Integrand size = 25, antiderivative size = 372 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e (3 A c d-5 a B e)+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (5 a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )-3 A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}-\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \left (5 a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (12 c^{3/2} d^2-6 \sqrt {a} c d e-3 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}} \]
1/4*(e*x+d)^(3/2)*(a*(A*e+B*d)+(A*c*d+B*a*e)*x)/a/c/(-c*x^2+a)^2+1/16*(a*e *(3*A*c*d-5*B*a*e)+c*(6*A*c*d^2-a*e*(3*A*e+5*B*d))*x)*(e*x+d)^(1/2)/a^2/c^ 2/(-c*x^2+a)+1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/ 2))*(5*a*B*e*(e*a^(1/2)+2*d*c^(1/2))-3*A*(4*c^(3/2)*d^2+2*c*d*e*a^(1/2)-a* e^2*c^(1/2)))*(-e*a^(1/2)+d*c^(1/2))^(1/2)/a^(5/2)/c^(9/4)-1/32*arctanh(c^ (1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(5*a*B*e*(-e*a^(1/2)+2*d* c^(1/2))-A*(12*c^(3/2)*d^2-6*c*d*e*a^(1/2)-3*a*e^2*c^(1/2)))*(e*a^(1/2)+d* c^(1/2))^(1/2)/a^(5/2)/c^(9/4)
Time = 2.89 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c} \sqrt {d+e x} \left (5 a^3 B e^2+6 A c^3 d^2 x^3-a c^2 x \left (5 B d e x^2+A \left (10 d^2+d e x+3 e^2 x^2\right )\right )-a^2 c \left (A e (7 d+e x)+B \left (4 d^2+3 d e x+9 e^2 x^2\right )\right )\right )}{\left (a-c x^2\right )^2}-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (5 a B e \left (-2 \sqrt {c} d+\sqrt {a} e\right )+3 A \left (4 c^{3/2} d^2-2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (-5 a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )+3 A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} c^{5/2}} \]
((-2*Sqrt[a]*Sqrt[c]*Sqrt[d + e*x]*(5*a^3*B*e^2 + 6*A*c^3*d^2*x^3 - a*c^2* x*(5*B*d*e*x^2 + A*(10*d^2 + d*e*x + 3*e^2*x^2)) - a^2*c*(A*e*(7*d + e*x) + B*(4*d^2 + 3*d*e*x + 9*e^2*x^2))))/(a - c*x^2)^2 - Sqrt[-(c*d) - Sqrt[a] *Sqrt[c]*e]*(5*a*B*e*(-2*Sqrt[c]*d + Sqrt[a]*e) + 3*A*(4*c^(3/2)*d^2 - 2*S qrt[a]*c*d*e - a*Sqrt[c]*e^2))*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sq rt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)] + Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*( -5*a*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) + 3*A*(4*c^(3/2)*d^2 + 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^2))*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/ (Sqrt[c]*d - Sqrt[a]*e)])/(32*a^(5/2)*c^(5/2))
Time = 0.72 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {684, 27, 685, 27, 654, 25, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}-\frac {\int -\frac {\sqrt {d+e x} \left (6 A c d^2-a e (5 B d+3 A e)+e (3 A c d-5 a B e) x\right )}{2 \left (a-c x^2\right )^2}dx}{4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (6 A c d^2-a e (5 B d+3 A e)+e (3 A c d-5 a B e) x\right )}{\left (a-c x^2\right )^2}dx}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 685 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}-\frac {\int -\frac {3 A c d \left (4 c d^2-3 a e^2\right )-5 a B e \left (2 c d^2-a e^2\right )+c e \left (6 A c d^2-a e (5 B d+3 A e)\right ) x}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 A c d \left (4 c d^2-3 a e^2\right )-5 a B e \left (2 c d^2-a e^2\right )+c e \left (6 A c d^2-a e (5 B d+3 A e)\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a c}+\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int -\frac {e \left ((6 A c d-5 a B e) \left (c d^2-a e^2\right )+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}+\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {e \left ((6 A c d-5 a B e) \left (c d^2-a e^2\right )+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}-\frac {e \int \frac {(6 A c d-5 a B e) \left (c d^2-a e^2\right )+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (5 a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-3 A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (5 a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-3 A \left (-2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a c}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \left (5 a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-3 A \left (-2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (5 a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-3 A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e}\right )}{2 a c}}{8 a c}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\) |
