Integrand size = 22, antiderivative size = 244 \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {d \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}-\frac {d^4 \left (4 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )^{3/2}} \]
-d*(-a*e^2+4*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^5-d^4*(5* a*e^2+4*c*d^2)*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e ^5/(a*e^2+c*d^2)^(3/2)+1/3*(-2*a*e^2+13*c*d^2)*(c*x^2+a)^(1/2)/c^2/e^4+d^5 *(c*x^2+a)^(1/2)/e^4/(a*e^2+c*d^2)/(e*x+d)-5/3*d*(e*x+d)*(c*x^2+a)^(1/2)/c /e^4+1/3*(e*x+d)^2*(c*x^2+a)^(1/2)/c/e^4
Time = 1.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.03 \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (-2 a^2 e^4 (d+e x)+a c e^2 \left (7 d^3+4 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+c^2 d^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{c^2 \left (c d^2+a e^2\right ) (d+e x)}+\frac {6 d^4 \left (4 c d^2+5 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {3 \left (4 c d^3-a d e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{3 e^5} \]
((e*Sqrt[a + c*x^2]*(-2*a^2*e^4*(d + e*x) + a*c*e^2*(7*d^3 + 4*d^2*e*x - 2 *d*e^2*x^2 + e^3*x^3) + c^2*d^2*(12*d^3 + 6*d^2*e*x - 2*d*e^2*x^2 + e^3*x^ 3)))/(c^2*(c*d^2 + a*e^2)*(d + e*x)) + (6*d^4*(4*c*d^2 + 5*a*e^2)*ArcTan[( Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e^2)^(3/2) + (3*(4*c*d^3 - a*d*e^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] ])/c^(3/2))/(3*e^5)
Time = 1.19 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {603, 25, 2185, 2185, 27, 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.
| \(\displaystyle \int \frac {x^5}{\sqrt {a+c x^2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 603 |
| \(\displaystyle \frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac {\int -\frac {\frac {a d^4}{e^3}-\frac {\left (c d^2+a e^2\right ) x d^3}{e^4}+\frac {\left (c d^2+a e^2\right ) x^2 d^2}{e^3}-\left (\frac {c d^2}{e^2}+a\right ) x^3 d+\frac {\left (c d^2+a e^2\right ) x^4}{e}}{(d+e x) \sqrt {c x^2+a}}dx}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {a d^4}{e^3}-\frac {\left (c d^2+a e^2\right ) x d^3}{e^4}+\frac {\left (c d^2+a e^2\right ) x^2 d^2}{e^3}-\left (\frac {c d^2}{e^2}+a\right ) x^3 d+\frac {\left (c d^2+a e^2\right ) x^4}{e}}{(d+e x) \sqrt {c x^2+a}}dx}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\int \frac {-10 c d e^2 \left (c d^2+a e^2\right ) x^3-2 e \left (c d^2+a e^2\right )^2 x^2-4 d \left (c d^2+a e^2\right )^2 x+a d^2 e \left (c d^2-2 a e^2\right )}{(d+e x) \sqrt {c x^2+a}}dx}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 \left (c \left (13 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x^2 e^4+3 a c d^2 \left (2 c d^2+a e^2\right ) e^4+c d \left (c d^2+a e^2\right )^2 x e^3\right )}{(d+e x) \sqrt {c x^2+a}}dx}{2 c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {c \left (13 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x^2 e^4+3 a c d^2 \left (2 c d^2+a e^2\right ) e^4+c d \left (c d^2+a e^2\right )^2 x e^3}{(d+e x) \sqrt {c x^2+a}}dx}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 c^2 d e^5 \left (a d e \left (2 c d^2+a e^2\right )-\left (4 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{c e^2}+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 c d e^3 \int \frac {a d e \left (2 c d^2+a e^2\right )-\left (4 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {c x^2+a}}dx+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {\frac {3 c d e^3 \left (\frac {c d^3 \left (5 a e^2+4 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (4 c d^2-a e^2\right ) \left (a e^2+c d^2\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{e}\right )+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {3 c d e^3 \left (\frac {c d^3 \left (5 a e^2+4 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (4 c d^2-a e^2\right ) \left (a e^2+c d^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}\right )+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 c d e^3 \left (\frac {c d^3 \left (5 a e^2+4 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (4 c d^2-a e^2\right ) \left (a e^2+c d^2\right )}{\sqrt {c} e}\right )+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {\frac {3 c d e^3 \left (-\frac {c d^3 \left (5 a e^2+4 c d^2\right ) \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (4 c d^2-a e^2\right ) \left (a e^2+c d^2\right )}{\sqrt {c} e}\right )+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 c d e^3 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (4 c d^2-a e^2\right ) \left (a e^2+c d^2\right )}{\sqrt {c} e}-\frac {c d^3 \left (5 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \sqrt {a e^2+c d^2}}\right )+e^3 \sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right ) \left (a e^2+c d^2\right )}{c e^3}-5 d \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}{3 c e^4}}{a e^2+c d^2}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}\) |
(d^5*Sqrt[a + c*x^2])/(e^4*(c*d^2 + a*e^2)*(d + e*x)) + (((c*d^2 + a*e^2)* (d + e*x)^2*Sqrt[a + c*x^2])/(3*c*e^4) + (-5*d*(c*d^2 + a*e^2)*(d + e*x)*S qrt[a + c*x^2] + (e^3*(13*c*d^2 - 2*a*e^2)*(c*d^2 + a*e^2)*Sqrt[a + c*x^2] + 3*c*d*e^3*(-(((4*c*d^2 - a*e^2)*(c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr t[a + c*x^2]])/(Sqrt[c]*e)) - (c*d^3*(4*c*d^2 + 5*a*e^2)*ArcTanh[(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])))/(c* e^3))/(3*c*e^4))/(c*d^2 + a*e^2)
3.4.42.3.1 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 ] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x) ^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(439\) vs. \(2(218)=436\).
