Integrand size = 25, antiderivative size = 237 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=-\frac {2 \left (B c d^2+2 A c d e+a B e^2\right ) \sqrt {d+e x}}{c^2}-\frac {2 (B d+A e) (d+e x)^{3/2}}{3 c}-\frac {2 B (d+e x)^{5/2}}{5 c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{9/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{9/4}} \]
-2/3*(A*e+B*d)*(e*x+d)^(3/2)/c-2/5*B*(e*x+d)^(5/2)/c+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))*(-e*a^(1/2)+d *c^(1/2))^(5/2)/c^(9/4)/a^(1/2)+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))*(e*a^(1/2)+d*c^(1/2))^(5/2)/c^(9/4) /a^(1/2)-2*(2*A*c*d*e+B*a*e^2+B*c*d^2)*(e*x+d)^(1/2)/c^2
Time = 0.68 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.20 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=\frac {-2 \sqrt {d+e x} \left (15 a B e^2+5 A c e (7 d+e x)+B c \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )+\frac {15 \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {15 \left (\sqrt {a} B-A \sqrt {c}\right ) \left (-\sqrt {c} d+\sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{15 c^2} \]
(-2*Sqrt[d + e*x]*(15*a*B*e^2 + 5*A*c*e*(7*d + e*x) + B*c*(23*d^2 + 11*d*e *x + 3*e^2*x^2)) + (15*(Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^3*A rcTan[(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]*Sqrt[c]*e]) - (15*(Sqrt[a]*B - A*Sqr t[c])*(-(Sqrt[c]*d) + Sqrt[a]*e)^3*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]* Sqrt[c]*e]))/(15*c^2)
Time = 0.67 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {653, 25, 653, 25, 27, 653, 25, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} (A c d+a B e+c (B d+A e) x)}{a-c x^2}dx}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} (A c d+a B e+c (B d+A e) x)}{a-c x^2}dx}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \sqrt {d+e x} \left (A c d^2+2 a B e d+a A e^2+\left (B c d^2+2 A c e d+a B e^2\right ) x\right )}{a-c x^2}dx}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \sqrt {d+e x} \left (A c d^2+2 a B e d+a A e^2+\left (B c d^2+2 A c e d+a B e^2\right ) x\right )}{a-c x^2}dx}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (A c d^2+2 a B e d+a A e^2+\left (B c d^2+2 A c e d+a B e^2\right ) x\right )}{a-c x^2}dx-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {-\frac {\int -\frac {a B e \left (3 c d^2+a e^2\right )+A c d \left (c d^2+3 a e^2\right )+c \left (B c d^3+3 A c e d^2+3 a B e^2 d+a A e^3\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 \sqrt {d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a B e \left (3 c d^2+a e^2\right )+A c d \left (c d^2+3 a e^2\right )+c \left (B c d^3+3 A c e d^2+3 a B e^2 d+a A e^3\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 \sqrt {d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (c d^2-a e^2\right ) \left (B c d^2+2 A c e d+a B e^2\right )-c \left (B c d^3+3 A c e d^2+3 a B e^2 d+a A e^3\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 \sqrt {d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {2 \left (-\frac {\sqrt {c} \left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^3 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}-\frac {\sqrt {c} \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}\right )}{c}-\frac {2 \sqrt {d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c}}\right )}{c}-\frac {2 \sqrt {d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c}-\frac {2}{3} (d+e x)^{3/2} (A e+B d)}{c}-\frac {2 B (d+e x)^{5/2}}{5 c}\) |
(-2*B*(d + e*x)^(5/2))/(5*c) + ((-2*(B*c*d^2 + 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/c - (2*(B*d + A*e)*(d + e*x)^(3/2))/3 + (2*(((Sqrt[a]*B - A*Sqrt[ c])*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqr t[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt [c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4))))/c)/c
3.15.47.3.1 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[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
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 = 0.92 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.51
| method | result | size |