((d + e*x)^(3/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2 ) + ((Sqrt[d + e*x]*(a*e*(3*A*c*d - 5*a*B*e) + c*(6*A*c*d^2 - a*e*(5*B*d + 3*A*e))*x))/(2*a*c*(a - c*x^2)) - (e*(-1/2*(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*( 5*a*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) - 3*A*(4*c^(3/2)*d^2 + 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]* e]])/(Sqrt[a]*c^(1/4)*e) + (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(5*a*B*e*(2*Sqrt[c ]*d - Sqrt[a]*e) - 3*A*(4*c^(3/2)*d^2 - 2*Sqrt[a]*c*d*e - a*Sqrt[c]*e^2))* ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c ^(1/4)*e)))/(2*a*c))/(8*a*c)
3.15.59.3.1 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)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c *(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[(d + e*x)^(m - 1)*(a + c*x^2) ^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.99 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {9 c \left (\frac {\left (2 A c \,d^{2}-e \left (A e +\frac {5 B d}{3}\right ) a \right ) \sqrt {a c \,e^{2}}}{3}-\frac {4 A \,d^{3} c^{2}}{3}+a d e \left (A e +\frac {10 B d}{9}\right ) c -\frac {5 a^{2} B \,e^{3}}{9}\right ) \left (-c \,x^{2}+a \right )^{2} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, e \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32}+\frac {7 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {9 c \left (-c \,x^{2}+a \right )^{2} \left (\frac {\left (-2 A c \,d^{2}+e \left (A e +\frac {5 B d}{3}\right ) a \right ) \sqrt {a c \,e^{2}}}{3}-\frac {4 A \,d^{3} c^{2}}{3}+a d e \left (A e +\frac {10 B d}{9}\right ) c -\frac {5 a^{2} B \,e^{3}}{9}\right ) e \,\operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{14}+\sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \left (-\frac {6 A \,c^{3} d^{2} x^{3}}{7}+\frac {10 x \left (A \,d^{2}+\frac {e x \left (5 B x +A \right ) d}{10}+\frac {3 A \,e^{2} x^{2}}{10}\right ) a \,c^{2}}{7}+\left (\frac {4 B \,d^{2}}{7}+e \left (\frac {3 B x}{7}+A \right ) d +\frac {e^{2} x \left (9 B x +A \right )}{7}\right ) a^{2} c -\frac {5 B \,e^{2} a^{3}}{7}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\right )}{16}}{a^{2} c^{2} \left (-c \,x^{2}+a \right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(420\) |
| default | \(2 e^{4} \left (\frac {\frac {\left (3 A a \,e^{2}-6 A c \,d^{2}+5 B a d e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3}}-\frac {\left (8 A a c d \,e^{2}-18 A \,d^{3} c^{2}-9 a^{2} B \,e^{3}+15 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{3} c}+\frac {\left (A \,a^{2} e^{4}+17 A a c \,d^{2} e^{2}-18 d^{4} A \,c^{2}-15 B \,a^{2} d \,e^{3}+15 B a c \,d^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{3} c}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (6 A a c d \,e^{2}-6 A \,d^{3} c^{2}-5 a^{2} B \,e^{3}+5 B a c \,d^{2} e \right ) \sqrt {e x +d}}{32 a^{2} e^{3} c^{2}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {-\frac {\left (9 A a c d \,e^{2}-12 A \,d^{3} c^{2}-5 a^{2} B \,e^{3}+10 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c \,d^{2}+5 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-9 A a c d \,e^{2}+12 A \,d^{3} c^{2}+5 a^{2} B \,e^{3}-10 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c \,d^{2}+5 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{3} c}\right )\) | \(546\) |
| derivativedivides | \(-2 e^{4} \left (-\frac {\frac {\left (3 A a \,e^{2}-6 A c \,d^{2}+5 B a d e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3}}-\frac {\left (8 A a c d \,e^{2}-18 A \,d^{3} c^{2}-9 a^{2} B \,e^{3}+15 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{3} c}+\frac {\left (A \,a^{2} e^{4}+17 A a c \,d^{2} e^{2}-18 d^{4} A \,c^{2}-15 B \,a^{2} d \,e^{3}+15 B a c \,d^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{3} c}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (6 A a c d \,e^{2}-6 A \,d^{3} c^{2}-5 a^{2} B \,e^{3}+5 B a c \,d^{2} e \right ) \sqrt {e x +d}}{32 a^{2} e^{3} c^{2}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {-\frac {\left (9 A a c d \,e^{2}-12 A \,d^{3} c^{2}-5 a^{2} B \,e^{3}+10 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c \,d^{2}+5 B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-9 A a c d \,e^{2}+12 A \,d^{3} c^{2}+5 a^{2} B \,e^{3}-10 B a c \,d^{2} e +3 A \sqrt {a c \,e^{2}}\, a \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c \,d^{2}+5 B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{3} c}\right )\) | \(547\) |
