Integrand size = 20, antiderivative size = 209 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\frac {(a e+c d x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (2 \sqrt {c} d+\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4}}+\frac {\left (2 \sqrt {c} d-\sqrt {a} e\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4}} \]
1/2*(c*d*x+a*e)*(e*x+d)^(1/2)/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)+2*d*c^(1/2))*(-e*a^(1/2)+d*c ^(1/2))^(1/2)/a^(3/2)/c^(5/4)+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)+2*d*c^(1/2))*(e*a^(1/2)+d*c^(1/2))^(1/2)/a^ (3/2)/c^(5/4)
Time = 1.44 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {c} (a e+c d x) \sqrt {d+e x}}{a-c x^2}-\left (2 \sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\left (2 \sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^{3/2}} \]
((2*Sqrt[a]*Sqrt[c]*(a*e + c*d*x)*Sqrt[d + e*x])/(a - c*x^2) - (2*Sqrt[c]* d - Sqrt[a]*e)*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) - Sqrt [a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)] + (2*Sqrt[c]*d + Sq rt[a]*e)*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sq rt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(4*a^(3/2)*c^(3/2))
Time = 0.40 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {495, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {\int -\frac {2 c d^2+c e x d-a e^2}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 c d^2+c e x d-a e^2}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\int -\frac {e \left (c d^2+c (d+e x) d-a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}+\frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {e \left (c d^2+c (d+e x) d-a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {e \int \frac {c d^2+c (d+e x) d-a e^2}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {1}{2} \sqrt {c} \left (\sqrt {c} d-\frac {2 c d^2-a e^2}{\sqrt {a} e}\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}+\frac {1}{2} \sqrt {c} \left (\frac {2 c d^2-a e^2}{\sqrt {a} e}+\sqrt {c} d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}\right )}{2 a c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )}-\frac {e \left (-\frac {\left (\sqrt {c} d-\frac {2 c d^2-a e^2}{\sqrt {a} e}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\frac {2 c d^2-a e^2}{\sqrt {a} e}+\sqrt {c} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt [4]{c} \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 a c}\) |
((a*e + c*d*x)*Sqrt[d + e*x])/(2*a*c*(a - c*x^2)) - (e*(-1/2*((Sqrt[c]*d - (2*c*d^2 - a*e^2)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[ c]*d - Sqrt[a]*e]])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((Sqrt[c]*d + (2*c*d^2 - a*e^2)/(Sqrt[a]*e))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c ]*d + Sqrt[a]*e]])/(2*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(2*a*c)
3.7.26.3.1 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_)^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 = 2.32 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(2 e^{3} \left (\frac {\frac {d \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}{4 a \,e^{2} c}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {\frac {\left (-e^{2} a +2 c \,d^{2}-\sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a -2 c \,d^{2}-\sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) | \(250\) |
| default | \(2 e^{3} \left (\frac {\frac {d \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}{4 a \,e^{2} c}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {\frac {\left (-e^{2} a +2 c \,d^{2}-\sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a -2 c \,d^{2}-\sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) | \(250\) |
| pseudoelliptic | \(\frac {-\frac {\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c \,x^{2}+a \right ) c e \left (e^{2} a -2 c \,d^{2}+\sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {c e \left (-c \,x^{2}+a \right ) \left (e^{2} a -2 c \,d^{2}-\sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (c d x +a e \right ) \sqrt {e x +d}\right )}{2 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a c \left (-c \,x^{2}+a \right )}\) | \(258\) |
2*e^3*((1/4*d/a/e^2*(e*x+d)^(3/2)+1/4*(a*e^2-c*d^2)/a/e^2/c*(e*x+d)^(1/2)) /(-c*(e*x+d)^2+2*c*d*(e*x+d)+e^2*a-c*d^2)+1/4/a/e^2*(1/2*(-e^2*a+2*c*d^2-( a*c*e^2)^(1/2)*d)/(a*c*e^2)^(1/2)/((-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))-1/2*(e^2*a-2*c*d^2-(a*c* e^2)^(1/2)*d)/(a*c*e^2)^(1/2)/((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))))
Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (154) = 308\).
