Integrand size = 25, antiderivative size = 202 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 (B d+A e) \sqrt {d+e x}}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}} \]
-2/3*B*(e*x+d)^(3/2)/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))^(3/2)/c^(7/4)/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))^(3/2)/c^(7/4)/a^(1/2)-2*(A*e+B*d)*(e*x+d)^( 1/2)/c
Time = 0.53 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 c \sqrt {d+e x} (4 B d+3 A e+B e x)+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a}}+\frac {3 \left (-\sqrt {a} B+A \sqrt {c}\right ) \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{3 c^2} \]
-1/3*(2*c*Sqrt[d + e*x]*(4*B*d + 3*A*e + B*e*x) + (3*(Sqrt[a]*B + A*Sqrt[c ])*(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)])/Sqrt[a ] + (3*(-(Sqrt[a]*B) + A*Sqrt[c])*Sqrt[c]*(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]*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]))/c^2
Time = 0.49 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {653, 25, 653, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {d+e x} (A c d+a B e+c (B d+A e) x)}{a-c x^2}dx}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (A c d+a B e+c (B d+A e) x)}{a-c x^2}dx}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (A c d^2+2 a B e d+a A e^2+\left (B c d^2+2 A c e d+a B e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (A c d^2+2 a B e d+a A e^2+\left (B c d^2+2 A c e d+a B e^2\right ) x\right )}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A c d^2+2 a B e d+a A e^2+\left (B c d^2+2 A c e d+a B e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx-2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 \int \frac {(B d+A e) \left (c d^2-a e^2\right )-\left (B c d^2+2 A c e d+a B e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}-2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 \left (-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}\right )-2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \left (\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4}}\right )-2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}\) |
(-2*B*(d + e*x)^(3/2))/(3*c) + (-2*(B*d + A*e)*Sqrt[d + e*x] + 2*(((Sqrt[a ]*B - A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e *x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)) + ((Sqrt[a]*B + A*S qrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt [Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4))))/c
3.15.48.3.1 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 = 0.53 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.34
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {2 \left (B e x +3 A e +4 B d \right ) \sqrt {e x +d}}{3}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a c d \,e^{2}+2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a c d \,e^{2}+2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{c}\) | \(271\) |
| risch | \(-\frac {2 \left (B e x +3 A e +4 B d \right ) \sqrt {e x +d}}{3 c}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a c d \,e^{2}+2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a c d \,e^{2}+2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(276\) |
| derivativedivides | \(-\frac {2 \left (\frac {B \left (e x +d \right )^{\frac {3}{2}}}{3}+A e \sqrt {e x +d}+B d \sqrt {e x +d}\right )}{c}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a c d \,e^{2}+2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a c d \,e^{2}+2 A \sqrt {a c \,e^{2}}\, c d e +B \sqrt {a c \,e^{2}}\, a \,e^{2}+B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(287\) |
| default | \(-\frac {2 \left (\frac {B \left (e x +d \right )^{\frac {3}{2}}}{3}+A e \sqrt {e x +d}+B d \sqrt {e x +d}\right )}{c}-\frac {\left (-A a c \,e^{3}-A \,c^{2} d^{2} e -2 B a c d \,e^{2}-2 A \sqrt {a c \,e^{2}}\, c d e -B \sqrt {a c \,e^{2}}\, a \,e^{2}-B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +2 B a c d \,e^{2}-2 A \sqrt {a c \,e^{2}}\, c d e -B \sqrt {a c \,e^{2}}\, a \,e^{2}-B \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(291\) |
1/c*(-2/3*(B*e*x+3*A*e+4*B*d)*(e*x+d)^(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)/(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))-(-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)/(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. 4480 vs. \(2 (148) = 296\).
Time = 3.93 (sec) , antiderivative size = 4480, normalized size of antiderivative = 22.18 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \]
1/6*(3*c*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + a*c^3*sqrt((4*A^2*B^2*c^4 *d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2* a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6* (B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^ 3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7 )) + (B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))*log(( 2*(A*B^3*a*c^3 - A^3*B*c^4)*d^5 + 3*(B^4*a^2*c^2 - A^4*c^4)*d^4*e + 4*(A*B ^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^2 - 2*(B^4*a^3*c - A^4*a*c^3)*d^2*e^3 - 6* (A*B^3*a^3*c - A^3*B*a^2*c^2)*d*e^4 - (B^4*a^4 - A^4*a^2*c^2)*e^5)*sqrt(e* x + d) + (2*A*B^2*a*c^4*d^4 + (3*B^3*a^2*c^3 + 5*A^2*B*a*c^4)*d^3*e + 3*(3 *A*B^2*a^2*c^3 + A^3*a*c^4)*d^2*e^2 + (B^3*a^3*c^2 + 7*A^2*B*a^2*c^3)*d*e^ 3 + (A*B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 - (A*a*c^6*d + B*a^2*c^5*e)*sqrt((4* A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d ^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3 *a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)* e^6)/(a*c^7)))*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + a*c^3*sqrt((4*A^2*B ^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^ 2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^ 3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a...
