Integrand size = 20, antiderivative size = 149 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 e \sqrt {d+e x}}{c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \]
-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)/c^(5/4)/a^(1/2)+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)/c^(5/4)/a^(1/2)-2*e*(e*x+d)^( 1/2)/c
Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\frac {-2 e \sqrt {d+e x}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{c} \]
(-2*e*Sqrt[d + e*x] + ((Sqrt[c]*d + Sqrt[a]*e)^2*ArcTan[(Sqrt[-(c*d) - Sqr t[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c *d) - Sqrt[a]*Sqrt[c]*e]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*Sq rt[-(c*d) + Sqrt[a]*Sqrt[c]*e]))/c
Time = 0.37 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {481, 25, 654, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx\) |
| \(\Big \downarrow \) 481 |
| \(\displaystyle -\frac {\int -\frac {c d^2+2 c e x d+a e^2}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 e \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c d^2+2 c e x d+a e^2}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 e \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 \int \frac {e \left (c d^2-2 c (d+e x) d-a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e \int \frac {c d^2-2 c (d+e x) d-a e^2}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 e \left (\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{c}-\frac {2 e \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 e \left (\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}\right )}{c}-\frac {2 e \sqrt {d+e x}}{c}\) |
(-2*e*Sqrt[d + e*x])/c + (2*e*(-1/2*((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh [(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*e) + ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqr t[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e)))/c
3.7.13.3.1 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), x_Symbol] :> Simp[d*((c + d*x)^(n - 1)/(b*(n - 1))), x] + Simp[1/b Int[(c + d*x)^(n - 2)*(Simp[b *c^2 - a*d^2 + 2*b*c*d*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 1]
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 = 2.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(-2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(168\) |
| pseudoelliptic | \(-e \left (\frac {2 \sqrt {e x +d}}{c}+\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(168\) |
| default | \(2 e \left (-\frac {\sqrt {e x +d}}{c}+\frac {\left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-e^{2} a -c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(169\) |
| risch | \(-\frac {2 e \sqrt {e x +d}}{c}-2 e \left (\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(171\) |
-2*e*(1/c*(e*x+d)^(1/2)+1/2*(-e^2*a-c*d^2+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+c*d^2+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. 974 vs. \(2 (107) = 214\).
Time = 0.31 (sec) , antiderivative size = 974, normalized size of antiderivative = 6.54 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\frac {c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} - a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} - a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} + a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} + a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - 4 \, \sqrt {e x + d} e}{2 \, c} \]
1/2*(c*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5 )*sqrt(e*x + d) + (3*a*c^2*d^2*e^2 + a^2*c*e^4 - a*c^4*d*sqrt((9*c^2*d^4*e ^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sq rt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt( (c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/ (a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 + a^2*c*e^4 - a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d ^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^ 4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) + c*sqrt((c*d^3 + 3*a *d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a *c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) + (3*a*c ^2*d^2*e^2 + a^2*c*e^4 + a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2 *e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a* c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt((c*d^3 + 3*a*d*e^2 - a*c ^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(- (3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 + a^2*c*e^4 + a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5 )))*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - 4*sqrt(e*x + d)*e)/c
\[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=- \int \frac {d \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {e x \sqrt {d + e x}}{- a + c x^{2}}\, dx \]
\[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\int { -\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (107) = 214\).
Time = 0.32 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 \, \sqrt {e x + d} e}{c} - \frac {{\left (\sqrt {a c} c^{3} d^{3} e - \sqrt {a c} a c^{2} d e^{3} + {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} c^{3} d^{3} e - \sqrt {a c} a c^{2} d e^{3} - {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \]
-2*sqrt(e*x + d)*e/c - (sqrt(a*c)*c^3*d^3*e - sqrt(a*c)*a*c^2*d*e^3 + (a*c ^2*d^2*e - a^2*c*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d + s qrt(c^4*d^2 - (c^2*d^2 - a*c*e^2)*c^2))/c^2))/((a*c^3*d - sqrt(a*c)*a*c^2* e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + (sqrt(a*c)*c^3*d^3*e - sqrt(a*c) *a*c^2*d*e^3 - (a*c^2*d^2*e - a^2*c*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d - sqrt(c^4*d^2 - (c^2*d^2 - a*c*e^2)*c^2))/c^2))/((a*c^3*d + sqrt(a*c)*a*c^2*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e))
Time = 9.63 (sec) , antiderivative size = 1581, normalized size of antiderivative = 10.61 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \]
2*atanh((32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) - ( e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1 /2))/(16*a^2*d*e^7 - 48*c^2*d^5*e^3 - (16*a*e^8*(a^3*c^5)^(1/2))/c^3 + 32* a*c*d^3*e^5 - (32*d^2*e^6*(a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(a^3*c^5)^(1/ 2))/(a*c)) - (32*d*e^5*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(a^3*c^5)^(1/2) )/(4*a^2*c^4))^(1/2))/(48*c^3*d^5*e^3 - 32*a*c^2*d^3*e^5 + (16*a*e^8*(a^3* c^5)^(1/2))/c^2 - 16*a^2*c*d*e^7 - (48*d^4*e^4*(a^3*c^5)^(1/2))/a + (32*d^ 2*e^6*(a^3*c^5)^(1/2))/c) + (96*d^3*e^3*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*(( 3*d*e^2)/(4*c^2) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2* e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^3*d*e^7 - 48*a*c^2*d^5*e^3 + 32*a^2*c*d^3*e^5 - (16*a^2*e^8*(a^3*c^5)^(1/2))/c^3 + (48*d^4*e^4*(a^3*c^5 )^(1/2))/c - (32*a*d^2*e^6*(a^3*c^5)^(1/2))/c^2) + (96*a*c^2*d^2*e^4*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) + d^3/(4*a*c) - (e^3*(a^3*c^5)^(1/2))/(4*a*c ^5) - (3*d^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7 - 48*c^2 *d^5*e^3 - (16*a*e^8*(a^3*c^5)^(1/2))/c^3 + 32*a*c*d^3*e^5 - (32*d^2*e^6*( a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(a^3*c^5)^(1/2))/(a*c)))*((a*c^4*d^3 - a *e^3*(a^3*c^5)^(1/2) + 3*a^2*c^3*d*e^2 - 3*c*d^2*e*(a^3*c^5)^(1/2))/(4*a^2 *c^5))^(1/2) - 2*atanh((32*d*e^5*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2 )/(4*c^2) + d^3/(4*a*c) + (e^3*(a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(a...