Time = 0.31 (sec) , antiderivative size = 679, normalized size of antiderivative = 3.25 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=-\frac {{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {e x + d} + {\left (2 \, a^{3} c^{4} d \sqrt {\frac {e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) - {\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {e x + d} - {\left (2 \, a^{3} c^{4} d \sqrt {\frac {e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) - {\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {e x + d} + {\left (2 \, a^{3} c^{4} d \sqrt {\frac {e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + {\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt {e x + d} - {\left (2 \, a^{3} c^{4} d \sqrt {\frac {e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + 4 \, {\left (c d x + a e\right )} \sqrt {e x + d}}{8 \, {\left (a c^{2} x^{2} - a^{2} c\right )}} \]
-1/8*((a*c^2*x^2 - a^2*c)*sqrt((a^3*c^2*sqrt(e^6/(a^3*c^5)) + 4*c*d^3 - 3* a*d*e^2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) + (2*a^3*c^4* d*sqrt(e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt((a^3*c^2*sqrt(e^6/(a^3*c^5)) + 4*c *d^3 - 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 - a^2*c)*sqrt((a^3*c^2*sqrt(e^6 /(a^3*c^5)) + 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*s qrt(e*x + d) - (2*a^3*c^4*d*sqrt(e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt((a^3*c^2 *sqrt(e^6/(a^3*c^5)) + 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 - a^2 *c)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c^5)) - 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))*l og(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c^5)) - 4*c*d^3 + 3*a*d*e^2)/(a ^3*c^2))) + (a*c^2*x^2 - a^2*c)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c^5)) - 4*c*d ^3 + 3*a*d*e^2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) - (2*a ^3*c^4*d*sqrt(e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c^5 )) - 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))) + 4*(c*d*x + a*e)*sqrt(e*x + d))/(a* c^2*x^2 - a^2*c)
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} - a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (154) = 308\).
Time = 0.36 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.98 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=\frac {{\left (2 \, a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3} - {\left (\sqrt {a c} c d^{2} e - \sqrt {a c} a e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e - \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (2 \, a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3} + {\left (\sqrt {a c} c d^{2} e - \sqrt {a c} a e^{3}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e + \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |} {\left | e \right |}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} c d e - \sqrt {e x + d} c d^{2} e + \sqrt {e x + d} a e^{3}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \]
1/4*(2*a*c^3*d^3*e - 2*a^2*c^2*d*e^3 - (sqrt(a*c)*c*d^2*e - sqrt(a*c)*a*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^2*e - sqrt(a*c)* a*c^2*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a)*abs(e)) + 1/4*(2*a*c^3*d^3*e - 2*a^2*c^2*d*e^3 + (sqrt(a*c)*c*d^2*e - sqrt(a*c)*a*e^3)*abs(a)*abs(c)*ab s(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^2*e + sqrt(a*c)*a*c^2*d)*sqrt(-c^2* d + sqrt(a*c)*c*e)*abs(a)*abs(e)) - 1/2*((e*x + d)^(3/2)*c*d*e - sqrt(e*x + d)*c*d^2*e + sqrt(e*x + d)*a*e^3)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c* d^2 - a*e^2)*a*c)
