Integrand size = 20, antiderivative size = 263 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}} \]
1/2*d*e*(e*x+d)^(3/2)/a/c+1/2*(c*d*x+a*e)*(e*x+d)^(5/2)/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))^(5/2)*(-5*e*a^(1/2)+2*d*c^(1/2))/a^(3/2)/c^(9/4)-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))^(5/2 )*(5*e*a^(1/2)+2*d*c^(1/2))/a^(3/2)/c^(9/4)+1/2*e*(5*a*e^2+c*d^2)*(e*x+d)^ (1/2)/a/c^2
Time = 1.43 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^2 e^3+c^2 d^3 x+a c e \left (3 d^2+3 d e x-4 e^2 x^2\right )\right )}{-a+c x^2}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2} c^2} \]
((-2*Sqrt[a]*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(3*d^2 + 3*d*e*x - 4*e^2*x^2)))/(-a + c*x^2) + ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + S qrt[a]*e)^3*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] - ((Sqrt[c]*d - Sqrt[ a]*e)^3*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*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])/(4*a^(3/2)*c^2)
Time = 0.59 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {495, 27, 653, 25, 27, 653, 25, 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.
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {\int -\frac {(d+e x)^{3/2} \left (2 c d^2-3 c e x d-5 a e^2\right )}{2 \left (a-c x^2\right )}dx}{2 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (2 c d^2-3 c e x d-5 a e^2\right )}{a-c x^2}dx}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {2 d e (d+e x)^{3/2}-\frac {\int -\frac {c \sqrt {d+e x} \left (2 d \left (c d^2-4 a e^2\right )-e \left (c d^2+5 a e^2\right ) x\right )}{a-c x^2}dx}{c}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \sqrt {d+e x} \left (2 d \left (c d^2-4 a e^2\right )-e \left (c d^2+5 a e^2\right ) x\right )}{a-c x^2}dx}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (2 d \left (c d^2-4 a e^2\right )-e \left (c d^2+5 a e^2\right ) x\right )}{a-c x^2}dx+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (c d^2-5 a e^2\right ) \left (2 c d^2+a e^2\right )+c d e \left (c d^2-13 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (c d^2-5 a e^2\right ) \left (2 c d^2+a e^2\right )+c d e \left (c d^2-13 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {2 \int -\frac {e \left (\left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )+c d \left (c d^2-13 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {2 \int \frac {e \left (\left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )+c d \left (c d^2-13 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 e \int \frac {\left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )+c d \left (c d^2-13 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {2 e \left (\frac {\sqrt {c} \left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {2 e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}-\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}\right )}{c}+\frac {2 e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{c}+2 d e (d+e x)^{3/2}}{4 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
((a*e + c*d*x)*(d + e*x)^(5/2))/(2*a*c*(a - c*x^2)) + ((2*e*(c*d^2 + 5*a*e ^2)*Sqrt[d + e*x])/c + 2*d*e*(d + e*x)^(3/2) - (2*e*(((Sqrt[c]*d - Sqrt[a] *e)^(5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt [Sqrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e) - ((2*Sqrt[c]*d - 5*Sqrt[a ]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sq rt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e)))/c)/(4*a*c)
3.7.24.3.1 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 = 2.56 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (\left (-c \,x^{2}+a \right ) c \left (\frac {\left (-13 a d \,e^{2}+d^{3} c \right ) \sqrt {a c \,e^{2}}}{5}+a^{2} e^{4}+\frac {9 a c \,d^{2} e^{2}}{5}-\frac {2 c^{2} d^{4}}{5}\right ) e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (-c \,x^{2}+a \right ) \left (\frac {\left (13 a d \,e^{2}-d^{3} c \right ) \sqrt {a c \,e^{2}}}{5}+a^{2} e^{4}+\frac {9 a c \,d^{2} e^{2}}{5}-\frac {2 c^{2} d^{4}}{5}\right ) c e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )-2 \sqrt {a c \,e^{2}}\, \left (\frac {c^{2} d^{3} x}{5}+\frac {3 \left (-\frac {4}{3} x^{2} e^{2}+d e x +d^{2}\right ) e a c}{5}+a^{2} e^{3}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\right )\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, c^{2} \left (-c \,x^{2}+a \right ) a}\) | \(341\) |
| derivativedivides | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}-\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-5 a^{2} e^{4}-9 a c \,d^{2} e^{2}+2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}+9 a c \,d^{2} e^{2}-2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}}{c^{2}}\right )\) | \(345\) |
| default | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}-\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-5 a^{2} e^{4}-9 a c \,d^{2} e^{2}+2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}+9 a c \,d^{2} e^{2}-2 c^{2} d^{4}-13 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}}{c^{2}}\right )\) | \(345\) |
| risch | \(\frac {2 e^{3} \sqrt {e x +d}}{c^{2}}+\frac {2 e^{3} \left (\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}-\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{c \left (e x +d \right )^{2}-2 c d \left (e x +d \right )-e^{2} a +c \,d^{2}}+\frac {c \left (\frac {\left (-5 a^{2} e^{4}-9 a c \,d^{2} e^{2}+2 c^{2} d^{4}+13 \sqrt {a c \,e^{2}}\, a d \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (5 a^{2} e^{4}+9 a c \,d^{2} e^{2}-2 c^{2} d^{4}+13 \sqrt {a c \,e^{2}}\, a d \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 e^{2} a}\right )}{c^{2}}\) | \(348\) |
-5/4/((-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)*c*(1/5*(-13*a*d*e^2+c*d^3)*(a*c*e^2)^(1/2)+a^2*e^4 +9/5*a*c*d^2*e^2-2/5*c^2*d^4)*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)*(1/5*(13*a*d*e^2-c*d^3)*(a*c*e^2)^(1/2)+a^2*e^4+9/5*a*c* d^2*e^2-2/5*c^2*d^4)*c*e*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c) ^(1/2))-2*(a*c*e^2)^(1/2)*(1/5*c^2*d^3*x+3/5*(-4/3*x^2*e^2+d*e*x+d^2)*e*a* c+a^2*e^3)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(e*x+d)^(1/2)))/c^2/(-c*x^2+a)/ a
Leaf count of result is larger than twice the leaf count of optimal. 2073 vs. \(2 (200) = 400\).
