Integrand size = 19, antiderivative size = 198 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \]
-1/2*c^2*d*(-3*a*e^2+2*c*d^2)*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c* x^2+a)^(1/2))/(a*e^2+c*d^2)^(7/2)-1/3*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x +d)^3-5/6*c*d*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^2-1/6*c*e*(-4*a*e^ 2+11*c*d^2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)
Time = 10.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\frac {-e \sqrt {c d^2+a e^2} \sqrt {a+c x^2} \left (2 \left (c d^2+a e^2\right )^2+5 c d \left (c d^2+a e^2\right ) (d+e x)+c \left (11 c d^2-4 a e^2\right ) (d+e x)^2\right )+3 c^2 d \left (2 c d^2-3 a e^2\right ) (d+e x)^3 \log (d+e x)-3 c^2 d \left (2 c d^2-3 a e^2\right ) (d+e x)^3 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{6 \left (c d^2+a e^2\right )^{7/2} (d+e x)^3} \]
(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(2*(c*d^2 + a*e^2)^2 + 5*c*d*(c*d ^2 + a*e^2)*(d + e*x) + c*(11*c*d^2 - 4*a*e^2)*(d + e*x)^2)) + 3*c^2*d*(2* c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[d + e*x] - 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^ 2 + a*e^2)^(7/2)*(d + e*x)^3)
Time = 0.36 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {498, 25, 688, 25, 679, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+c x^2} (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {c \int -\frac {3 d-2 e x}{(d+e x)^3 \sqrt {c x^2+a}}dx}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \int \frac {3 d-2 e x}{(d+e x)^3 \sqrt {c x^2+a}}dx}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {c \left (-\frac {\int -\frac {2 \left (3 c d^2-2 a e^2\right )-5 c d e x}{(d+e x)^2 \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {5 d e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {2 \left (3 c d^2-2 a e^2\right )-5 c d e x}{(d+e x)^2 \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {5 d e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {c \left (\frac {\frac {3 c d \left (2 c d^2-3 a e^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {e \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{(d+e x) \left (a e^2+c d^2\right )}}{2 \left (a e^2+c d^2\right )}-\frac {5 d e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {c \left (\frac {-\frac {3 c d \left (2 c d^2-3 a e^2\right ) \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{a e^2+c d^2}-\frac {e \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{(d+e x) \left (a e^2+c d^2\right )}}{2 \left (a e^2+c d^2\right )}-\frac {5 d e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {c \left (\frac {-\frac {3 c d \left (2 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {e \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{(d+e x) \left (a e^2+c d^2\right )}}{2 \left (a e^2+c d^2\right )}-\frac {5 d e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )}\) |
-1/3*(e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)^3) + (c*((-5*d*e*Sqrt[ a + c*x^2])/(2*(c*d^2 + a*e^2)*(d + e*x)^2) + (-((e*(11*c*d^2 - 4*a*e^2)*S qrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x))) - (3*c*d*(2*c*d^2 - 3*a*e^2)* ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e ^2)^(3/2))/(2*(c*d^2 + a*e^2))))/(3*(c*d^2 + a*e^2))
3.6.68.3.1 Defintions of rubi rules used
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/(((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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(761\) vs. \(2(178)=356\).
Time = 2.11 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.85
| method | result | size |
| default | \(\frac {-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{3 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {5 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \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 {c \left (x +\frac {d}{e}\right )^{2}-\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 )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}+\frac {c \,e^{2} \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 {c \left (x +\frac {d}{e}\right )^{2}-\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 )}{2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{3 \left (e^{2} a +c \,d^{2}\right )}-\frac {2 c \,e^{2} \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \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 {c \left (x +\frac {d}{e}\right )^{2}-\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 )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{3 \left (e^{2} a +c \,d^{2}\right )}}{e^{4}}\) | \(762\) |
1/e^4*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^ 2+c*d^2)/e^2)^(1/2)+5/3*c*d*e/(a*e^2+c*d^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e )^2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+3/2*c*d*e/(a*e^2 +c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+ c*d^2)/e^2)^(1/2)-c*d*e/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e ^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c *d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(a*e^2+c*d^2)*e^2/( (a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2 +c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/ (x+d/e)))-2/3*c/(a*e^2+c*d^2)*e^2*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e) ^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c*d*e/(a*e^2+c*d^2)/((a*e^2+c* d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e ^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))) )
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (179) = 358\).
