Integrand size = 20, antiderivative size = 231 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}} \]
1/2*(c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(-c*x^2+a)+1/4*arctanh(c^(1/4)*(e*x+d)^( 1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(e*a^(1/2)+d*c^(1/2))^(3/2)*(-3*e*a^(1/2 )+2*d*c^(1/2))/a^(3/2)/c^(7/4)-1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/ 2)+d*c^(1/2))^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(3/2)*(3*e*a^(1/2)+2*d*c^(1/2) )/a^(3/2)/c^(7/4)+1/2*d*e*(e*x+d)^(1/2)/a/c
Time = 1.69 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} c \sqrt {d+e x} \left (c d^2 x+a e (2 d+e x)\right )}{-a+c x^2}-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 c d^2-\sqrt {a} \sqrt {c} d e-3 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (2 c d^2+\sqrt {a} \sqrt {c} d e-3 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^2} \]
((-2*Sqrt[a]*c*Sqrt[d + e*x]*(c*d^2*x + a*e*(2*d + e*x)))/(-a + c*x^2) - S qrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*(2*c*d^2 - Sqrt[a]*Sqrt[c]*d*e - 3*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]*Sqrt[c]*e]*(2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c] *d - Sqrt[a]*e)])/(4*a^(3/2)*c^2)
Time = 0.46 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {495, 27, 653, 25, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {\int -\frac {\sqrt {d+e x} \left (2 c d^2-c e x d-3 a e^2\right )}{2 \left (a-c x^2\right )}dx}{2 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (2 c d^2-c e x d-3 a e^2\right )}{a-c x^2}dx}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {2 d e \sqrt {d+e x}-\frac {\int -\frac {c \left (2 d \left (c d^2-2 a e^2\right )+e \left (c d^2-3 a e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (2 d \left (c d^2-2 a e^2\right )+e \left (c d^2-3 a e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}+2 d e \sqrt {d+e x}}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 d \left (c d^2-2 a e^2\right )+e \left (c d^2-3 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx+2 d e \sqrt {d+e x}}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 \int -\frac {e \left (d \left (c d^2-a e^2\right )+\left (c d^2-3 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}+2 d e \sqrt {d+e x}}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 d e \sqrt {d+e x}-2 \int \frac {e \left (d \left (c d^2-a e^2\right )+\left (c d^2-3 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 d e \sqrt {d+e x}-2 e \int \frac {d \left (c d^2-a e^2\right )+\left (c d^2-3 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 d e \sqrt {d+e x}-2 e \left (\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (3 \sqrt {a} e+2 \sqrt {c} d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 d e \sqrt {d+e x}-2 e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (3 \sqrt {a} e+2 \sqrt {c} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} e}-\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e}\right )}{4 a c}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
((a*e + c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a - c*x^2)) + (2*d*e*Sqrt[d + e*x] - 2*e*(((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTanh [(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)* e) - ((2*Sqrt[c]*d - 3*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c ^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*e)) )/(4*a*c)
3.7.25.3.1 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*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
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_)^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 = 2.44 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(\frac {-\left (-c \,x^{2}+a \right ) \left (\frac {\left (-3 e^{2} a +c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{4}+c d \left (e^{2} a -\frac {c \,d^{2}}{2}\right )\right ) e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\left (-c \,x^{2}+a \right ) \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{4}+c d \left (e^{2} a -\frac {c \,d^{2}}{2}\right )\right ) e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (\frac {c \,d^{2} x}{2}+a e \left (\frac {e x}{2}+d \right )\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a c \left (-c \,x^{2}+a \right )}\) | \(296\) |
| derivativedivides | \(2 e^{3} \left (\frac {\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) d \sqrt {e x +d}}{4 a c \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {-\frac {\left (4 d \,e^{2} a c -2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 d \,e^{2} a c +2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) | \(313\) |
| default | \(2 e^{3} \left (\frac {\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) d \sqrt {e x +d}}{4 a c \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {-\frac {\left (4 d \,e^{2} a c -2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-4 d \,e^{2} a c +2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) | \(313\) |
1/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c*e^ 2)^(1/2)*(-(-c*x^2+a)*(1/4*(-3*a*e^2+c*d^2)*(a*c*e^2)^(1/2)+c*d*(e^2*a-1/2 *c*d^2))*e*((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))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-(-c*x^2+a)*(1 /4*(3*a*e^2-c*d^2)*(a*c*e^2)^(1/2)+c*d*(e^2*a-1/2*c*d^2))*e*arctanh(c*(e*x +d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(1/2*c*d^2*x+a*e*(1/2*e*x+d))*( a*c*e^2)^(1/2)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)))/a/c/(-c*x^2 +a)
Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (172) = 344\).
