Integrand size = 22, antiderivative size = 226 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (c d \left (d^2+\frac {3 a e^2}{c}\right )+e \left (3 c d^2+a e^2\right ) x^2\right )}{2 a c \sqrt {a-c x^4}}-\frac {3 e \left (c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {\left (c^{3/2} d^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{7/4} \sqrt {a-c x^4}} \] Output:
1/2*x*(c*d*(d^2+3*a*e^2/c)+e*(a*e^2+3*c*d^2)*x^2)/a/c/(-c*x^4+a)^(1/2)-3/2 *e*(a*e^2+c*d^2)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/ c^(7/4)/(-c*x^4+a)^(1/2)+1/2*(c^(3/2)*d^3+3*a^(1/2)*c*d^2*e-3*a*c^(1/2)*d* e^2+3*a^(3/2)*e^3)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4 )/c^(7/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.63 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {c d^3 x+a e^2 x \left (3 d-2 e x^2\right )+d \left (c d^2-3 a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+2 e \left (c d^2+a e^2\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{2 a c \sqrt {a-c x^4}} \] Input:
Integrate[(d + e*x^2)^3/(a - c*x^4)^(3/2),x]
Output:
(c*d^3*x + a*e^2*x*(3*d - 2*e*x^2) + d*(c*d^2 - 3*a*e^2)*x*Sqrt[1 - (c*x^4 )/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + 2*e*(c*d^2 + a*e^2)*x^3 *Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, (c*x^4)/a])/(2*a*c*S qrt[a - c*x^4])
Time = 1.00 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1519, 25, 2397, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1519 |
| \(\displaystyle -\frac {\int -\frac {3 c d e^2 x^4+3 e \left (c d^2+a e^2\right ) x^2+c d^3}{\left (a-c x^4\right )^{3/2}}dx}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 c d e^2 x^4+3 e \left (c d^2+a e^2\right ) x^2+c d^3}{\left (a-c x^4\right )^{3/2}}dx}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (d \left (c d^2-3 a e^2\right )-3 e \left (c d^2+a e^2\right ) x^2\right )}{\sqrt {a-c x^4}}dx}{2 a c}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (c d^2-3 a e^2\right )-3 e \left (c d^2+a e^2\right ) x^2}{\sqrt {a-c x^4}}dx}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {\frac {\left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \int \frac {1}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-\frac {3 \sqrt {a} e \left (a e^2+c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \int \frac {1}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-\frac {3 e \left (a e^2+c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}-\frac {3 e \left (a e^2+c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {3 e \left (a e^2+c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {3 e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {3 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+c d^2\right ) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (3 a^{3/2} e^3+3 \sqrt {a} c d^2 e-3 a \sqrt {c} d e^2+c^{3/2} d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+c d^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}}{2 a}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (3 a e^2+c d^2\right )\right )}{2 a \sqrt {a-c x^4}}}{c}-\frac {e^3 x^3}{c \sqrt {a-c x^4}}\) |
Input:
Int[(d + e*x^2)^3/(a - c*x^4)^(3/2),x]
Output:
-((e^3*x^3)/(c*Sqrt[a - c*x^4])) + ((x*(d*(c*d^2 + 3*a*e^2) + 3*e*(c*d^2 + a*e^2)*x^2))/(2*a*Sqrt[a - c*x^4]) + ((-3*a^(3/4)*e*(c*d^2 + a*e^2)*Sqrt[ 1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4) ], -1])/(c^(3/4)*Sqrt[a - c*x^4]))/(2*a))/c
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Sim p[e^q*x^(2*q - 3)*((a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c *(4*p + 2*q + 1)) Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e* x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x ], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Time = 1.85 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.23
| method | result | size |
| elliptic | \(\frac {2 c \left (\frac {e \left (a \,e^{2}+3 c \,d^{2}\right ) x^{3}}{4 c^{2} a}+\frac {d \left (3 a \,e^{2}+c \,d^{2}\right ) x}{4 c^{2} a}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (-\frac {3 d \,e^{2}}{c}+\frac {d \left (3 a \,e^{2}+c \,d^{2}\right )}{2 a c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (-\frac {e^{3}}{c}-\frac {e \left (a \,e^{2}+3 c \,d^{2}\right )}{2 a c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(278\) |