| pseudoelliptic | \(-\frac {14 \left (-\frac {9 c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (\left (-A \,d^{2} e -\frac {1}{3} B \,d^{3}\right ) c -\frac {a \,e^{2} \left (A e +3 B d \right )}{3}\right ) \sqrt {a c \,e^{2}}+\left (\frac {A \,d^{3} c^{2}}{3}+a d e \left (A e +B d \right ) c +\frac {a^{2} B \,e^{3}}{3}\right ) e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{14}+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {9 c \left (\left (d^{2} \left (A e +\frac {B d}{3}\right ) c +\frac {a \,e^{2} \left (A e +3 B d \right )}{3}\right ) \sqrt {a c \,e^{2}}+\left (\frac {A \,d^{3} c^{2}}{3}+a d e \left (A e +B d \right ) c +\frac {a^{2} B \,e^{3}}{3}\right ) e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{14}+\sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (\frac {x \left (\frac {3 B x}{5}+A \right ) e^{2}}{7}+d \left (\frac {11 B x}{35}+A \right ) e +\frac {23 B \,d^{2}}{35}\right ) c +\frac {3 B a \,e^{2}}{7}\right )\right )\right )}{3 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c^{2}}\) | \(357\) |
| risch | \(-\frac {2 \left (3 B c \,x^{2} e^{2}+5 A c \,e^{2} x +11 B c d e x +35 A c d e +15 B a \,e^{2}+23 B c \,d^{2}\right ) \sqrt {e x +d}}{15 c^{2}}-\frac {2 \left (-\frac {\left (3 A a c d \,e^{3}+A \,c^{2} d^{3} e +B \,e^{4} a^{2}+3 B a c \,d^{2} e^{2}+A \sqrt {a c \,e^{2}}\, a \,e^{3}+3 A \sqrt {a c \,e^{2}}\, c \,d^{2} e +3 B \sqrt {a c \,e^{2}}\, a d \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 A a c d \,e^{3}-A \,c^{2} d^{3} e -B \,e^{4} a^{2}-3 B a c \,d^{2} e^{2}+A \sqrt {a c \,e^{2}}\, a \,e^{3}+3 A \sqrt {a c \,e^{2}}\, c \,d^{2} e +3 B \sqrt {a c \,e^{2}}\, a d \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c}\) | \(370\) |
| derivativedivides | \(-\frac {2 \left (\frac {B c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {A c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A c d e \sqrt {e x +d}+a \,e^{2} B \sqrt {e x +d}+B c \,d^{2} \sqrt {e x +d}\right )}{c^{2}}-\frac {2 \left (-\frac {\left (3 A a c d \,e^{3}+A \,c^{2} d^{3} e +B \,e^{4} a^{2}+3 B a c \,d^{2} e^{2}+A \sqrt {a c \,e^{2}}\, a \,e^{3}+3 A \sqrt {a c \,e^{2}}\, c \,d^{2} e +3 B \sqrt {a c \,e^{2}}\, a d \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 A a c d \,e^{3}-A \,c^{2} d^{3} e -B \,e^{4} a^{2}-3 B a c \,d^{2} e^{2}+A \sqrt {a c \,e^{2}}\, a \,e^{3}+3 A \sqrt {a c \,e^{2}}\, c \,d^{2} e +3 B \sqrt {a c \,e^{2}}\, a d \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c}\) | \(392\) |
| default | \(-\frac {2 \left (\frac {B c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {A c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A c d e \sqrt {e x +d}+a \,e^{2} B \sqrt {e x +d}+B c \,d^{2} \sqrt {e x +d}\right )}{c^{2}}+\frac {-\frac {\left (-3 A a c d \,e^{3}-A \,c^{2} d^{3} e -B \,e^{4} a^{2}-3 B a c \,d^{2} e^{2}-A \sqrt {a c \,e^{2}}\, a \,e^{3}-3 A \sqrt {a c \,e^{2}}\, c \,d^{2} e -3 B \sqrt {a c \,e^{2}}\, a d \,e^{2}-B \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (3 A a c d \,e^{3}+A \,c^{2} d^{3} e +B \,e^{4} a^{2}+3 B a c \,d^{2} e^{2}-A \sqrt {a c \,e^{2}}\, a \,e^{3}-3 A \sqrt {a c \,e^{2}}\, c \,d^{2} e -3 B \sqrt {a c \,e^{2}}\, a d \,e^{2}-B \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{c}\) | \(396\) |
-14/3*(-9/14*c*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(((-A*d^2*e-1/3*B*d^3)*c-1/ 3*a*e^2*(A*e+3*B*d))*(a*c*e^2)^(1/2)+(1/3*A*d^3*c^2+a*d*e*(A*e+B*d)*c+1/3* a^2*B*e^3)*e)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+((- c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-9/14*c*((d^2*(A*e+1/3*B*d)*c+1/3*a*e^2*(A* e+3*B*d))*(a*c*e^2)^(1/2)+(1/3*A*d^3*c^2+a*d*e*(A*e+B*d)*c+1/3*a^2*B*e^3)* e)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(e*x+d)^(1/2)* (a*c*e^2)^(1/2)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*((1/7*x*(3/5*B*x+A)*e^2+d* (11/35*B*x+A)*e+23/35*B*d^2)*c+3/7*B*a*e^2)))/((-c*d+(a*c*e^2)^(1/2))*c)^( 1/2)/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/c^2
Leaf count of result is larger than twice the leaf count of optimal. 7410 vs. \(2 (179) = 358\).