Time = 0.35 (sec) , antiderivative size = 1370, normalized size of antiderivative = 5.93 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
-1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7))) /(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) + (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 - (2*a^3 *c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e ^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3 *sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3 ))) - (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7))) /(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 - (2*a^3 *c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e ^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3 *sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3 ))) + (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7))) /(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) + (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 + (2*a^3 *c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e ^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*...
Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} - a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (172) = 344\).
Time = 0.38 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.19 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\frac {{\left (2 \, a c^{4} d^{4} e - 4 \, a^{2} c^{3} d^{2} e^{3} - {\left (c d^{2} e - 3 \, a e^{3}\right )} a^{2} c^{2} e^{2} - {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (2 \, \sqrt {a c} a c^{4} d^{4} e - 4 \, \sqrt {a c} a^{2} c^{3} d^{2} e^{3} - {\left (\sqrt {a c} c d^{2} e - 3 \, \sqrt {a c} a e^{3}\right )} a^{2} c^{2} e^{2} + {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{4} d + \sqrt {a c} a^{2} c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} c d^{2} e - \sqrt {e x + d} c d^{3} e + {\left (e x + d\right )}^{\frac {3}{2}} a e^{3} + \sqrt {e x + d} a d e^{3}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \]
1/4*(2*a*c^4*d^4*e - 4*a^2*c^3*d^2*e^3 - (c*d^2*e - 3*a*e^3)*a^2*c^2*e^2 - (sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c*d*e^3)*abs(a)*abs(c)*abs(e))*arctan( sqrt(e*x + d)/sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)* a*c^2))/(a*c^2)))/((a^2*c^3*e - sqrt(a*c)*a*c^3*d)*sqrt(-c^2*d - sqrt(a*c) *c*e)*abs(a)*abs(e)) + 1/4*(2*sqrt(a*c)*a*c^4*d^4*e - 4*sqrt(a*c)*a^2*c^3* d^2*e^3 - (sqrt(a*c)*c*d^2*e - 3*sqrt(a*c)*a*e^3)*a^2*c^2*e^2 + (a*c^3*d^3 *e - a^2*c^2*d*e^3)*abs(a)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(a*c^ 2*d - sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c ^4*d + sqrt(a*c)*a^2*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/2*((e*x + d)^(3/2)*c*d^2*e - sqrt(e*x + d)*c*d^3*e + (e*x + d)^(3/2)*a*e ^3 + sqrt(e*x + d)*a*d*e^3)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a* e^2)*a*c)
Time = 9.98 (sec) , antiderivative size = 1988, normalized size of antiderivative = 8.61 \[ \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
2*atanh((18*a*e^8*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) + (5*d^ 2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((15*d^2*e^9)/c - (43*d^4*e^7) /(4*a) - (27*a*e^11)/(4*c^2) + (5*c*d^6*e^5)/(2*a^2) + (9*d*e^10*(a^9*c^7) ^(1/2))/(4*a^4*c^5) - (7*d^3*e^8*(a^9*c^7)^(1/2))/(2*a^5*c^4) + (5*d^5*e^6 *(a^9*c^7)^(1/2))/(4*a^6*c^3)) - (10*c*d^2*e^6*(d + e*x)^(1/2)*(d^5/(16*a^ 3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(a^9*c^7 )^(1/2))/(64*a^5*c^7) + (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/( (15*d^2*e^9)/c - (43*d^4*e^7)/(4*a) - (27*a*e^11)/(4*c^2) + (5*c*d^6*e^5)/ (2*a^2) + (9*d*e^10*(a^9*c^7)^(1/2))/(4*a^4*c^5) - (7*d^3*e^8*(a^9*c^7)^(1 /2))/(2*a^5*c^4) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^6*c^3)) + (18*d*e^7*(a ^9*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(16*a^3*c) + (15*d*e^4)/(64*a*c^3) - (1 5*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(a^9*c^7)^(1/2))/(64*a^5*c^7) + (5*d^2*e^ 3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2))/((5*a^2*c^4*d^6*e^5)/2 - (27*a^5*c *e^11)/4 - (43*a^3*c^3*d^4*e^7)/4 + 15*a^4*c^2*d^2*e^9 + (9*d*e^10*(a^9*c^ 7)^(1/2))/(4*c^2) + (5*d^5*e^6*(a^9*c^7)^(1/2))/(4*a^2) - (7*d^3*e^8*(a^9* c^7)^(1/2))/(2*a*c)) - (10*d^3*e^5*(a^9*c^7)^(1/2)*(d + e*x)^(1/2)*(d^5/(1 6*a^3*c) + (15*d*e^4)/(64*a*c^3) - (15*d^3*e^2)/(64*a^2*c^2) - (9*e^5*(a^9 *c^7)^(1/2))/(64*a^5*c^7) + (5*d^2*e^3*(a^9*c^7)^(1/2))/(64*a^6*c^6))^(1/2 ))/(15*a^5*c*d^2*e^9 - (27*a^6*e^11)/4 + (5*a^3*c^3*d^6*e^5)/2 - (43*a^...