Integrand size = 25, antiderivative size = 243 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=-\frac {2 (B d-A e)}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (B c d^2-2 A c d e+a B e^2\right )}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \]
-2/3*(-A*e+B*d)/(-a*e^2+c*d^2)/(e*x+d)^(3/2)+c^(1/4)*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))/a^(1/2)/(-e*a ^(1/2)+d*c^(1/2))^(5/2)+c^(1/4)*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))/a^(1/2)/(e*a^(1/2)+d*c^(1/2))^(5/2) -2*(-2*A*c*d*e+B*a*e^2+B*c*d^2)/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)
Time = 0.89 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=-\frac {2 \left (a A e^3+a B e^2 (2 d+3 e x)+B c d^2 (4 d+3 e x)-A c d e (7 d+6 e x)\right )}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^3}+\frac {\left (\sqrt {a} B \sqrt {c}-A c\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-c d+\sqrt {a} \sqrt {c} e}} \]
(-2*(a*A*e^3 + a*B*e^2*(2*d + 3*e*x) + B*c*d^2*(4*d + 3*e*x) - A*c*d*e*(7* d + 6*e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - ((Sqrt[a]*B + A*Sqrt[ c])*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)])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a] *e)^3) + ((Sqrt[a]*B*Sqrt[c] - A*c)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]* e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]* e)^2*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])
Time = 0.56 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.31, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {655, 25, 655, 25, 27, 654, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right ) (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle -\frac {\int -\frac {A c d-a B e+c (B d-A e) x}{(d+e x)^{3/2} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {A c d-a B e+c (B d-A e) x}{(d+e x)^{3/2} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (A c d^2-2 a B e d+a A e^2+\left (B c d^2-2 A c e d+a B e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (A c d^2-2 a B e d+a A e^2+\left (B c d^2-2 A c e d+a B e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \int \frac {A c d^2-2 a B e d+a A e^2+\left (B c d^2-2 A c e d+a B e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}-\frac {2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {2 c \int \frac {B c d^3-3 A c e d^2+3 a B e^2 d-a A e^3-\left (B c d^2-2 A c e d+a B e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {2 c \left (-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}\right )}{c d^2-a e^2}-\frac {2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 c \left (\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{c d^2-a e^2}-\frac {2 \left (a B e^2-2 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{c d^2-a e^2}-\frac {2 (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
(-2*(B*d - A*e))/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + ((-2*(B*c*d^2 - 2*A *c*d*e + a*B*e^2))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + (2*c*(((Sqrt[a]*B - A *Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^2*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S qrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^2*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(c*d^2 - a*e^2))/(c*d^2 - a*e^2)
3.15.52.3.1 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[((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 = 0.56 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(-\frac {2 \left (A e -B d \right )}{3 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A c d e +B a \,e^{2}+B c \,d^{2}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {2 c^{2} \left (-\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e -2 B a c d \,e^{2}-2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e +2 B a c d \,e^{2}-2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\) | \(345\) |
| derivativedivides | \(\frac {-\frac {2 A e}{3}+\frac {2 B d}{3}}{\left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 A c d e -2 B a \,e^{2}-2 B c \,d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {2 c^{2} \left (-\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e -2 B a c d \,e^{2}-2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e +2 B a c d \,e^{2}-2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\) | \(347\) |
| pseudoelliptic | \(\frac {c \left (e x +d \right )^{\frac {3}{2}} \left (\left (\left (2 A d e -B \,d^{2}\right ) c -B a \,e^{2}\right ) \sqrt {a c \,e^{2}}+\left (A c \,d^{2}+a e \left (A e -2 B d \right )\right ) c e \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\left (c \left (\left (\left (-2 A d e +B \,d^{2}\right ) c +B a \,e^{2}\right ) \sqrt {a c \,e^{2}}+\left (A c \,d^{2}+a e \left (A e -2 B d \right )\right ) c e \right ) \left (e x +d \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )-\frac {2 \sqrt {a c \,e^{2}}\, \left (\left (-6 A d \,e^{2} x -7 \left (-\frac {3 B x}{7}+A \right ) d^{2} e +4 B \,d^{3}\right ) c +\left (\left (3 B x +A \right ) e +2 B d \right ) e^{2} a \right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}{3}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (e^{2} a -c \,d^{2}\right )^{2}}\) | \(349\) |
-2/3*(A*e-B*d)/(a*e^2-c*d^2)/(e*x+d)^(3/2)-2/(a*e^2-c*d^2)^2*(-2*A*c*d*e+B *a*e^2+B*c*d^2)/(e*x+d)^(1/2)-2*c^2/(a*e^2-c*d^2)^2*(-1/2*(A*a*c*e^3+A*c^2 *d^2*e-2*B*a*c*d*e^2-2*A*(a*c*e^2)^(1/2)*c*d*e+B*(a*c*e^2)^(1/2)*a*e^2+B*( a*c*e^2)^(1/2)*c*d^2)/c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*ar ctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+1/2*(-A*a*c*e^3-A*c ^2*d^2*e+2*B*a*c*d*e^2-2*A*(a*c*e^2)^(1/2)*c*d*e+B*(a*c*e^2)^(1/2)*a*e^2+B *(a*c*e^2)^(1/2)*c*d^2)/c/(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. 11231 vs. \(2 (189) = 378\).
Time = 30.85 (sec) , antiderivative size = 11231, normalized size of antiderivative = 46.22 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\int { -\frac {B x + A}{{\left (c x^{2} - a\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1751 vs. \(2 (189) = 378\).
