Integrand size = 21, antiderivative size = 386 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {4 d e \left (5 c d^2-3 a e^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {4 \sqrt [4]{a} d e \left (5 c d^2-3 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {\left (84 \sqrt {a} \sqrt {c} d e \left (5 c d^2-3 a e^2\right )+5 \left (21 c^2 d^4-42 a c d^2 e^2+5 a^2 e^4\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}} \] Output:
1/21*e^2*(-5*a*e^2+42*c*d^2)*x*(c*x^4+a)^(1/2)/c^2+4/5*d*e^3*x^3*(c*x^4+a) ^(1/2)/c+1/7*e^4*x^5*(c*x^4+a)^(1/2)/c+4/5*d*e*(-3*a*e^2+5*c*d^2)*x*(c*x^4 +a)^(1/2)/c^(3/2)/(a^(1/2)+c^(1/2)*x^2)-4/5*a^(1/4)*d*e*(-3*a*e^2+5*c*d^2) *(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE (sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/c^(7/4)/(c*x^4+a)^(1/2)+1/2 10*(84*a^(1/2)*c^(1/2)*d*e*(-3*a*e^2+5*c*d^2)+25*a^2*e^4-210*a*c*d^2*e^2+1 05*c^2*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2 )*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/c^(9/4) /(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {5 \left (21 c^2 d^4-42 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+e x \left (-e \left (a+c x^4\right ) \left (25 a e^2-3 c \left (70 d^2+28 d e x^2+5 e^2 x^4\right )\right )+28 c d \left (5 c d^2-3 a e^2\right ) x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{105 c^2 \sqrt {a+c x^4}} \] Input:
Integrate[(d + e*x^2)^4/Sqrt[a + c*x^4],x]
Output:
(5*(21*c^2*d^4 - 42*a*c*d^2*e^2 + 5*a^2*e^4)*x*Sqrt[1 + (c*x^4)/a]*Hyperge ometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + e*x*(-(e*(a + c*x^4)*(25*a*e^2 - 3*c*(70*d^2 + 28*d*e*x^2 + 5*e^2*x^4))) + 28*c*d*(5*c*d^2 - 3*a*e^2)*x^2* Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)]))/(105* c^2*Sqrt[a + c*x^4])
Time = 1.22 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1519, 2427, 2427, 27, 1512, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx\) |
| \(\Big \downarrow \) 1519 |
| \(\displaystyle \frac {\int \frac {28 c d e^3 x^6+e^2 \left (42 c d^2-5 a e^2\right ) x^4+28 c d^3 e x^2+7 c d^4}{\sqrt {c x^4+a}}dx}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {\frac {\int \frac {35 c^2 d^4+28 c e \left (5 c d^2-3 a e^2\right ) x^2 d+5 c e^2 \left (42 c d^2-5 a e^2\right ) x^4}{\sqrt {c x^4+a}}dx}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {c \left (84 c d e \left (5 c d^2-3 a e^2\right ) x^2+5 \left (21 c^2 d^4-42 a c e^2 d^2+5 a^2 e^4\right )\right )}{\sqrt {c x^4+a}}dx}{3 c}+\frac {5}{3} e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \int \frac {84 c d e \left (5 c d^2-3 a e^2\right ) x^2+5 \left (21 c^2 d^4-42 a c e^2 d^2+5 a^2 e^4\right )}{\sqrt {c x^4+a}}dx+\frac {5}{3} e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (\left (-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2+105 c^2 d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-84 \sqrt {a} \sqrt {c} d e \left (5 c d^2-3 a e^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx\right )+\frac {5}{3} e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (\left (-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2+105 c^2 d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-84 \sqrt {c} d e \left (5 c d^2-3 a e^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx\right )+\frac {5}{3} e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2+105 c^2 d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-84 \sqrt {c} d e \left (5 c d^2-3 a e^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx\right )+\frac {5}{3} e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2+105 c^2 d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}-84 \sqrt {c} d e \left (5 c d^2-3 a e^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )+\frac {5}{3} e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{5 c}+\frac {28}{5} d e^3 x^3 \sqrt {a+c x^4}}{7 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}\) |
Input:
Int[(d + e*x^2)^4/Sqrt[a + c*x^4],x]
Output:
(e^4*x^5*Sqrt[a + c*x^4])/(7*c) + ((28*d*e^3*x^3*Sqrt[a + c*x^4])/5 + ((5* e^2*(42*c*d^2 - 5*a*e^2)*x*Sqrt[a + c*x^4])/3 + (-84*Sqrt[c]*d*e*(5*c*d^2 - 3*a*e^2)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqr t[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[ 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])) + ((105*c^ 2*d^4 + 420*Sqrt[a]*c^(3/2)*d^3*e - 210*a*c*d^2*e^2 - 252*a^(3/2)*Sqrt[c]* d*e^3 + 25*a^2*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sq rt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*c^ (1/4)*Sqrt[a + c*x^4]))/3)/(5*c))/(7*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/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] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
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]
Result contains complex when optimal does not.
