Integrand size = 20, antiderivative size = 315 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d \left (c d^2-2 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {3 \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \]
-3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d^2 +7*a*e^2-10*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(1/4)/(-e*a^(1/2)+d*c^(1/2))^(5 /2)+3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d ^2+7*a*e^2+10*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(1/4)/(e*a^(1/2)+d*c^(1/2))^( 5/2)-1/4*(-c*d*x+a*e)*(e*x+d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^2+a)^2-1/16*(a* e*(-7*a*e^2+c*d^2)-6*c*d*(-2*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)/a^2/(-a*e^2+c*d ^2)^2/(-c*x^2+a)
Time = 1.66 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (-11 a^3 e^3+6 c^3 d^3 x^3+a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )-a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )\right )}{\left (c d^2-a e^2\right )^2 \left (a-c x^2\right )^2}+\frac {3 \left (4 c d^2+10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {3 \left (4 c d^2-10 \sqrt {a} \sqrt {c} d e+7 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \]
((-2*Sqrt[a]*Sqrt[d + e*x]*(-11*a^3*e^3 + 6*c^3*d^3*x^3 + a^2*c*e*(5*d^2 + 16*d*e*x + 7*e^2*x^2) - a*c^2*d*x*(10*d^2 + d*e*x + 12*e^2*x^2)))/((c*d^2 - a*e^2)^2*(a - c*x^2)^2) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^ 2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sq rt[a]*e)])/((Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt [a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/((Sqrt[c]*d - Sqrt [a]*e)^2*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]))/(32*a^(5/2))
Time = 0.62 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {496, 27, 686, 27, 654, 25, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {1}{\left (a-c x^2\right )^3 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\int \frac {6 c d^2+5 c e x d-7 a e^2}{2 \sqrt {d+e x} \left (a-c x^2\right )^2}dx}{4 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 c d^2+5 c e x d-7 a e^2}{\sqrt {d+e x} \left (a-c x^2\right )^2}dx}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 c \left (4 c^2 d^4-9 a c e^2 d^2+2 c e \left (c d^2-2 a e^2\right ) x d+7 a^2 e^4\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {4 c^2 d^4-9 a c e^2 d^2+2 c e \left (c d^2-2 a e^2\right ) x d+7 a^2 e^4}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {3 \int -\frac {e \left (2 c^2 d^4-5 a c e^2 d^2+2 c \left (c d^2-2 a e^2\right ) (d+e x) d+7 a^2 e^4\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 \int \frac {e \left (2 c^2 d^4-5 a c e^2 d^2+2 c \left (c d^2-2 a e^2\right ) (d+e x) d+7 a^2 e^4\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 e \int \frac {2 c^2 d^4-5 a c e^2 d^2+2 c \left (c d^2-2 a e^2\right ) (d+e x) d+7 a^2 e^4}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {3 e \left (\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {3 e \left (\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (-10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (10 \sqrt {a} \sqrt {c} d e+7 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
-1/4*((a*e - c*d*x)*Sqrt[d + e*x])/(a*(c*d^2 - a*e^2)*(a - c*x^2)^2) + (-1 /2*(Sqrt[d + e*x]*(a*e*(c*d^2 - 7*a*e^2) - 6*c*d*(c*d^2 - 2*a*e^2)*x))/(a* (c*d^2 - a*e^2)*(a - c*x^2)) - (3*e*(((Sqrt[c]*d + Sqrt[a]*e)^2*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sq rt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*A rcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^ (1/4)*e*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(2*a*(c*d^2 - a*e^2)))/(8*a*(c*d^2 - a*e^2))
3.7.42.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 2.95 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(-2 e^{5} c^{3} \left (-\frac {\frac {-\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d +3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d +\sqrt {a c \,e^{2}}\right )}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {3 \left (7 e^{2} a +4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}+\frac {\frac {\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d -3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d -\sqrt {a c \,e^{2}}\right )}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {3 \left (-7 e^{2} a -4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}\right )\) | \(505\) |
| default | \(-2 e^{5} c^{3} \left (-\frac {\frac {-\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d +3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d +\sqrt {a c \,e^{2}}\right )}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {3 \left (7 e^{2} a +4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}+\frac {\frac {\frac {3 \sqrt {a c \,e^{2}}\, \left (2 c d -3 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{3} \left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -11 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3} \left (c d -\sqrt {a c \,e^{2}}\right )}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {3 \left (-7 e^{2} a -4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}\right )\) | \(505\) |