Time = 0.51 (sec) , antiderivative size = 1751, normalized size of antiderivative = 7.21 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
-(2*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)^2*sqrt(a*c)*A*a*c*d*e*abs(c) - ( c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)^2*(sqrt(a*c)*a*c*d^2 + sqrt(a*c)*a^2* e^2)*B*abs(c) - (3*a*c^4*d^6*e - 5*a^2*c^3*d^4*e^3 + a^3*c^2*d^2*e^5 + a^4 *c*e^7)*A*abs(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*abs(c) + (a*c^4*d^7 + a ^2*c^3*d^5*e^2 - 5*a^3*c^2*d^3*e^4 + 3*a^4*c*d*e^6)*B*abs(c^2*d^4*e - 2*a* c*d^2*e^3 + a^2*e^5)*abs(c) + (sqrt(a*c)*c^6*d^11*e - 3*sqrt(a*c)*a*c^5*d^ 9*e^3 + 2*sqrt(a*c)*a^2*c^4*d^7*e^5 + 2*sqrt(a*c)*a^3*c^3*d^5*e^7 - 3*sqrt (a*c)*a^4*c^2*d^3*e^9 + sqrt(a*c)*a^5*c*d*e^11)*A*abs(c) - 2*(sqrt(a*c)*a* c^5*d^10*e^2 - 4*sqrt(a*c)*a^2*c^4*d^8*e^4 + 6*sqrt(a*c)*a^3*c^3*d^6*e^6 - 4*sqrt(a*c)*a^4*c^2*d^4*e^8 + sqrt(a*c)*a^5*c*d^2*e^10)*B*abs(c))*arctan( sqrt(e*x + d)/sqrt(-(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 + sqrt((c^3*d ^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2 *c*d^2*e^4 - a^3*e^6)*(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c^5*d^9 - 4*a^2*c^4*d^7*e^2 + 6*a^3*c^ 3*d^5*e^4 - 4*a^4*c^2*d^3*e^6 + a^5*c*d*e^8 - sqrt(a*c)*a*c^4*d^8*e + 4*sq rt(a*c)*a^2*c^3*d^6*e^3 - 6*sqrt(a*c)*a^3*c^2*d^4*e^5 + 4*sqrt(a*c)*a^4*c* d^2*e^7 - sqrt(a*c)*a^5*e^9)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)) + (2*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)^2*sqr t(a*c)*A*a*c*d*e*abs(c) - (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)^2*(sqrt(a* c)*a*c*d^2 + sqrt(a*c)*a^2*e^2)*B*abs(c) + (3*a*c^4*d^6*e - 5*a^2*c^3*d...
Time = 16.27 (sec) , antiderivative size = 17610, normalized size of antiderivative = 72.47 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
- atan((((d + e*x)^(1/2)*(16*A^2*a^8*c^5*e^18 + 16*B^2*a^9*c^4*e^18 + 16*A ^2*c^13*d^16*e^2 - 320*A^2*a^2*c^11*d^12*e^6 + 1024*A^2*a^3*c^10*d^10*e^8 - 1440*A^2*a^4*c^9*d^8*e^10 + 1024*A^2*a^5*c^8*d^6*e^12 - 320*A^2*a^6*c^7* d^4*e^14 - 320*B^2*a^3*c^10*d^12*e^6 + 1024*B^2*a^4*c^9*d^10*e^8 - 1440*B^ 2*a^5*c^8*d^8*e^10 + 1024*B^2*a^6*c^7*d^6*e^12 - 320*B^2*a^7*c^6*d^4*e^14 + 16*B^2*a*c^12*d^16*e^2 - 128*A*B*a*c^12*d^15*e^3 - 128*A*B*a^8*c^5*d*e^1 7 + 640*A*B*a^2*c^11*d^13*e^5 - 1152*A*B*a^3*c^10*d^11*e^7 + 640*A*B*a^4*c ^9*d^9*e^9 + 640*A*B*a^5*c^8*d^7*e^11 - 1152*A*B*a^6*c^7*d^5*e^13 + 640*A* B*a^7*c^6*d^3*e^15) - (-(B^2*a^2*c^3*d^5 + B^2*a^3*e^5*(a^3*c)^(1/2) + A^2 *a*c^4*d^5 + 10*A^2*a^2*c^3*d^3*e^2 + 10*B^2*a^3*c^2*d^3*e^2 - 2*A*B*c^3*d ^5*(a^3*c)^(1/2) + 5*B^2*a^4*c*d*e^4 + 5*A^2*a^3*c^2*d*e^4 + A^2*a^2*c*e^5 *(a^3*c)^(1/2) - 2*A*B*a^4*c*e^5 + 5*A^2*c^3*d^4*e*(a^3*c)^(1/2) + 5*B^2*a *c^2*d^4*e*(a^3*c)^(1/2) + 10*A^2*a*c^2*d^2*e^3*(a^3*c)^(1/2) - 10*A*B*a^2 *c^3*d^4*e + 10*B^2*a^2*c*d^2*e^3*(a^3*c)^(1/2) - 20*A*B*a^3*c^2*d^2*e^3 - 10*A*B*a^2*c*d*e^4*(a^3*c)^(1/2) - 20*A*B*a*c^2*d^3*e^2*(a^3*c)^(1/2))/(4 *(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c ^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(B^2*a^2*c^3*d ^5 + B^2*a^3*e^5*(a^3*c)^(1/2) + A^2*a*c^4*d^5 + 10*A^2*a^2*c^3*d^3*e^2 + 10*B^2*a^3*c^2*d^3*e^2 - 2*A*B*c^3*d^5*(a^3*c)^(1/2) + 5*B^2*a^4*c*d*e^4 + 5*A^2*a^3*c^2*d*e^4 + A^2*a^2*c*e^5*(a^3*c)^(1/2) - 2*A*B*a^4*c*e^5 + ...