Time = 3.94 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.74
| method | result | size |
| elliptic | \(\frac {e^{4} x^{5} \sqrt {c \,x^{4}+a}}{7 c}+\frac {4 d \,e^{3} x^{3} \sqrt {c \,x^{4}+a}}{5 c}+\frac {\left (6 d^{2} e^{2}-\frac {5 a \,e^{4}}{7 c}\right ) x \sqrt {c \,x^{4}+a}}{3 c}+\frac {\left (d^{4}-\frac {a \left (6 d^{2} e^{2}-\frac {5 a \,e^{4}}{7 c}\right )}{3 c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (4 d^{3} e -\frac {12 a d \,e^{3}}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(286\) |
| risch | \(-\frac {x \,e^{2} \left (-15 c \,x^{4} e^{2}-84 d e \,x^{2} c +25 a \,e^{2}-210 c \,d^{2}\right ) \sqrt {c \,x^{4}+a}}{105 c^{2}}+\frac {-\frac {84 i e \sqrt {c}\, d \left (3 a \,e^{2}-5 c \,d^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {25 a^{2} e^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {105 c^{2} d^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {210 a c \,d^{2} e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{105 c^{2}}\) | \(397\) |
| default | \(\frac {d^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+e^{4} \left (\frac {x^{5} \sqrt {c \,x^{4}+a}}{7 c}-\frac {5 a x \sqrt {c \,x^{4}+a}}{21 c^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+4 d \,e^{3} \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5 c}-\frac {3 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+6 d^{2} e^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {4 i d^{3} e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(506\) |
Input:
int((e*x^2+d)^4/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/7*e^4*x^5*(c*x^4+a)^(1/2)/c+4/5*d*e^3*x^3*(c*x^4+a)^(1/2)/c+1/3*(6*d^2*e ^2-5/7*a/c*e^4)/c*x*(c*x^4+a)^(1/2)+(d^4-1/3*a/c*(6*d^2*e^2-5/7*a/c*e^4))/ (I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^( 1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+I *(4*d^3*e-12/5*a/c*d*e^3)*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c ^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)* (EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^ (1/2),I))
Time = 0.08 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.53 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {84 \, {\left (5 \, a c d^{3} e - 3 \, a^{2} d e^{3}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (105 \, c^{2} d^{4} - 420 \, a c d^{3} e - 210 \, a c d^{2} e^{2} + 252 \, a^{2} d e^{3} + 25 \, a^{2} e^{4}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (15 \, a c e^{4} x^{6} + 84 \, a c d e^{3} x^{4} + 420 \, a c d^{3} e - 252 \, a^{2} d e^{3} + 5 \, {\left (42 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{105 \, a c^{2} x} \] Input:
integrate((e*x^2+d)^4/(c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
1/105*(84*(5*a*c*d^3*e - 3*a^2*d*e^3)*sqrt(c)*x*(-a/c)^(3/4)*elliptic_e(ar csin((-a/c)^(1/4)/x), -1) + (105*c^2*d^4 - 420*a*c*d^3*e - 210*a*c*d^2*e^2 + 252*a^2*d*e^3 + 25*a^2*e^4)*sqrt(c)*x*(-a/c)^(3/4)*elliptic_f(arcsin((- a/c)^(1/4)/x), -1) + (15*a*c*e^4*x^6 + 84*a*c*d*e^3*x^4 + 420*a*c*d^3*e - 252*a^2*d*e^3 + 5*(42*a*c*d^2*e^2 - 5*a^2*e^4)*x^2)*sqrt(c*x^4 + a))/(a*c^ 2*x)
Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {d^{4} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d^{3} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d^{2} e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {e^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((e*x**2+d)**4/(c*x**4+a)**(1/2),x)
Output:
d**4*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(I*pi)/a)/(4*s qrt(a)*gamma(5/4)) + d**3*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x* *4*exp_polar(I*pi)/a)/(sqrt(a)*gamma(7/4)) + 3*d**2*e**2*x**5*gamma(5/4)*h yper((1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(9/4)) + d*e**3*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), c*x**4*exp_polar(I*pi) /a)/(sqrt(a)*gamma(11/4)) + e**4*x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,) , c*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(13/4))
\[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{4}}{\sqrt {c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^4/(c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^4/sqrt(c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{4}}{\sqrt {c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^4/(c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((e*x^2 + d)^4/sqrt(c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^4}{\sqrt {c\,x^4+a}} \,d x \] Input:
int((d + e*x^2)^4/(a + c*x^4)^(1/2),x)
Output:
int((d + e*x^2)^4/(a + c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {-25 \sqrt {c \,x^{4}+a}\, a \,e^{4} x +210 \sqrt {c \,x^{4}+a}\, c \,d^{2} e^{2} x +84 \sqrt {c \,x^{4}+a}\, c d \,e^{3} x^{3}+15 \sqrt {c \,x^{4}+a}\, c \,e^{4} x^{5}+25 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{2} e^{4}-210 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a c \,d^{2} e^{2}+105 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) c^{2} d^{4}-252 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) a c d \,e^{3}+420 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) c^{2} d^{3} e}{105 c^{2}} \] Input:
int((e*x^2+d)^4/(c*x^4+a)^(1/2),x)
Output:
( - 25*sqrt(a + c*x**4)*a*e**4*x + 210*sqrt(a + c*x**4)*c*d**2*e**2*x + 84 *sqrt(a + c*x**4)*c*d*e**3*x**3 + 15*sqrt(a + c*x**4)*c*e**4*x**5 + 25*int (sqrt(a + c*x**4)/(a + c*x**4),x)*a**2*e**4 - 210*int(sqrt(a + c*x**4)/(a + c*x**4),x)*a*c*d**2*e**2 + 105*int(sqrt(a + c*x**4)/(a + c*x**4),x)*c**2 *d**4 - 252*int((sqrt(a + c*x**4)*x**2)/(a + c*x**4),x)*a*c*d*e**3 + 420*i nt((sqrt(a + c*x**4)*x**2)/(a + c*x**4),x)*c**2*d**3*e)/(105*c**2)