| default | \(d^{3} \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e^{3} \left (\frac {x^{3}}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {3 \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+3 d \,e^{2} \left (\frac {x}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+3 d^{2} e \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\) | \(422\) |
Input:
int((e*x^2+d)^3/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/4*e*(a*e^2+3*c*d^2)/c^2/a*x^3+1/4*d*(3*a*e^2+c*d^2)/c^2/a*x)/(-(x^4 -1/c*a)*c)^(1/2)+(-3*d*e^2/c+1/2*d*(3*a*e^2+c*d^2)/a/c)/(1/a^(1/2)*c^(1/2) )^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(- c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-(-e^3/c-1/2*e*(a*e ^2+3*c*d^2)/a/c)*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^ 2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(Ellipti cF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I) )
Time = 0.10 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 \, {\left ({\left (a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{5} - {\left (a^{2} c d^{2} e + a^{3} e^{3}\right )} x\right )} \sqrt {-c} \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left ({\left (c^{3} d^{3} - 3 \, a c^{2} d^{2} e - 3 \, a c^{2} d e^{2} - 3 \, a^{2} c e^{3}\right )} x^{5} - {\left (a c^{2} d^{3} - 3 \, a^{2} c d^{2} e - 3 \, a^{2} c d e^{2} - 3 \, a^{3} e^{3}\right )} x\right )} \sqrt {-c} \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (2 \, a^{2} c e^{3} x^{4} - 3 \, a^{2} c d^{2} e - 3 \, a^{3} e^{3} - {\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{2 \, {\left (a^{2} c^{3} x^{5} - a^{3} c^{2} x\right )}} \] Input:
integrate((e*x^2+d)^3/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/2*(3*((a*c^2*d^2*e + a^2*c*e^3)*x^5 - (a^2*c*d^2*e + a^3*e^3)*x)*sqrt(-c )*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) + ((c^3*d^3 - 3*a*c^2* d^2*e - 3*a*c^2*d*e^2 - 3*a^2*c*e^3)*x^5 - (a*c^2*d^3 - 3*a^2*c*d^2*e - 3* a^2*c*d*e^2 - 3*a^3*e^3)*x)*sqrt(-c)*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^( 1/4)/x), -1) + (2*a^2*c*e^3*x^4 - 3*a^2*c*d^2*e - 3*a^3*e^3 - (a*c^2*d^3 + 3*a^2*c*d*e^2)*x^2)*sqrt(-c*x^4 + a))/(a^2*c^3*x^5 - a^3*c^2*x)
\[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{3}}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)**3/(-c*x**4+a)**(3/2),x)
Output:
Integral((d + e*x**2)**3/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^3/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^3/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^3/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^3/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int((d + e*x^2)^3/(a - c*x^4)^(3/2),x)
Output:
int((d + e*x^2)^3/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^3}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 \sqrt {-c \,x^{4}+a}\, d \,e^{2} x -\sqrt {-c \,x^{4}+a}\, e^{3} x^{3}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} d \,e^{2}+\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a c \,d^{3}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a c d \,e^{2} x^{4}-\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) c^{2} d^{3} x^{4}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} e^{3}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a c \,d^{2} e -3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a c \,e^{3} x^{4}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) c^{2} d^{2} e \,x^{4}}{c \left (-c \,x^{4}+a \right )} \] Input:
int((e*x^2+d)^3/(-c*x^4+a)^(3/2),x)
Output:
(3*sqrt(a - c*x**4)*d*e**2*x - sqrt(a - c*x**4)*e**3*x**3 - 3*int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*d*e**2 + int(sqrt(a - c*x **4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c*d**3 + 3*int(sqrt(a - c*x**4)/ (a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c*d*e**2*x**4 - int(sqrt(a - c*x**4)/ (a**2 - 2*a*c*x**4 + c**2*x**8),x)*c**2*d**3*x**4 + 3*int((sqrt(a - c*x**4 )*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*e**3 + 3*int((sqrt(a - c*x **4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c*d**2*e - 3*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c*e**3*x**4 - 3*int((s qrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*c**2*d**2*e*x**4) /(c*(a - c*x**4))