Integrand size = 25, antiderivative size = 197 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]
arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(B*a^(1/2)-A*c ^(1/2))/c^(1/4)/a^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(3/2)+arctanh(c^(1/4)*(e*x+ d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(B*a^(1/2)+A*c^(1/2))/c^(1/4)/a^(1/2 )/(e*a^(1/2)+d*c^(1/2))^(3/2)-2*(-A*e+B*d)/(-a*e^2+c*d^2)/(e*x+d)^(1/2)
Time = 0.61 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\frac {-2 B d+2 A e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e}} \]
(-2*B*d + 2*A*e)/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B + A*Sqrt[c] )*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqr t[a]*e)])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e ]) + ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqr t[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)*Sqr t[-(c*d) + Sqrt[a]*Sqrt[c]*e])
Time = 0.45 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {655, 25, 654, 25, 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.
\(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right ) (d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 655 |
\(\displaystyle -\frac {\int -\frac {A c d-a B e+c (B d-A e) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {A c d-a B e+c (B d-A e) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 654 |
\(\displaystyle \frac {2 \int -\frac {2 A c d e-B \left (c d^2+a e^2\right )+c (B d-A e) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \int \frac {2 A c d e-B \left (c d^2+a e^2\right )+c (B d-A e) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle \frac {2 \left (-\frac {\sqrt {c} \left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}-\frac {\sqrt {c} \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}\right )}{c d^2-a e^2}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \left (\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{c d^2-a e^2}-\frac {2 (B d-A e)}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
(-2*(B*d - A*e))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + (2*(((Sqrt[a]*B - A*Sqr t[c])*(Sqrt[c]*d + Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c] *d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((Sqrt [a]*B + A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x]) /Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[a] *e])))/(c*d^2 - a*e^2)
3.15.51.3.1 Defintions of rubi rules used
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[((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), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) ), x] + Simp[1/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1]
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 = 0.45 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {-2 A e +2 B d}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}-\frac {2 c \left (-\frac {\left (-A c d e +B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{e^{2} a -c \,d^{2}}\) | \(229\) |
default | \(-\frac {2 \left (A e -B d \right )}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}+\frac {2 c \left (-\frac {\left (A c d e -B a \,e^{2}-A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A c d e +B a \,e^{2}-A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{e^{2} a -c \,d^{2}}\) | \(229\) |
pseudoelliptic | \(-\frac {\sqrt {e x +d}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c \left (A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (c \left (\left (-A e +B d \right ) \sqrt {a c \,e^{2}}+e \left (A c d -B a e \right )\right ) \sqrt {e x +d}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (A e -B d \right )\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}{\sqrt {e x +d}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (e^{2} a -c \,d^{2}\right )}\) | \(272\) |
2*(-A*e+B*d)/(a*e^2-c*d^2)/(e*x+d)^(1/2)-2/(a*e^2-c*d^2)*c*(-1/2*(-A*c*d*e +B*a*e^2+A*(a*c*e^2)^(1/2)*e-B*(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))+1/2*(A*c*d*e-B*a*e^2+A*(a*c*e^2)^(1/2)*e-B*(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)))
Leaf count of result is larger than twice the leaf count of optimal. 6448 vs. \(2 (147) = 294\).
Time = 4.30 (sec) , antiderivative size = 6448, normalized size of antiderivative = 32.73 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=- \int \frac {A}{- a d \sqrt {d + e x} - a e x \sqrt {d + e x} + c d x^{2} \sqrt {d + e x} + c e x^{3} \sqrt {d + e x}}\, dx - \int \frac {B x}{- a d \sqrt {d + e x} - a e x \sqrt {d + e x} + c d x^{2} \sqrt {d + e x} + c e x^{3} \sqrt {d + e x}}\, dx \]
-Integral(A/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x) - Integral(B*x/(-a*d*sqrt(d + e*x) - a* e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\int { -\frac {B x + A}{{\left (c x^{2} - a\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 941 vs. \(2 (147) = 294\).