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=- \int \frac {A d \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {A e x \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {B d x \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {B e x^{2} \sqrt {d + e x}}{- a + c x^{2}}\, dx \]
-Integral(A*d*sqrt(d + e*x)/(-a + c*x**2), x) - Integral(A*e*x*sqrt(d + e* x)/(-a + c*x**2), x) - Integral(B*d*x*sqrt(d + e*x)/(-a + c*x**2), x) - In tegral(B*e*x**2*sqrt(d + e*x)/(-a + c*x**2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=\int { -\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (148) = 296\).
Time = 0.33 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.72 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {{\left (2 \, \sqrt {a c} B a c^{3} d^{2} e^{2} - 2 \, \sqrt {a c} A a c^{3} d e^{3} - {\left (\sqrt {a c} a c d^{2} + \sqrt {a c} a^{2} e^{2}\right )} B c^{2} e^{2} + {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} {\left | e \right |} + {\left (a c^{3} d^{3} - a^{2} c^{2} d e^{2}\right )} B {\left | c \right |} {\left | e \right |} + {\left (\sqrt {a c} c^{4} d^{3} e + \sqrt {a c} a c^{3} d e^{3}\right )} A\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (2 \, \sqrt {a c} B a c^{3} d^{2} e^{2} - 2 \, \sqrt {a c} A a c^{3} d e^{3} - {\left (\sqrt {a c} a c d^{2} + \sqrt {a c} a^{2} e^{2}\right )} B c^{2} e^{2} - {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} {\left | e \right |} - {\left (a c^{3} d^{3} - a^{2} c^{2} d e^{2}\right )} B {\left | c \right |} {\left | e \right |} + {\left (\sqrt {a c} c^{4} d^{3} e + \sqrt {a c} a c^{3} d e^{3}\right )} A\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B c^{2} + 3 \, \sqrt {e x + d} B c^{2} d + 3 \, \sqrt {e x + d} A c^{2} e\right )}}{3 \, c^{3}} \]
-(2*sqrt(a*c)*B*a*c^3*d^2*e^2 - 2*sqrt(a*c)*A*a*c^3*d*e^3 - (sqrt(a*c)*a*c *d^2 + sqrt(a*c)*a^2*e^2)*B*c^2*e^2 + (a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c) *abs(e) + (a*c^3*d^3 - a^2*c^2*d*e^2)*B*abs(c)*abs(e) + (sqrt(a*c)*c^4*d^3 *e + sqrt(a*c)*a*c^3*d*e^3)*A)*arctan(sqrt(e*x + d)/sqrt(-(c^4*d + sqrt(c^ 8*d^2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c^4))/((a*c^4*d - sqrt(a*c)*a*c^3*e)*s qrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + (2*sqrt(a*c)*B*a*c^3*d^2*e^2 - 2*sqr t(a*c)*A*a*c^3*d*e^3 - (sqrt(a*c)*a*c*d^2 + sqrt(a*c)*a^2*e^2)*B*c^2*e^2 - (a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c)*abs(e) - (a*c^3*d^3 - a^2*c^2*d*e^2) *B*abs(c)*abs(e) + (sqrt(a*c)*c^4*d^3*e + sqrt(a*c)*a*c^3*d*e^3)*A)*arctan (sqrt(e*x + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c ^4))/((a*c^4*d + sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 2/3*((e*x + d)^(3/2)*B*c^2 + 3*sqrt(e*x + d)*B*c^2*d + 3*sqrt(e*x + d)*A* c^2*e)/c^3
Time = 11.31 (sec) , antiderivative size = 7560, normalized size of antiderivative = 37.43 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \]
- ((2*A*e - 2*B*d)/c + (4*B*d)/c)*(d + e*x)^(1/2) - atan(((((8*(4*A*a^2*c^ 4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^( 1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e ^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B *a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^ 2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^ 4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a ^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) + (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A ^2*c^3*d^4*e^2 + 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^ 4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2*c*d*e^5 + 128*A*B*a*c^2*d^3*e^3 ))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B *a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^( 1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^( 1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d *e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*1i - (((8*(4*A*a^2*c^4*e^5 - 4*A* a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 + 64*a*c^4*...