Integrand size = 22, antiderivative size = 195 \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d \left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac {d^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \sqrt {c d^2+a e^2}} \]
-1/2*d*(-a*e^2+2*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^4-d^4 *arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^4/(a*e^2+c*d^ 2)^(1/2)+1/6*(-4*a*e^2+11*c*d^2)*(c*x^2+a)^(1/2)/c^2/e^3-7/6*d*(e*x+d)*(c* x^2+a)^(1/2)/c/e^3+1/3*(e*x+d)^2*(c*x^2+a)^(1/2)/c/e^3
Time = 0.51 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (-4 a e^2+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )}{c^2}-\frac {12 d^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {3 d \left (2 c d^2-a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{6 e^4} \]
((e*Sqrt[a + c*x^2]*(-4*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)))/c^2 - (1 2*d^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2 ]])/Sqrt[-(c*d^2) - a*e^2] + (3*d*(2*c*d^2 - a*e^2)*Log[-(Sqrt[c]*x) + Sqr t[a + c*x^2]])/c^(3/2))/(6*e^4)
Time = 0.73 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {604, 25, 2185, 25, 2185, 27, 719, 224, 219, 488, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {a+c x^2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle \frac {\int -\frac {7 c d e^3 x^3+e^2 \left (5 c d^2+2 a e^2\right ) x^2+d e \left (c d^2+4 a e^2\right ) x+2 a d^2 e^2}{(d+e x) \sqrt {c x^2+a}}dx}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\int \frac {7 c d e^3 x^3+e^2 \left (5 c d^2+2 a e^2\right ) x^2+d e \left (c d^2+4 a e^2\right ) x+2 a d^2 e^2}{(d+e x) \sqrt {c x^2+a}}dx}{3 c e^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {\int -\frac {3 a c d^2 e^5+c \left (11 c d^2-4 a e^2\right ) x^2 e^5+c d \left (5 c d^2-a e^2\right ) x e^4}{(d+e x) \sqrt {c x^2+a}}dx}{2 c e^3}+\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)}{3 c e^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {\int \frac {3 a c d^2 e^5+c \left (11 c d^2-4 a e^2\right ) x^2 e^5+c d \left (5 c d^2-a e^2\right ) x e^4}{(d+e x) \sqrt {c x^2+a}}dx}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {\frac {\int \frac {3 c^2 d e^6 \left (a d e-\left (2 c d^2-a e^2\right ) x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{c e^2}+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {3 c d e^4 \int \frac {a d e-\left (2 c d^2-a e^2\right ) x}{(d+e x) \sqrt {c x^2+a}}dx+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {3 c d e^4 \left (\frac {2 c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (2 c d^2-a e^2\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{e}\right )+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {3 c d e^4 \left (\frac {2 c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (2 c d^2-a e^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}\right )+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {3 c d e^4 \left (\frac {2 c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{\sqrt {c} e}\right )+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {3 c d e^4 \left (-\frac {2 c d^3 \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{\sqrt {c} e}\right )+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3}-\frac {\frac {7}{2} d e \sqrt {a+c x^2} (d+e x)-\frac {3 c d e^4 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{\sqrt {c} e}-\frac {2 c d^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \sqrt {a e^2+c d^2}}\right )+e^4 \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{2 c e^3}}{3 c e^4}\) |
((d + e*x)^2*Sqrt[a + c*x^2])/(3*c*e^3) - ((7*d*e*(d + e*x)*Sqrt[a + c*x^2 ])/2 - (e^4*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2] + 3*c*d*e^4*(-(((2*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*e)) - (2*c*d^3*Arc Tanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e*Sqrt[c*d^2 + a*e^2])))/(2*c*e^3))/(3*c*e^4)
3.4.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(b*d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b* x^2)^p*ExpandToSum[b*d^m*(m + n + 2*p + 1)*x^m - b*(m + n + 2*p + 1)*(c + d *x)^m - (c + d*x)^(m - 2)*(a*d^2*(m + n - 1) - b*c^2*(m + n + 2*p + 1) - 2* b*c*d*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.44 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-\frac {\left (-2 c \,e^{2} x^{2}+3 c d e x +4 e^{2} a -6 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{2} e^{3}}+\frac {d \left (\frac {\left (e^{2} a -2 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {2 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 e^{3} c}\) | \(222\) |
| default | \(\frac {\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}}{e}+\frac {d^{2} \sqrt {c \,x^{2}+a}}{e^{3} c}-\frac {d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}-\frac {d \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{e^{2}}-\frac {d^{4} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(258\) |
-1/6*(-2*c*e^2*x^2+3*c*d*e*x+4*a*e^2-6*c*d^2)*(c*x^2+a)^(1/2)/c^2/e^3+1/2* d/e^3/c*((a*e^2-2*c*d^2)/e*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-2*c*d^3/e ^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2/e*c*d*(x+d/e)+2*((a *e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/ 2))/(x+d/e)))
Time = 2.39 (sec) , antiderivative size = 1060, normalized size of antiderivative = 5.44 \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {6 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, -\frac {12 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, \frac {3 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, -\frac {6 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}\right ] \]
[1/12*(6*sqrt(c*d^2 + a*e^2)*c^2*d^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^ 2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c *x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 3*(2*c^2*d^5 + a*c*d^3*e^2 - a^2*d *e^4)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(6*c^2*d ^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 - 3*(c^2* d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^4 + a*c^2*e^6), -1/12* (12*sqrt(-c*d^2 - a*e^2)*c^2*d^4*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e) *sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + 3*(2*c^2 *d^5 + a*c*d^3*e^2 - a^2*d*e^4)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*s qrt(c)*x - a) - 2*(6*c^2*d^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 + 2*(c^2*d^2*e^ 3 + a*c*e^5)*x^2 - 3*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^ 2*e^4 + a*c^2*e^6), 1/6*(3*sqrt(c*d^2 + a*e^2)*c^2*d^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c *d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 3*(2*c^2*d^5 + a *c*d^3*e^2 - a^2*d*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (6*c ^2*d^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 - 3*( c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^4 + a*c^2*e^6), -1 /6*(6*sqrt(-c*d^2 - a*e^2)*c^2*d^4*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a* e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - 3*(2*c ^2*d^5 + a*c*d^3*e^2 - a^2*d*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2...
\[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^{4}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
Exception generated. \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^4}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]