Time = 0.41 (sec) , antiderivative size = 941, normalized size of antiderivative = 4.78 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=-\frac {2 \, {\left (B d - A e\right )}}{{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d}} + \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} B a d {\left | c \right |} - {\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} A a e {\left | c \right |} + 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (a c^{2} d^{4} - a^{3} e^{4}\right )} B {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} A {\left | c \right |} + {\left (\sqrt {a c} a c^{2} d^{5} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{3} e^{4} + \sqrt {a c} a^{3} d e^{6}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} + \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4} - \sqrt {a c} a c^{2} d^{4} e + 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} - \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} - \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} B a d {\left | c \right |} - {\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} A a e {\left | c \right |} - 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} + {\left (a c^{2} d^{4} - a^{3} e^{4}\right )} B {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} A {\left | c \right |} + {\left (\sqrt {a c} a c^{2} d^{5} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{3} e^{4} + \sqrt {a c} a^{3} d e^{6}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} - \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4} + \sqrt {a c} a c^{2} d^{4} e - 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} + \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} \]
-2*(B*d - A*e)/((c*d^2 - a*e^2)*sqrt(e*x + d)) + ((c*d^2*e - a*e^3)^2*sqrt (a*c)*B*a*d*abs(c) - (c*d^2*e - a*e^3)^2*sqrt(a*c)*A*a*e*abs(c) + 2*(a*c^2 *d^3*e - a^2*c*d*e^3)*A*abs(c*d^2*e - a*e^3)*abs(c) - (a*c^2*d^4 - a^3*e^4 )*B*abs(c*d^2*e - a*e^3)*abs(c) - (sqrt(a*c)*c^3*d^6*e - 2*sqrt(a*c)*a*c^2 *d^4*e^3 + sqrt(a*c)*a^2*c*d^2*e^5)*A*abs(c) + (sqrt(a*c)*a*c^2*d^5*e^2 - 2*sqrt(a*c)*a^2*c*d^3*e^4 + sqrt(a*c)*a^3*d*e^6)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d^3 - a*c*d*e^2 + sqrt((c^2*d^3 - a*c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/(( a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt(a*c)*a*c^2*d^4*e + 2*sq rt(a*c)*a^2*c*d^2*e^3 - sqrt(a*c)*a^3*e^5)*sqrt(-c^2*d - sqrt(a*c)*c*e)*ab s(c*d^2*e - a*e^3)) - ((c*d^2*e - a*e^3)^2*sqrt(a*c)*B*a*d*abs(c) - (c*d^2 *e - a*e^3)^2*sqrt(a*c)*A*a*e*abs(c) - 2*(a*c^2*d^3*e - a^2*c*d*e^3)*A*abs (c*d^2*e - a*e^3)*abs(c) + (a*c^2*d^4 - a^3*e^4)*B*abs(c*d^2*e - a*e^3)*ab s(c) - (sqrt(a*c)*c^3*d^6*e - 2*sqrt(a*c)*a*c^2*d^4*e^3 + sqrt(a*c)*a^2*c* d^2*e^5)*A*abs(c) + (sqrt(a*c)*a*c^2*d^5*e^2 - 2*sqrt(a*c)*a^2*c*d^3*e^4 + sqrt(a*c)*a^3*d*e^6)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d^3 - a*c* d*e^2 - sqrt((c^2*d^3 - a*c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4) *(c^2*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^3*d^5 - 2*a^2*c^2*d^3*e ^2 + a^3*c*d*e^4 + sqrt(a*c)*a*c^2*d^4*e - 2*sqrt(a*c)*a^2*c*d^2*e^3 + sqr t(a*c)*a^3*e^5)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(c*d^2*e - a*e^3))
Time = 14.35 (sec) , antiderivative size = 10288, normalized size of antiderivative = 52.22 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
atan((((-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1/2) + A^2*a*c^3*d^3 - 2* A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2) - 6*A*B*a*c*d*e^2*(a^3 *c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 - 3*a^4*c^2*d^2 *e^4)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^2*d^3 + B^2*a^2*e^3*(a^3*c)^(1 /2) + A^2*a*c^3*d^3 - 2*A*B*c^2*d^3*(a^3*c)^(1/2) + 3*B^2*a^3*c*d*e^2 + A^ 2*a*c*e^3*(a^3*c)^(1/2) + 3*A^2*a^2*c^2*d*e^2 - 2*A*B*a^3*c*e^3 + 3*A^2*c^ 2*d^2*e*(a^3*c)^(1/2) - 6*A*B*a^2*c^2*d^2*e + 3*B^2*a*c*d^2*e*(a^3*c)^(1/2 ) - 6*A*B*a*c*d*e^2*(a^3*c)^(1/2))/(4*(a^5*c*e^6 - a^2*c^4*d^6 + 3*a^3*c^3 *d^4*e^2 - 3*a^4*c^2*d^2*e^4)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^ 12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320 *a^5*c^5*d^3*e^10) - 32*B*a^6*c^3*e^12 + 64*A*a*c^8*d^9*e^3 + 64*A*a^5*c^4 *d*e^11 - 32*B*a*c^8*d^10*e^2 - 256*A*a^2*c^7*d^7*e^5 + 384*A*a^3*c^6*d^5* e^7 - 256*A*a^4*c^5*d^3*e^9 + 96*B*a^2*c^7*d^8*e^4 - 64*B*a^3*c^6*d^6*e^6 - 64*B*a^4*c^5*d^4*e^8 + 96*B*a^5*c^4*d^2*e^10) + (d + e*x)^(1/2)*(16*A^2* a^4*c^4*e^10 + 16*B^2*a^5*c^3*e^10 - 16*A^2*c^8*d^8*e^2 - 32*A^2*a^3*c^5*d ^2*e^8 + 32*B^2*a^2*c^6*d^6*e^4 - 32*B^2*a^4*c^4*d^2*e^8 + 32*A^2*a*c^7*d^ 6*e^4 - 16*B^2*a*c^7*d^8*e^2 + 64*A*B*a*c^7*d^7*e^3 - 64*A*B*a^4*c^4*d*e^9 - 192*A*B*a^2*c^6*d^5*e^5 + 192*A*B*a^3*c^5*d^3*e^7))*(-(B^2*a^2*c^2*d...