Time = 0.53 (sec) , antiderivative size = 2073, normalized size of antiderivative = 7.88 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
1/8*((a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d ^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6* e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^ 9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3* d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt( e*x + d) + (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 - 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 - 4*a^4*c^7*d*e^2)*sqrt((1225*c^4*d^8* e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d ^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6* e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^ 9)))/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^ 2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10 780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4 *e^14)/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 + 625*a ^5*e^13)*sqrt(e*x + d) - (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^ 4*c^3*d^2*e^8 - 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 - 4*a^4*c^7*d*e^2)*sqrt( (1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^ 3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5...
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} - a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (200) = 400\).
Time = 0.38 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.24 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e^{3}}{c^{2}} + \frac {{\left ({\left (\sqrt {a c} c d^{3} e - 13 \, \sqrt {a c} a d e^{3}\right )} a^{2} e^{2} {\left | c \right |} + {\left (a c^{2} d^{4} e + 4 \, a^{2} c d^{2} e^{3} - 5 \, a^{3} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} - {\left (2 \, \sqrt {a c} a c^{2} d^{5} e - 9 \, \sqrt {a c} a^{2} c d^{3} e^{3} - 5 \, \sqrt {a c} a^{3} d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d - \sqrt {a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left ({\left (c^{2} d^{3} e - 13 \, a c d e^{3}\right )} a^{2} e^{2} {\left | c \right |} - {\left (\sqrt {a c} c^{2} d^{4} e + 4 \, \sqrt {a c} a c d^{2} e^{3} - 5 \, \sqrt {a c} a^{2} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} - {\left (2 \, a c^{3} d^{5} e - 9 \, a^{2} c^{2} d^{3} e^{3} - 5 \, a^{3} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - \sqrt {e x + d} c^{2} d^{4} e + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d e^{3} + \sqrt {e x + d} a^{2} e^{5}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c^{2}} \]
2*sqrt(e*x + d)*e^3/c^2 + 1/4*((sqrt(a*c)*c*d^3*e - 13*sqrt(a*c)*a*d*e^3)* a^2*e^2*abs(c) + (a*c^2*d^4*e + 4*a^2*c*d^2*e^3 - 5*a^3*e^5)*abs(a)*abs(c) *abs(e) - (2*sqrt(a*c)*a*c^2*d^5*e - 9*sqrt(a*c)*a^2*c*d^3*e^3 - 5*sqrt(a* c)*a^3*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^3*d + sqrt(a^2*c^6*d ^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*d - sqrt(a*c)*a ^2*c^2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/4*((c^2*d^3*e - 13*a*c*d*e^3)*a^2*e^2*abs(c) - (sqrt(a*c)*c^2*d^4*e + 4*sqrt(a*c)*a*c*d^2* e^3 - 5*sqrt(a*c)*a^2*e^5)*abs(a)*abs(c)*abs(e) - (2*a*c^3*d^5*e - 9*a^2*c ^2*d^3*e^3 - 5*a^3*c*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^3*d - sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((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)) - 1/2*(( e*x + d)^(3/2)*c^2*d^3*e - sqrt(e*x + d)*c^2*d^4*e + 3*(e*x + d)^(3/2)*a*c *d*e^3 + sqrt(e*x + d)*a^2*e^5)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)*a*c^2)
Time = 10.35 (sec) , antiderivative size = 4090, normalized size of antiderivative = 15.55 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (3 5*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2 ))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3* (a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((491*a*d^3*e^11)/(2*c) - (885*d ^5*e^9)/2 + (329*c*d^7*e^7)/(2*a) + (50*a^2*d*e^13)/c^2 - (35*c^2*d^9*e^5) /(2*a^2) + (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^7) + (335*d^2*e^12*(a^9*c^9 )^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e ^8*(a^9*c^9)^(1/2))/(2*a^5*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^3) ) - (d^3*e^7*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/( 16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7* (a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*a^3*c^4*d^7 *e^7)/2 - (35*a^2*c^5*d^9*e^5)/2 - (885*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2* d^3*e^11)/2 + 50*a^6*c*d*e^13 + (125*a^2*e^14*(a^9*c^9)^(1/2))/(4*c^4) + ( 35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^2) - (204*d^4*e^10*(a^9*c^9)^(1/2))/c^2 + (335*a*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2 *a*c)) + (d^5*e^5*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25 *e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5* c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/(50*a^7*d*...