Time = 12.62 (sec) , antiderivative size = 7410, normalized size of antiderivative = 31.27 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=\int { -\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{c x^{2} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (179) = 358\).
Time = 0.34 (sec) , antiderivative size = 680, normalized size of antiderivative = 2.87 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=\frac {{\left ({\left (3 \, \sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} A c^{2} e^{2} + {\left (\sqrt {a c} a c d^{3} + 3 \, \sqrt {a c} a^{2} d e^{2}\right )} B c^{2} e^{2} - 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} A {\left | c \right |} {\left | e \right |} - {\left (a c^{3} d^{4} - a^{3} c e^{4}\right )} B {\left | c \right |} {\left | e \right |} - {\left (\sqrt {a c} c^{4} d^{4} e + 3 \, \sqrt {a c} a c^{3} d^{2} e^{3}\right )} A - {\left (3 \, \sqrt {a c} a c^{3} d^{3} e^{2} + \sqrt {a c} a^{2} c^{2} d e^{4}\right )} B\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{6} d + \sqrt {c^{12} d^{2} - {\left (c^{6} d^{2} - a c^{5} e^{2}\right )} c^{6}}}{c^{6}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} - \frac {{\left ({\left (3 \, \sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} A c^{2} e^{2} + {\left (\sqrt {a c} a c d^{3} + 3 \, \sqrt {a c} a^{2} d e^{2}\right )} B c^{2} e^{2} + 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} A {\left | c \right |} {\left | e \right |} + {\left (a c^{3} d^{4} - a^{3} c e^{4}\right )} B {\left | c \right |} {\left | e \right |} - {\left (\sqrt {a c} c^{4} d^{4} e + 3 \, \sqrt {a c} a c^{3} d^{2} e^{3}\right )} A - {\left (3 \, \sqrt {a c} a c^{3} d^{3} e^{2} + \sqrt {a c} a^{2} c^{2} d e^{4}\right )} B\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{6} d - \sqrt {c^{12} d^{2} - {\left (c^{6} d^{2} - a c^{5} e^{2}\right )} c^{6}}}{c^{6}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{4} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{4} d + 15 \, \sqrt {e x + d} B c^{4} d^{2} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{4} e + 30 \, \sqrt {e x + d} A c^{4} d e + 15 \, \sqrt {e x + d} B a c^{3} e^{2}\right )}}{15 \, c^{5}} \]