Time = 0.62 (sec) , antiderivative size = 1139, normalized size of antiderivative = 5.75 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\left [-\frac {3 \, {\left (2 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \sqrt {c d^{2} + a e^{2}} \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 ) + 2 \, {\left (18 \, c^{3} d^{6} e + 23 \, a c^{2} d^{4} e^{3} + 7 \, a^{2} c d^{2} e^{5} + 2 \, a^{3} e^{7} + {\left (11 \, c^{3} d^{4} e^{3} + 7 \, a c^{2} d^{2} e^{5} - 4 \, a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (9 \, c^{3} d^{5} e^{2} + 8 \, a c^{2} d^{3} e^{4} - a^{2} c d e^{6}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{4} d^{11} + 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} + 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} + 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} + 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e + 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} + 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}}, -\frac {3 \, {\left (2 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \sqrt {-c d^{2} - a e^{2}} \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 ) + {\left (18 \, c^{3} d^{6} e + 23 \, a c^{2} d^{4} e^{3} + 7 \, a^{2} c d^{2} e^{5} + 2 \, a^{3} e^{7} + {\left (11 \, c^{3} d^{4} e^{3} + 7 \, a c^{2} d^{2} e^{5} - 4 \, a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (9 \, c^{3} d^{5} e^{2} + 8 \, a c^{2} d^{3} e^{4} - a^{2} c d e^{6}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{4} d^{11} + 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} + 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} + 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} + 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e + 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} + 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}}\right ] \]
[-1/12*(3*(2*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x ^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 + 3*(2*c^3*d^5*e - 3*a*c^2*d^ 3*e^3)*x)*sqrt(c*d^2 + a*e^2)*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)) + 2*(18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^ 2*c*d^2*e^5 + 2*a^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*e^7) *x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e^6)*x)*sqrt(c*x^2 + a ))/(c^4*d^11 + 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4 *d^3*e^8 + (c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^3*c*d^ 2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e ^6 + 4*a^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x), -1/6*(3*(2*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^ 2 - 3*a*c^2*d^2*e^4)*x^2 + 3*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*sqrt(-c*d^ 2 - a*e^2)*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)) + (18*c^3*d^6*e + 23*a*c^2*d^4*e ^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a ^2*c*e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e^6)*x)*sqrt( c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5* e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 ...
\[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (179) = 358\).
Time = 0.28 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.99 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\frac {1}{3} \, c^{\frac {3}{2}} {\left (\frac {3 \, {\left (2 \, c^{\frac {3}{2}} d^{3} - 3 \, a \sqrt {c} d e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{\frac {3}{2}} d^{3} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a \sqrt {c} d e^{4} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{2} d^{4} e - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c d^{2} e^{3} + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{\frac {5}{2}} d^{5} - 82 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{\frac {3}{2}} d^{3} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} \sqrt {c} d e^{4} - 102 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{2} d^{4} e + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} e^{5} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{\frac {3}{2}} d^{3} e^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} \sqrt {c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} + 4 \, a^{4} e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}}\right )} \]
1/3*c^(3/2)*(3*(2*c^(3/2)*d^3 - 3*a*sqrt(c)*d*e^2)*arctan(-((sqrt(c)*x - s qrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d ^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) - (6*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^(3/2)*d^3*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a *sqrt(c)*d*e^4 + 30*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^2*d^4*e - 45*(sqrt(c )*x - sqrt(c*x^2 + a))^4*a*c*d^2*e^3 + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3* c^(5/2)*d^5 - 82*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^(3/2)*d^3*e^2 + 24*(s qrt(c)*x - sqrt(c*x^2 + a))^3*a^2*sqrt(c)*d*e^4 - 102*(sqrt(c)*x - sqrt(c* x^2 + a))^2*a*c^2*d^4*e + 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c*d^2*e^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*e^5 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^(3/2)*d^3*e^2 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*sqrt(c)* d*e^4 - 11*a^3*c*d^2*e^3 + 4*a^4*e^5)/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2* c*d^2*e^4 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - s qrt(c*x^2 + a))*sqrt(c)*d - a*e)^3))
Timed out. \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^4} \,d x \]