Integrand size = 19, antiderivative size = 811 \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=-\frac {c^{3/2} d^3 x \sqrt {a+c x^4}}{e^2 \left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {d x \sqrt {a+c x^4}}{\left (d^2-e^2 x^2\right )^2}-\frac {c d^3 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right ) \left (d^2-e^2 x^2\right )}+\frac {\left (d^2 \left (c d^4-a e^4\right )-e^2 \left (3 c d^4+a e^4\right ) x^2\right ) \sqrt {a+c x^4}}{2 e \left (c d^4+a e^4\right ) \left (d^2-e^2 x^2\right )^2}+\frac {c d^2 \left (c d^4+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 e^3 \left (c d^4+a e^4\right )^{3/2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 e^3}-\frac {c d^2 \left (c d^4+3 a e^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{2 e^3 \left (c d^4+a e^4\right )^{3/2}}+\frac {\sqrt [4]{a} c^{5/4} d^3 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^2 \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} c^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {c^{3/4} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+3 a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}} \] Output:
-c^(3/2)*d^3*x*(c*x^4+a)^(1/2)/e^2/(a*e^4+c*d^4)/(a^(1/2)+c^(1/2)*x^2)+d*x *(c*x^4+a)^(1/2)/(-e^2*x^2+d^2)^2-c*d^3*x*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(- e^2*x^2+d^2)+1/2*(d^2*(-a*e^4+c*d^4)-e^2*(a*e^4+3*c*d^4)*x^2)*(c*x^4+a)^(1 /2)/e/(a*e^4+c*d^4)/(-e^2*x^2+d^2)^2+1/2*c*d^2*(3*a*e^4+c*d^4)*arctanh((a* e^4+c*d^4)^(1/2)*x/d/e/(c*x^4+a)^(1/2))/e^3/(a*e^4+c*d^4)^(3/2)+1/2*c^(1/2 )*arctanh(c^(1/2)*x^2/(c*x^4+a)^(1/2))/e^3-1/2*c*d^2*(3*a*e^4+c*d^4)*arcta nh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/2))/e^3/(a*e^4+c*d^4 )^(3/2)+a^(1/4)*c^(5/4)*d^3*(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))/ e^2/(a*e^4+c*d^4)/(c*x^4+a)^(1/2)-a^(1/4)*c^(3/4)*d*(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))/e^2/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+a)^(1/2)-1/4 *c^(3/4)*d*(c^(1/2)*d^2-a^(1/2)*e^2)*(3*a*e^4+c*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)*EllipticPi(sin(2*arctan(c^(1/4) *x/a^(1/4))),1/4*(c^(1/2)*d^2+a^(1/2)*e^2)^2/a^(1/2)/c^(1/2)/d^2/e^2,1/2*2 ^(1/2))/a^(1/4)/e^4/(c^(1/2)*d^2+a^(1/2)*e^2)/(a*e^4+c*d^4)/(c*x^4+a)^(1/2 )
Result contains complex when optimal does not.
Time = 12.01 (sec) , antiderivative size = 989, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[a + c*x^4]/(d + e*x)^3,x]
Output:
(-(e^3*(c*d^4 + a*e^4)^2*(a + c*x^4)) + 2*c*d^3*e^3*(c*d^4 + a*e^4)*(d + e *x)*(a + c*x^4) - 2*c^2*d^6*e*Sqrt[-(c*d^4) - a*e^4]*(d + e*x)^2*Sqrt[a + c*x^4]*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4 ) - a*e^4]] - 6*a*c*d^2*e^5*Sqrt[-(c*d^4) - a*e^4]*(d + e*x)^2*Sqrt[a + c* x^4]*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]] + (2*I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^3*e^2*(c*d^4 + a*e^4)*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a] ]*x], -1] + ((2*I)*c^2*d^5*(c*d^4 + a*e^4)*(d + e*x)^2*Sqrt[1 + (c*x^4)/a] *EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/S qrt[a]] - (2*I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^3*e^2*(c*d^4 + a*e^4)*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]* x], -1] + ((4*I)*a*c*d*e^4*(c*d^4 + a*e^4)*(d + e*x)^2*Sqrt[1 + (c*x^4)/a] *EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/S qrt[a]] - 2*(-1)^(1/4)*a^(1/4)*c^(7/4)*d^5*(c*d^4 + a*e^4)*(d + e*x)^2*Sqr t[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3 /4)*c^(1/4)*x)/a^(1/4)], -1] - 6*(-1)^(1/4)*a^(5/4)*c^(3/4)*d*e^4*(c*d^4 + a*e^4)*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c ]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] - c^(5/2)*d^8*e*(d + e *x)^2*Sqrt[a + c*x^4]*Log[-(Sqrt[c]*x^2) + Sqrt[a + c*x^4]] - 2*a*c^(3/2)* d^4*e^5*(d + e*x)^2*Sqrt[a + c*x^4]*Log[-(Sqrt[c]*x^2) + Sqrt[a + c*x^4...