Time = 0.44 (sec) , antiderivative size = 704, normalized size of antiderivative = 3.37 \[ \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^2} \, dx=2\,\mathrm {atanh}\left (\frac {2\,c\,e^6\,\sqrt {d+e\,x}\,\sqrt {\frac {d^3}{16\,a^3\,c}-\frac {3\,d\,e^2}{64\,a^2\,c^2}-\frac {e^3\,\sqrt {a^9\,c^5}}{64\,a^6\,c^5}}}{\frac {d\,e^7}{2\,a}-\frac {c\,d^3\,e^5}{2\,a^2}+\frac {e^8\,\sqrt {a^9\,c^5}}{4\,a^5\,c^3}-\frac {d^2\,e^6\,\sqrt {a^9\,c^5}}{4\,a^6\,c^2}}+\frac {2\,d\,e^5\,\sqrt {a^9\,c^5}\,\sqrt {d+e\,x}\,\sqrt {\frac {d^3}{16\,a^3\,c}-\frac {3\,d\,e^2}{64\,a^2\,c^2}-\frac {e^3\,\sqrt {a^9\,c^5}}{64\,a^6\,c^5}}}{\frac {e^8\,\sqrt {a^9\,c^5}}{4\,c^2}-\frac {a^3\,c^2\,d^3\,e^5}{2}+\frac {a^4\,c\,d\,e^7}{2}-\frac {d^2\,e^6\,\sqrt {a^9\,c^5}}{4\,a\,c}}\right )\,\sqrt {-\frac {e^3\,\sqrt {a^9\,c^5}-4\,a^3\,c^4\,d^3+3\,a^4\,c^3\,d\,e^2}{64\,a^6\,c^5}}-\frac {\frac {\left (a\,e^3-c\,d^2\,e\right )\,\sqrt {d+e\,x}}{2\,a\,c}+\frac {d\,e\,{\left (d+e\,x\right )}^{3/2}}{2\,a}}{c\,{\left (d+e\,x\right )}^2-a\,e^2+c\,d^2-2\,c\,d\,\left (d+e\,x\right )}+2\,\mathrm {atanh}\left (\frac {2\,c\,e^6\,\sqrt {d+e\,x}\,\sqrt {\frac {d^3}{16\,a^3\,c}-\frac {3\,d\,e^2}{64\,a^2\,c^2}+\frac {e^3\,\sqrt {a^9\,c^5}}{64\,a^6\,c^5}}}{\frac {d\,e^7}{2\,a}-\frac {c\,d^3\,e^5}{2\,a^2}-\frac {e^8\,\sqrt {a^9\,c^5}}{4\,a^5\,c^3}+\frac {d^2\,e^6\,\sqrt {a^9\,c^5}}{4\,a^6\,c^2}}+\frac {2\,d\,e^5\,\sqrt {a^9\,c^5}\,\sqrt {d+e\,x}\,\sqrt {\frac {d^3}{16\,a^3\,c}-\frac {3\,d\,e^2}{64\,a^2\,c^2}+\frac {e^3\,\sqrt {a^9\,c^5}}{64\,a^6\,c^5}}}{\frac {e^8\,\sqrt {a^9\,c^5}}{4\,c^2}+\frac {a^3\,c^2\,d^3\,e^5}{2}-\frac {a^4\,c\,d\,e^7}{2}-\frac {d^2\,e^6\,\sqrt {a^9\,c^5}}{4\,a\,c}}\right )\,\sqrt {\frac {e^3\,\sqrt {a^9\,c^5}+4\,a^3\,c^4\,d^3-3\,a^4\,c^3\,d\,e^2}{64\,a^6\,c^5}} \]
2*atanh((2*c*e^6*(d + e*x)^(1/2)*(d^3/(16*a^3*c) - (3*d*e^2)/(64*a^2*c^2) - (e^3*(a^9*c^5)^(1/2))/(64*a^6*c^5))^(1/2))/((d*e^7)/(2*a) - (c*d^3*e^5)/ (2*a^2) + (e^8*(a^9*c^5)^(1/2))/(4*a^5*c^3) - (d^2*e^6*(a^9*c^5)^(1/2))/(4 *a^6*c^2)) + (2*d*e^5*(a^9*c^5)^(1/2)*(d + e*x)^(1/2)*(d^3/(16*a^3*c) - (3 *d*e^2)/(64*a^2*c^2) - (e^3*(a^9*c^5)^(1/2))/(64*a^6*c^5))^(1/2))/((e^8*(a ^9*c^5)^(1/2))/(4*c^2) - (a^3*c^2*d^3*e^5)/2 + (a^4*c*d*e^7)/2 - (d^2*e^6* (a^9*c^5)^(1/2))/(4*a*c)))*(-(e^3*(a^9*c^5)^(1/2) - 4*a^3*c^4*d^3 + 3*a^4* c^3*d*e^2)/(64*a^6*c^5))^(1/2) - (((a*e^3 - c*d^2*e)*(d + e*x)^(1/2))/(2*a *c) + (d*e*(d + e*x)^(3/2))/(2*a))/(c*(d + e*x)^2 - a*e^2 + c*d^2 - 2*c*d* (d + e*x)) + 2*atanh((2*c*e^6*(d + e*x)^(1/2)*(d^3/(16*a^3*c) - (3*d*e^2)/ (64*a^2*c^2) + (e^3*(a^9*c^5)^(1/2))/(64*a^6*c^5))^(1/2))/((d*e^7)/(2*a) - (c*d^3*e^5)/(2*a^2) - (e^8*(a^9*c^5)^(1/2))/(4*a^5*c^3) + (d^2*e^6*(a^9*c ^5)^(1/2))/(4*a^6*c^2)) + (2*d*e^5*(a^9*c^5)^(1/2)*(d + e*x)^(1/2)*(d^3/(1 6*a^3*c) - (3*d*e^2)/(64*a^2*c^2) + (e^3*(a^9*c^5)^(1/2))/(64*a^6*c^5))^(1 /2))/((e^8*(a^9*c^5)^(1/2))/(4*c^2) + (a^3*c^2*d^3*e^5)/2 - (a^4*c*d*e^7)/ 2 - (d^2*e^6*(a^9*c^5)^(1/2))/(4*a*c)))*((e^3*(a^9*c^5)^(1/2) + 4*a^3*c^4* d^3 - 3*a^4*c^3*d*e^2)/(64*a^6*c^5))^(1/2)