d**2*e** 
4*x**3 - 840*sqrt(d + e*x**2)*c**2*d**5*e*x - 1120*sqrt(d + e*x**2)*c**2*d 
**4*e**2*x**3 - 168*sqrt(d + e*x**2)*c**2*d**3*e**3*x**5 + 48*sqrt(d + e*x 
**2)*c**2*d**2*e**4*x**7 - 384*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/ 
sqrt(d))*a*c*d**4*e**2 - 768*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sq 
rt(d))*a*c*d**3*e**3*x**2 - 384*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x) 
/sqrt(d))*a*c*d**2*e**4*x**4 + 840*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e) 
*x)/sqrt(d))*c**2*d**6 + 1680*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/s 
qrt(d))*c**2*d**5*e*x**2 + 840*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/ 
sqrt(d))*c**2*d**4*e**2*x**4 - 128*sqrt(e)*a**2*d**2*e**4 - 256*sqrt(e)*a* 
*2*d*e**5*x**2 - 128*sqrt(e)*a**2*e**6*x**4 + 175*sqrt(e)*c**2*d**6 + 350* 
sqrt(e)*c**2*d**5*e*x**2 + 175*sqrt(e)*c**2*d**4*e**2*x**4)/(192*d**2*e**5 
*(d**2 + 2*d*e*x**2 + e**2*x**4))