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.
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} \left (d^3-3 d^2 e x+3 d e^2 x^2-e^3 x^3\right )}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4} (d-e x)^3}{\left (d^2-e^2 x^2\right )^3}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{(d+e x)^3}dx\) |
Input:
Int[Sqrt[a + c*x^4]/(d + e*x)^3,x]
Output:
$Aborted
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbo l] :> Int[ExpandToSum[(c - d*x^n)^(-q), x]*((a + b*x^nn)^p/(c^2 - d^2*x^(2* n))^(-q)), x] /; FreeQ[{a, b, c, d, n, nn, p}, x] && !IntegerQ[p] && ILtQ[ q, 0] && IGtQ[Log[2, nn/n], 0]
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.63
| method | result | size |
| default | \(-\frac {\sqrt {c \,x^{4}+a}}{2 e \left (e x +d \right )^{2}}+\frac {c \,d^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +c \,d^{4}\right ) e \left (e x +d \right )}+\frac {\left (-\frac {3 c d}{e^{4}}+\frac {c d \left (e^{4} a +2 c \,d^{4}\right )}{e^{4} \left (e^{4} a +c \,d^{4}\right )}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\sqrt {c}\, \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 e^{3}}-\frac {i d^{3} c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \left (e^{4} a +c \,d^{4}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {c \,d^{2} \left (3 e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +c \,d^{4}\right ) e^{5}}\) | \(509\) |
| elliptic | \(-\frac {\sqrt {c \,x^{4}+a}}{2 e \left (e x +d \right )^{2}}+\frac {c \,d^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +c \,d^{4}\right ) e \left (e x +d \right )}+\frac {\left (-\frac {3 c d}{e^{4}}+\frac {c d \left (e^{4} a +2 c \,d^{4}\right )}{e^{4} \left (e^{4} a +c \,d^{4}\right )}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\sqrt {c}\, \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 e^{3}}-\frac {i d^{3} c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \left (e^{4} a +c \,d^{4}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {c \,d^{2} \left (3 e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +c \,d^{4}\right ) e^{5}}\) | \(509\) |
Input:
int((c*x^4+a)^(1/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/2/e*(c*x^4+a)^(1/2)/(e*x+d)^2+c*d^3/(a*e^4+c*d^4)/e*(c*x^4+a)^(1/2)/(e* x+d)+(-3*c*d/e^4+c*d*(a*e^4+2*c*d^4)/e^4/(a*e^4+c*d^4))/(I*c^(1/2)/a^(1/2) )^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c *x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+1/2*c^(1/2)/e^3*ln( 2*c^(1/2)*x^2+2*(c*x^4+a)^(1/2))-I*d^3/e^2*c^(3/2)/(a*e^4+c*d^4)*a^(1/2)/( I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/ a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*(EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-E llipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I))+c*d^2*(3*a*e^4+c*d^4)/(a*e^4+c*d^ 4)/e^5*(-1/2/(a+c*d^4/e^4)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a)/(a+c*d^ 4/e^4)^(1/2)/(c*x^4+a)^(1/2))+1/(I*c^(1/2)/a^(1/2))^(1/2)/d*e*(1-I*c^(1/2) *x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*Ellipt icPi(x*(I*c^(1/2)/a^(1/2))^(1/2),-I/c^(1/2)*a^(1/2)/d^2*e^2,(-I/a^(1/2)*c^ (1/2))^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)))
Timed out. \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=\text {Timed out} \] Input:
integrate((c*x^4+a)^(1/2)/(e*x+d)^3,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a + c x^{4}}}{\left (d + e x\right )^{3}}\, dx \] Input:
integrate((c*x**4+a)**(1/2)/(e*x+d)**3,x)
Output:
Integral(sqrt(a + c*x**4)/(d + e*x)**3, x)
\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/(e*x+d)^3,x, algorithm="maxima")
Output:
integrate(sqrt(c*x^4 + a)/(e*x + d)^3, x)
\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/(e*x+d)^3,x, algorithm="giac")
Output:
integrate(sqrt(c*x^4 + a)/(e*x + d)^3, x)
Timed out. \[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=\int \frac {\sqrt {c\,x^4+a}}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int((a + c*x^4)^(1/2)/(d + e*x)^3,x)
Output:
int((a + c*x^4)^(1/2)/(d + e*x)^3, x)
\[ \int \frac {\sqrt {a+c x^4}}{(d+e x)^3} \, dx=\int \frac {\sqrt {c \,x^{4}+a}}{\left (e x +d \right )^{3}}d x \] Input:
int((c*x^4+a)^(1/2)/(e*x+d)^3,x)
Output:
int((c*x^4+a)^(1/2)/(e*x+d)^3,x)