Integrand size = 20, antiderivative size = 294 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}} \]
1/4*(c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(-c*x^2+a)^2+1/32*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)*(12*c*d^2 +5*a*e^2-18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(9/4)-1/32*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)*(12*c *d^2+5*a*e^2+18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(9/4)+1/16*(a*e*(-5*a*e^2+7 *c*d^2)+2*c*d*(-2*a*e^2+3*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c^2/(-c*x^2+a)
Time = 2.88 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^3 e^3+6 c^3 d^3 x^3-a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )\right )}{\left (a-c x^2\right )^2}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c^2} \]
((-2*Sqrt[a]*Sqrt[d + e*x]*(5*a^3*e^3 + 6*c^3*d^3*x^3 - a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) - a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a - c*x^2 )^2 + ((Sqrt[c]*d + Sqrt[a]*e)^2*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*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] - ((Sqrt[c]*d - Sqrt[a]*e)^2 *(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*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])/(32*a^(5/2)*c^2)
Time = 0.56 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {495, 27, 684, 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.
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}-\frac {\int -\frac {(d+e x)^{3/2} \left (6 c d^2+c e x d-5 a e^2\right )}{2 \left (a-c x^2\right )^2}dx}{4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (6 c d^2+c e x d-5 a e^2\right )}{\left (a-c x^2\right )^2}dx}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {\int -\frac {\left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+2 c d e \left (3 c d^2-4 a e^2\right ) x}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+2 c d e \left (3 c d^2-4 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a c}+\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int -\frac {e \left (\left (6 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+2 c d \left (3 c d^2-4 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}+\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {e \left (\left (6 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+2 c d \left (3 c d^2-4 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {e \int \frac {\left (6 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+2 c d \left (3 c d^2-4 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}\right )}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2}\) |
((a*e + c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a - c*x^2)^2) + ((Sqrt[d + e*x]*(a *e*(7*c*d^2 - 5*a*e^2) + 2*c*d*(3*c*d^2 - 2*a*e^2)*x))/(2*a*c*(a - c*x^2)) - (e*(((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2 *Sqrt[a]*c^(1/4)*e) - ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a ]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e)))/(2*a*c))/(8*a*c)
3.7.38.3.1 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[((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)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
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 = 2.64 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (-\frac {\left (\frac {2 \left (4 a d \,e^{2}-3 d^{3} c \right ) \sqrt {a c \,e^{2}}}{5}+\left (e^{2} a -\frac {4 c \,d^{2}}{5}\right ) \left (e^{2} a -3 c \,d^{2}\right )\right ) c e \left (-c \,x^{2}+a \right )^{2} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\left (-\frac {c e \left (-c \,x^{2}+a \right )^{2} \left (\frac {2 \left (-4 a d \,e^{2}+3 d^{3} c \right ) \sqrt {a c \,e^{2}}}{5}+\left (e^{2} a -\frac {4 c \,d^{2}}{5}\right ) \left (e^{2} a -3 c \,d^{2}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\left (\frac {6 c^{3} d^{3} x^{3}}{5}-2 x \left (\frac {4}{5} x^{2} e^{2}+\frac {1}{10} d e x +d^{2}\right ) d a \,c^{2}-\frac {11 e \,a^{2} \left (\frac {9}{11} x^{2} e^{2}+\frac {4}{11} d e x +d^{2}\right ) c}{5}+a^{3} e^{3}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\right )}{16 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, c^{2} a^{2} \left (-c \,x^{2}+a \right )^{2}}\) | \(377\) |
| default | \(2 e^{5} \left (\frac {\frac {d \left (4 e^{2} a -3 c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (9 a^{2} e^{4}-23 a c \,d^{2} e^{2}+18 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4} c}-\frac {d \left (7 a^{2} e^{4}-16 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4} c}-\frac {\left (e^{2} a -c \,d^{2}\right ) \left (5 a^{2} e^{4}-11 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c^{2}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {-\frac {\left (-5 a^{2} e^{4}+19 a c \,d^{2} e^{2}-12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}-19 a c \,d^{2} e^{2}+12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{4} c}\right )\) | \(449\) |
| derivativedivides | \(-2 e^{5} \left (-\frac {\frac {d \left (4 e^{2} a -3 c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (9 a^{2} e^{4}-23 a c \,d^{2} e^{2}+18 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4} c}-\frac {d \left (7 a^{2} e^{4}-16 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4} c}-\frac {\left (e^{2} a -c \,d^{2}\right ) \left (5 a^{2} e^{4}-11 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c^{2}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {-\frac {\left (-5 a^{2} e^{4}+19 a c \,d^{2} e^{2}-12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}-19 a c \,d^{2} e^{2}+12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{4} c}\right )\) | \(450\) |
-5/16/((-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)*(-1/2*(2/5*(4*a*d*e^2-3*c*d^3)*(a*c*e^2)^(1/2)+(e^2*a-4/5*c*d ^2)*(a*e^2-3*c*d^2))*c*e*(-c*x^2+a)^2*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arct an(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(-1/2*c*e*(-c*x^2+a)^ 2*(2/5*(-4*a*d*e^2+3*c*d^3)*(a*c*e^2)^(1/2)+(e^2*a-4/5*c*d^2)*(a*e^2-3*c*d ^2))*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(6/5*c^3*d^3 *x^3-2*x*(4/5*x^2*e^2+1/10*d*e*x+d^2)*d*a*c^2-11/5*e*a^2*(9/11*x^2*e^2+4/1 1*d*e*x+d^2)*c+a^3*e^3)*(a*c*e^2)^(1/2)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(e *x+d)^(1/2))*((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))/c^2/a^2/(-c*x^2+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 1730 vs. \(2 (236) = 472\).