| pseudoelliptic | \(-\frac {e^{5} c^{3} \left (\frac {-\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, \left (-9 a c \,e^{3} x -6 c^{2} d^{2} e x -15 \sqrt {a c \,e^{2}}\, c d e x +19 d \,e^{2} a c +11 \sqrt {a c \,e^{2}}\, a \,e^{2}+8 \sqrt {a c \,e^{2}}\, c \,d^{2}\right )}{\left (c d +\sqrt {a c \,e^{2}}\right ) \left (-c e x +\sqrt {a c \,e^{2}}\right )^{2}}-\frac {3 \left (7 e^{2} a +4 c \,d^{2}+10 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 c^{2} e^{4} a^{2} \sqrt {a c \,e^{2}}\, \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right )}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} d}{2 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) e^{4} a^{2}}+\frac {9 \left (e x +d \right )^{\frac {3}{2}}}{4 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c a \,e^{2} \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {a c \,e^{2}}}+\frac {3 \sqrt {e x +d}\, d}{2 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c \left (-c d +\sqrt {a c \,e^{2}}\right ) e^{4} a^{2}}-\frac {11 \sqrt {e x +d}}{4 \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} c a \,e^{2} \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {a c \,e^{2}}}+\frac {21 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c^{2} \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, e^{2} a \sqrt {a c \,e^{2}}}+\frac {3 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d^{2}}{\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c \,e^{4} a^{2} \sqrt {a c \,e^{2}}}-\frac {15 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d}{2 c^{2} \left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, e^{4} a^{2}}\right )}{8}\) | \(733\) |
-2*e^5*c^3*(-1/16/c/e^4/a^2/(a*c*e^2)^(1/2)*((-3/4*(a*c*e^2)^(1/2)/c^3*(2* c*d+3*(a*c*e^2)^(1/2))/(e^2*a+c*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)+1/4 *(a*c*e^2)^(1/2)/c^3*(6*c*d+11*(a*c*e^2)^(1/2))/(c*d+(a*c*e^2)^(1/2))*(e*x +d)^(1/2))/(-e*x+(a*c*e^2)^(1/2)/c)^2+3/4*(7*e^2*a+4*c*d^2+10*(a*c*e^2)^(1 /2)*d)/c/(e^2*a+c*d^2+2*(a*c*e^2)^(1/2)*d)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2) *arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)))+1/16/c/e^4/a^2/ (a*c*e^2)^(1/2)*((3/4*(a*c*e^2)^(1/2)/c^3*(2*c*d-3*(a*c*e^2)^(1/2))/(e^2*a +c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)-1/4*(a*c*e^2)^(1/2)/c^3*(6*c*d-1 1*(a*c*e^2)^(1/2))/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2))/(-e*x-(a*c*e^2)^(1 /2)/c)^2-3/4*(-7*e^2*a-4*c*d^2+10*(a*c*e^2)^(1/2)*d)/c/(-e^2*a-c*d^2+2*(a* c*e^2)^(1/2)*d)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/(( -c*d+(a*c*e^2)^(1/2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 5776 vs. \(2 (258) = 516\).
Time = 2.45 (sec) , antiderivative size = 5776, normalized size of antiderivative = 18.34 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\int { -\frac {1}{{\left (c x^{2} - a\right )}^{3} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (258) = 516\).
Time = 0.43 (sec) , antiderivative size = 1642, normalized size of antiderivative = 5.21 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
-3/32*(2*(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(c^2*d^3*e - 2*a*c* d*e^3)*abs(c) + (2*sqrt(a*c)*a*c^4*d^8*e - 9*sqrt(a*c)*a^2*c^3*d^6*e^3 + 1 9*sqrt(a*c)*a^3*c^2*d^4*e^5 - 19*sqrt(a*c)*a^4*c*d^2*e^7 + 7*sqrt(a*c)*a^5 *e^9)*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*abs(c) - (4*a^3*c^7*d ^13*e - 25*a^4*c^6*d^11*e^3 + 67*a^5*c^5*d^9*e^5 - 98*a^6*c^4*d^7*e^7 + 82 *a^7*c^3*d^5*e^9 - 37*a^8*c^2*d^3*e^11 + 7*a^9*c*d*e^13)*abs(c))*arctan(sq rt(e*x + d)/sqrt(-(a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4 + sqrt((a ^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 - 3*a^3*c^2 *d^4*e^2 + 3*a^4*c*d^2*e^4 - a^5*e^6)*(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a ^4*c*e^4)))/(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/((a^4*c^5*d^8* e - 4*a^5*c^4*d^6*e^3 + 6*a^6*c^3*d^4*e^5 - 4*a^7*c^2*d^2*e^7 + a^8*c*e^9 - sqrt(a*c)*a^3*c^5*d^9 + 4*sqrt(a*c)*a^4*c^4*d^7*e^2 - 6*sqrt(a*c)*a^5*c^ 3*d^5*e^4 + 4*sqrt(a*c)*a^6*c^2*d^3*e^6 - sqrt(a*c)*a^7*c*d*e^8)*sqrt(-c^2 *d - sqrt(a*c)*c*e)*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)) - 3/32 *(2*(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(sqrt(a*c)*c*d^3*e - 2*s qrt(a*c)*a*d*e^3)*abs(c) - (2*a^2*c^4*d^8*e - 9*a^3*c^3*d^6*e^3 + 19*a^4*c ^2*d^4*e^5 - 19*a^5*c*d^2*e^7 + 7*a^6*e^9)*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2 *e^3 + a^4*e^5)*abs(c) - (4*sqrt(a*c)*a^3*c^6*d^13*e - 25*sqrt(a*c)*a^4*c^ 5*d^11*e^3 + 67*sqrt(a*c)*a^5*c^4*d^9*e^5 - 98*sqrt(a*c)*a^6*c^3*d^7*e^7 + 82*sqrt(a*c)*a^7*c^2*d^5*e^9 - 37*sqrt(a*c)*a^8*c*d^3*e^11 + 7*sqrt(a*...
Time = 12.81 (sec) , antiderivative size = 8961, normalized size of antiderivative = 28.45 \[ \int \frac {1}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
- atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 - 18432*a^6*c^6*d^ 6*e^5 + 38912*a^7*c^5*d^4*e^7 - 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6 *c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5* (a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c)^(1/2)))/(4096*( a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12*c^4*d^6*e^4 + 1 0*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 40 96*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 - 16384 *a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^ 3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c )^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + 54*a*c*d^2*e^7*(a^15*c) ^(1/2)))/(4096*(a^15*c*e^10 - a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 - 10*a^12 *c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 - 5*a^14*c^2*d^2*e^8)))^(1/2) + ((d + e *x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 - 612*a*c^6*d^6*e^4 + 1089*a ^2*c^5*d^4*e^6 - 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7* c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(a^15*c)^(1/2) - 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 - 2 10*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(a^15*c)^(1/2) + 105*a^9*c*d*e^8 + ...