Time = 0.48 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.80
| method | result | size |
| risch | \(-\frac {\left (-c \,e^{2} x^{2}+3 c d e x +2 e^{2} a -9 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{3 c^{2} e^{4}}+\frac {d \left (\frac {\left (e^{2} a -4 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {5 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {c \,d^{4} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\right )}{e^{4} c}\) | \(440\) |
| default | \(\frac {\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}}{e^{2}}-\frac {4 d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{5} \sqrt {c}}-\frac {2 d \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{e^{3}}+\frac {3 d^{2} \sqrt {c \,x^{2}+a}}{e^{4} c}-\frac {d^{5} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{7}}-\frac {5 d^{4} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(477\) |
-1/3*(-c*e^2*x^2+3*c*d*e*x+2*a*e^2-9*c*d^2)*(c*x^2+a)^(1/2)/c^2/e^4+d/e^4/ c*((a*e^2-4*c*d^2)/e*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-5*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))-c*d^4/e^3*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c-2/e*c*d*(x+d/e )+(a*e^2+c*d^2)/e^2)^(1/2)-e*c*d/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*l n((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))))
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (219) = 438\).
Time = 25.37 (sec) , antiderivative size = 2025, normalized size of antiderivative = 8.30 \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]
[1/6*(3*(4*c^3*d^8 + 7*a*c^2*d^6*e^2 + 2*a^2*c*d^4*e^4 - a^3*d^2*e^6 + (4* c^3*d^7*e + 7*a*c^2*d^5*e^3 + 2*a^2*c*d^3*e^5 - a^3*d*e^7)*x)*sqrt(c)*log( -2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*(4*c^3*d^7 + 5*a*c^2*d^5*e ^2 + (4*c^3*d^6*e + 5*a*c^2*d^4*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*(12*c^3*d ^7*e + 19*a*c^2*d^5*e^3 + 5*a^2*c*d^3*e^5 - 2*a^3*d*e^7 + (c^3*d^4*e^4 + 2 *a*c^2*d^2*e^6 + a^2*c*e^8)*x^3 - 2*(c^3*d^5*e^3 + 2*a*c^2*d^3*e^5 + a^2*c *d*e^7)*x^2 + 2*(3*c^3*d^6*e^2 + 5*a*c^2*d^4*e^4 + a^2*c*d^2*e^6 - a^3*e^8 )*x)*sqrt(c*x^2 + a))/(c^4*d^5*e^5 + 2*a*c^3*d^3*e^7 + a^2*c^2*d*e^9 + (c^ 4*d^4*e^6 + 2*a*c^3*d^2*e^8 + a^2*c^2*e^10)*x), -1/6*(6*(4*c^3*d^7 + 5*a*c ^2*d^5*e^2 + (4*c^3*d^6*e + 5*a*c^2*d^4*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arcta n(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*(4*c^3*d^8 + 7*a*c^2*d^6*e^2 + 2*a^2*c*d^4*e ^4 - a^3*d^2*e^6 + (4*c^3*d^7*e + 7*a*c^2*d^5*e^3 + 2*a^2*c*d^3*e^5 - a^3* d*e^7)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(12* c^3*d^7*e + 19*a*c^2*d^5*e^3 + 5*a^2*c*d^3*e^5 - 2*a^3*d*e^7 + (c^3*d^4*e^ 4 + 2*a*c^2*d^2*e^6 + a^2*c*e^8)*x^3 - 2*(c^3*d^5*e^3 + 2*a*c^2*d^3*e^5 + a^2*c*d*e^7)*x^2 + 2*(3*c^3*d^6*e^2 + 5*a*c^2*d^4*e^4 + a^2*c*d^2*e^6 - a^ 3*e^8)*x)*sqrt(c*x^2 + a))/(c^4*d^5*e^5 + 2*a*c^3*d^3*e^7 + a^2*c^2*d*e...
\[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^{5}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {x^{5}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^5}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]