Time = 0.38 (sec) , antiderivative size = 1730, normalized size of antiderivative = 5.88 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a*c^ 2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^4* e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((3024* c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2 *e^11 + 625*a^4*e^13)*sqrt(e*x + d) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d ^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2)*sqrt((441* c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3 *d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqr t((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c ^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420*a* c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*sqrt((441*c^2*d^ 4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((302 4*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d ^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3 *d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2)*sqrt((44 1*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c ^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^5*c^4*s qrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5 *c^4))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - 420* a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^5*c^4*sqrt((441*c...
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\int { -\frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} - a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (236) = 472\).
Time = 0.43 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.43 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=-\frac {{\left (2 \, {\left (3 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} e^{2} {\left | c \right |} + {\left (6 \, \sqrt {a c} c^{2} d^{4} e - 11 \, \sqrt {a c} a c d^{2} e^{3} + 5 \, \sqrt {a c} a^{2} e^{5}\right )} {\left | c \right |} {\left | e \right |} - {\left (12 \, c^{3} d^{5} e - 19 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, {\left (a^{3} c^{3} e - \sqrt {a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} - \frac {{\left (2 \, {\left (3 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} e^{2} {\left | c \right |} - {\left (6 \, \sqrt {a c} c^{2} d^{4} e - 11 \, \sqrt {a c} a c d^{2} e^{3} + 5 \, \sqrt {a c} a^{2} e^{5}\right )} {\left | c \right |} {\left | e \right |} - {\left (12 \, c^{3} d^{5} e - 19 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, {\left (a^{3} c^{3} e + \sqrt {a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {e x + d} c^{3} d^{6} e - 8 \, {\left (e x + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} + 23 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} - 32 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} + 17 \, \sqrt {e x + d} a c^{2} d^{4} e^{3} - 9 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 14 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 16 \, \sqrt {e x + d} a^{2} c d^{2} e^{5} + 5 \, \sqrt {e x + d} a^{3} e^{7}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c^{2}} \]
-1/32*(2*(3*a*c^2*d^3*e - 4*a^2*c*d*e^3)*e^2*abs(c) + (6*sqrt(a*c)*c^2*d^4 *e - 11*sqrt(a*c)*a*c*d^2*e^3 + 5*sqrt(a*c)*a^2*e^5)*abs(c)*abs(e) - (12*c ^3*d^5*e - 19*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/ sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3) )/(a^2*c^3)))/((a^3*c^3*e - sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d - sqrt(a*c)*c *e)*abs(e)) - 1/32*(2*(3*a*c^2*d^3*e - 4*a^2*c*d*e^3)*e^2*abs(c) - (6*sqrt (a*c)*c^2*d^4*e - 11*sqrt(a*c)*a*c*d^2*e^3 + 5*sqrt(a*c)*a^2*e^5)*abs(c)*a bs(e) - (12*c^3*d^5*e - 19*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*abs(c))*arctan(s qrt(e*x + d)/sqrt(-(a^2*c^3*d - sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2* e^2)*a^2*c^3))/(a^2*c^3)))/((a^3*c^3*e + sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 1/16*(6*(e*x + d)^(7/2)*c^3*d^3*e - 18*(e*x + d )^(5/2)*c^3*d^4*e + 18*(e*x + d)^(3/2)*c^3*d^5*e - 6*sqrt(e*x + d)*c^3*d^6 *e - 8*(e*x + d)^(7/2)*a*c^2*d*e^3 + 23*(e*x + d)^(5/2)*a*c^2*d^2*e^3 - 32 *(e*x + d)^(3/2)*a*c^2*d^3*e^3 + 17*sqrt(e*x + d)*a*c^2*d^4*e^3 - 9*(e*x + d)^(5/2)*a^2*c*e^5 + 14*(e*x + d)^(3/2)*a^2*c*d*e^5 - 16*sqrt(e*x + d)*a^ 2*c*d^2*e^5 + 5*sqrt(e*x + d)*a^3*e^7)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 - a*e^2)^2*a^2*c^2)
Time = 0.74 (sec) , antiderivative size = 2518, normalized size of antiderivative = 8.56 \[ \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
- ((e*(3*c*d^3 - 4*a*d*e^2)*(d + e*x)^(7/2))/(8*a^2) + ((d + e*x)^(3/2)*(7 *a^2*d*e^5 + 9*c^2*d^5*e - 16*a*c*d^3*e^3))/(8*a^2*c) + ((d + e*x)^(1/2)*( 5*a^3*e^7 - 6*c^3*d^6*e + 17*a*c^2*d^4*e^3 - 16*a^2*c*d^2*e^5))/(16*a^2*c^ 2) - (e*(d + e*x)^(5/2)*(9*a^2*e^4 + 18*c^2*d^4 - 23*a*c*d^2*e^2))/(16*a^2 *c))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e *x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c* d^2*e^2) - 2*atanh((25*e^10*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d* e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a ^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2*e^5*(a^15*c^9 )^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^5*e^9)/(2048*a^3) + (325*d*e^ 13)/(2048*a*c^2) - (63*c*d^7*e^7)/(512*a^4) - (449*d^3*e^11)/(1024*a^2*c) + (125*e^14*(a^15*c^9)^(1/2))/(2048*a^8*c^7) - (95*d^2*e^12*(a^15*c^9)^(1/ 2))/(512*a^9*c^6) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^10*c^5) - (63* d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^11*c^4))) - (21*d^2*e^8*(d + e*x)^(1/2)* ((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^ 3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^ 9*c^9) + (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d *e^13)/(2048*c^3) - (63*d^7*e^7)/(512*a^3) - (449*d^3*e^11)/(1024*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) + (125*e^14*(a^15*c^9)^(1/2))/(2048*a^7*c^8) - (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a^8*c^7) + (381*d^4*e^10*(a^15*c^9...