((3*sqrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*A*c^2*e^2 + (sqrt(a*c)*a*c*d^ 3 + 3*sqrt(a*c)*a^2*d*e^2)*B*c^2*e^2 - 2*(a*c^3*d^3*e - a^2*c^2*d*e^3)*A*a bs(c)*abs(e) - (a*c^3*d^4 - a^3*c*e^4)*B*abs(c)*abs(e) - (sqrt(a*c)*c^4*d^ 4*e + 3*sqrt(a*c)*a*c^3*d^2*e^3)*A - (3*sqrt(a*c)*a*c^3*d^3*e^2 + sqrt(a*c )*a^2*c^2*d*e^4)*B)*arctan(sqrt(e*x + d)/sqrt(-(c^6*d + sqrt(c^12*d^2 - (c ^6*d^2 - a*c^5*e^2)*c^6))/c^6))/((a*c^4*d - sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) - ((3*sqrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*A *c^2*e^2 + (sqrt(a*c)*a*c*d^3 + 3*sqrt(a*c)*a^2*d*e^2)*B*c^2*e^2 + 2*(a*c^ 3*d^3*e - a^2*c^2*d*e^3)*A*abs(c)*abs(e) + (a*c^3*d^4 - a^3*c*e^4)*B*abs(c )*abs(e) - (sqrt(a*c)*c^4*d^4*e + 3*sqrt(a*c)*a*c^3*d^2*e^3)*A - (3*sqrt(a *c)*a*c^3*d^3*e^2 + sqrt(a*c)*a^2*c^2*d*e^4)*B)*arctan(sqrt(e*x + d)/sqrt( -(c^6*d - sqrt(c^12*d^2 - (c^6*d^2 - a*c^5*e^2)*c^6))/c^6))/((a*c^4*d + sq rt(a*c)*a*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 2/15*(3*(e*x + d)^ (5/2)*B*c^4 + 5*(e*x + d)^(3/2)*B*c^4*d + 15*sqrt(e*x + d)*B*c^4*d^2 + 5*( e*x + d)^(3/2)*A*c^4*e + 30*sqrt(e*x + d)*A*c^4*d*e + 15*sqrt(e*x + d)*B*a *c^3*e^2)/c^5
Time = 11.86 (sec) , antiderivative size = 11383, normalized size of antiderivative = 48.03 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx=\text {Too large to display} \]
- (2*d*((2*A*e - 2*B*d)/c + (4*B*d)/c) + (2*B*(a*e^2 - c*d^2))/c^2)*(d + e *x)^(1/2) - ((2*A*e - 2*B*d)/(3*c) + (4*B*d)/(3*c))*(d + e*x)^(3/2) - atan (((((8*(4*B*a^3*c^4*e^6 - 8*A*a*c^6*d^3*e^3 + 8*A*a^2*c^5*d*e^5 - 4*B*a*c^ 6*d^4*e^2))/c^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^7*d^5 + B^2*a ^3*e^5*(a^3*c^9)^(1/2) + A^2*a*c^8*d^5 + 10*A^2*a^2*c^7*d^3*e^2 + 10*B^2*a ^3*c^6*d^3*e^2 + A^2*a^2*c*e^5*(a^3*c^9)^(1/2) + 2*A*B*a^4*c^5*e^5 + 5*A^2 *c^3*d^4*e*(a^3*c^9)^(1/2) + 2*A*B*c^3*d^5*(a^3*c^9)^(1/2) + 5*A^2*a^3*c^6 *d*e^4 + 5*B^2*a^4*c^5*d*e^4 + 10*A*B*a^2*c^7*d^4*e + 5*B^2*a*c^2*d^4*e*(a ^3*c^9)^(1/2) + 10*A^2*a*c^2*d^2*e^3*(a^3*c^9)^(1/2) + 20*A*B*a^3*c^6*d^2* e^3 + 10*B^2*a^2*c*d^2*e^3*(a^3*c^9)^(1/2) + 10*A*B*a^2*c*d*e^4*(a^3*c^9)^ (1/2) + 20*A*B*a*c^2*d^3*e^2*(a^3*c^9)^(1/2))/(4*a^2*c^9))^(1/2))*((B^2*a^ 2*c^7*d^5 + B^2*a^3*e^5*(a^3*c^9)^(1/2) + A^2*a*c^8*d^5 + 10*A^2*a^2*c^7*d ^3*e^2 + 10*B^2*a^3*c^6*d^3*e^2 + A^2*a^2*c*e^5*(a^3*c^9)^(1/2) + 2*A*B*a^ 4*c^5*e^5 + 5*A^2*c^3*d^4*e*(a^3*c^9)^(1/2) + 2*A*B*c^3*d^5*(a^3*c^9)^(1/2 ) + 5*A^2*a^3*c^6*d*e^4 + 5*B^2*a^4*c^5*d*e^4 + 10*A*B*a^2*c^7*d^4*e + 5*B ^2*a*c^2*d^4*e*(a^3*c^9)^(1/2) + 10*A^2*a*c^2*d^2*e^3*(a^3*c^9)^(1/2) + 20 *A*B*a^3*c^6*d^2*e^3 + 10*B^2*a^2*c*d^2*e^3*(a^3*c^9)^(1/2) + 10*A*B*a^2*c *d*e^4*(a^3*c^9)^(1/2) + 20*A*B*a*c^2*d^3*e^2*(a^3*c^9)^(1/2))/(4*a^2*c^9) )^(1/2) + (16*(d + e*x)^(1/2)*(B^2*a^4*e^8 + A^2*c^4*d^6*e^2 + A^2*a^3*c*e ^8 + 15*A^2*a^2*c^2*d^2*e^6 + 15*B^2*a^2*c^2*d^4*e^4 + 15*A^2*a*c^3*d^4...