Integrand size = 22, antiderivative size = 213 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {3 a^{3/4} e \left (5 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{7/4} \sqrt {a-c x^4}} \] Output:
-d*e^2*x*(-c*x^4+a)^(1/2)/c-1/5*e^3*x^3*(-c*x^4+a)^(1/2)/c+3/5*a^(3/4)*e*( a*e^2+5*c*d^2)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(7/4)/(- c*x^4+a)^(1/2)+1/5*a^(1/4)*(5*c^(1/2)*d*(a*e^2+c*d^2)-3*a^(1/2)*e*(a*e^2+5 *c*d^2))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/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 = 10.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\frac {5 d \left (c d^2+a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+e x \left (e \left (5 d+e x^2\right ) \left (-a+c x^4\right )+\left (5 c d^2+a e^2\right ) x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{5 c \sqrt {a-c x^4}} \] Input:
Integrate[(d + e*x^2)^3/Sqrt[a - c*x^4],x]
Output:
(5*d*(c*d^2 + a*e^2)*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4 , (c*x^4)/a] + e*x*(e*(5*d + e*x^2)*(-a + c*x^4) + (5*c*d^2 + a*e^2)*x^2*S qrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(5*c*Sqrt [a - c*x^4])
Time = 0.93 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1519, 25, 2427, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx\) |
| \(\Big \downarrow \) 1519 |
| \(\displaystyle -\frac {\int -\frac {15 c d e^2 x^4+3 e \left (5 c d^2+a e^2\right ) x^2+5 c d^3}{\sqrt {a-c x^4}}dx}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {15 c d e^2 x^4+3 e \left (5 c d^2+a e^2\right ) x^2+5 c d^3}{\sqrt {a-c x^4}}dx}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 c \left (3 e \left (5 c d^2+a e^2\right ) x^2+5 d \left (c d^2+a e^2\right )\right )}{\sqrt {a-c x^4}}dx}{3 c}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 e \left (5 c d^2+a e^2\right ) x^2+5 d \left (c d^2+a e^2\right )}{\sqrt {a-c x^4}}dx-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+\frac {3 e \left (a e^2+5 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {3 e \left (a e^2+5 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {3 e \left (a e^2+5 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {3 e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+5 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {3 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+5 c d^2\right ) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+5 c d^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (5 d \left (a e^2+c d^2\right )-\frac {3 \sqrt {a} e \left (a e^2+5 c d^2\right )}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-5 d e^2 x \sqrt {a-c x^4}}{5 c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}\) |
Input:
Int[(d + e*x^2)^3/Sqrt[a - c*x^4],x]
Output:
-1/5*(e^3*x^3*Sqrt[a - c*x^4])/c + (-5*d*e^2*x*Sqrt[a - c*x^4] + (3*a^(3/4 )*e*(5*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)*(5*d*(c*d^2 + a*e^2) - (3 *Sqrt[a]*e*(5*c*d^2 + a*e^2))/Sqrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSi n[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]))/(5*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 ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 2.56 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04
| method | result | size |
| elliptic | \(-\frac {e^{3} x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {d \,e^{2} x \sqrt {-c \,x^{4}+a}}{c}+\frac {\left (d^{3}+\frac {a d \,e^{2}}{c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (3 d^{2} e +\frac {3 a \,e^{3}}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(222\) |
| risch | \(-\frac {x \,e^{2} \left (e \,x^{2}+5 d \right ) \sqrt {-c \,x^{4}+a}}{5 c}+\frac {-\frac {3 e \left (a \,e^{2}+5 c \,d^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {5 d^{3} c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 d \,e^{2} a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{5 c}\) | \(273\) |
| default | \(\frac {d^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+e^{3} \left (-\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+3 d \,e^{2} \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-\frac {3 d^{2} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(360\) |
Input:
int((e*x^2+d)^3/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/5*e^3*x^3*(-c*x^4+a)^(1/2)/c-d*e^2*x*(-c*x^4+a)^(1/2)/c+(d^3+a/c*d*e^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)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I) -(3*d^2*e+3/5*a/c*e^3)*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)*(E llipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*c^(1/2))^(1 /2),I))
Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=-\frac {3 \, {\left (5 \, a c d^{2} e + a^{2} e^{3}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (5 \, c^{2} d^{3} + 15 \, a c d^{2} e + 5 \, a c d e^{2} + 3 \, a^{2} e^{3}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a c e^{3} x^{4} + 5 \, a c d e^{2} x^{2} + 15 \, a c d^{2} e + 3 \, a^{2} e^{3}\right )} \sqrt {-c x^{4} + a}}{5 \, a c^{2} x} \] Input:
integrate((e*x^2+d)^3/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/5*(3*(5*a*c*d^2*e + a^2*e^3)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_e(arcsin(( a/c)^(1/4)/x), -1) - (5*c^2*d^3 + 15*a*c*d^2*e + 5*a*c*d*e^2 + 3*a^2*e^3)* sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) + (a*c*e^3*x^ 4 + 5*a*c*d*e^2*x^2 + 15*a*c*d^2*e + 3*a^2*e^3)*sqrt(-c*x^4 + a))/(a*c^2*x )
Time = 1.83 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\frac {d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {3 d^{2} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)**3/(-c*x**4+a)**(1/2),x)
Output:
d**3*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4 *sqrt(a)*gamma(5/4)) + 3*d**2*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + 3*d*e**2*x**5*gamma(5 /4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma (9/4)) + e**3*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), c*x**4*exp_polar( 2*I*pi)/a)/(4*sqrt(a)*gamma(11/4))
\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^3/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^3/sqrt(-c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^3/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^3/sqrt(-c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{\sqrt {a-c\,x^4}} \,d x \] Input:
int((d + e*x^2)^3/(a - c*x^4)^(1/2),x)
Output:
int((d + e*x^2)^3/(a - c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\frac {-5 \sqrt {-c \,x^{4}+a}\, d \,e^{2} x -\sqrt {-c \,x^{4}+a}\, e^{3} x^{3}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a d \,e^{2}+5 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) c \,d^{3}+3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a \,e^{3}+15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) c \,d^{2} e}{5 c} \] Input:
int((e*x^2+d)^3/(-c*x^4+a)^(1/2),x)
Output:
( - 5*sqrt(a - c*x**4)*d*e**2*x - sqrt(a - c*x**4)*e**3*x**3 + 5*int(sqrt( a - c*x**4)/(a - c*x**4),x)*a*d*e**2 + 5*int(sqrt(a - c*x**4)/(a - c*x**4) ,x)*c*d**3 + 3*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a*e**3 + 15*int ((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*c*d**2*e)/(5*c)