7/16*(-9/14*c*(1/3*(2*A*c*d^2-e*(A*e+5/3*B*d)*a)*(a*c*e^2)^(1/2)-4/3*A*d^3 *c^2+a*d*e*(A*e+10/9*B*d)*c-5/9*a^2*B*e^3)*(-c*x^2+a)^2*((c*d+(a*c*e^2)^(1 /2))*c)^(1/2)*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*(-c*x^2+a)^2*(1/3*(-2*A*c*d^2+e*( A*e+5/3*B*d)*a)*(a*c*e^2)^(1/2)-4/3*A*d^3*c^2+a*d*e*(A*e+10/9*B*d)*c-5/9*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)*(-6/7*A*c^3*d^2*x^3+10/7*x*(A*d^2+1/10*e*x*(5*B*x +A)*d+3/10*A*e^2*x^2)*a*c^2+(4/7*B*d^2+e*(3/7*B*x+A)*d+1/7*e^2*x*(9*B*x+A) )*a^2*c-5/7*B*e^2*a^3)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)))/((c*d+(a*c*e^2)^( 1/2))*c)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c*e^2)^(1/2)/a^2/c^2/(- c*x^2+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 3382 vs. \(2 (302) = 604\).
Time = 1.48 (sec) , antiderivative size = 3382, normalized size of antiderivative = 9.09 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^3*d^5 - 240 *A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 + a^5*c^4*sqrt(( 900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4 *a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))*log( -(4320*A^3*B*c^4*d^5*e^4 - 432*(25*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^5 + 36 0*(25*A*B^3*a^2*c^2 - 3*A^3*B*a*c^3)*d^3*e^6 - 4*(625*B^4*a^3*c - 1350*A^2 *B^2*a^2*c^2 - 243*A^4*a*c^3)*d^2*e^7 - 30*(125*A*B^3*a^3*c + 27*A^3*B*a^2 *c^2)*d*e^8 + (625*B^4*a^4 - 81*A^4*a^2*c^2)*e^9)*sqrt(e*x + d) + (180*A^2 *B*a^3*c^4*d^2*e^5 - 6*(50*A*B^2*a^4*c^3 + 9*A^3*a^3*c^4)*d*e^6 + 5*(25*B^ 3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 - (12*A*a^5*c^8*d^2 - 10*B*a^6*c^7*d*e - 3*A*a^6*c^7*e^2)*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3* B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9 )))*sqrt((144*A^2*c^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 + a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/ (a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^ 2*a^2*c)*d*e^4)/(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt ((144*A^2*c^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a ^3*e^5 + a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\int { -\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} - a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 941 vs. \(2 (302) = 604\).
Time = 0.45 (sec) , antiderivative size = 941, normalized size of antiderivative = 2.53 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\frac {{\left (5 \, B a^{2} c d e^{4} {\left | c \right |} - 3 \, {\left (2 \, a c^{2} d^{2} e - a^{2} c e^{3}\right )} A e^{2} {\left | c \right |} - 6 \, {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} A {\left | c \right |} {\left | e \right |} + 5 \, {\left (\sqrt {a c} a c d^{2} e^{2} - \sqrt {a c} a^{2} e^{4}\right )} B {\left | c \right |} {\left | e \right |} + 3 \, {\left (4 \, c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} A {\left | c \right |} - 5 \, {\left (2 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, {\left (a^{3} c^{3} e - \sqrt {a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (5 \, B a^{2} c d e^{4} {\left | c \right |} - 3 \, {\left (2 \, a c^{2} d^{2} e - a^{2} c e^{3}\right )} A e^{2} {\left | c \right |} + 6 \, {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} A {\left | c \right |} {\left | e \right |} - 5 \, {\left (\sqrt {a c} a c d^{2} e^{2} - \sqrt {a c} a^{2} e^{4}\right )} B {\left | c \right |} {\left | e \right |} + 3 \, {\left (4 \, c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} A {\left | c \right |} - 5 \, {\left (2 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, {\left (a^{3} c^{3} e + \sqrt {a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} - \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} A c^{3} d^{2} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{3} d^{3} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{3} d^{4} e - 6 \, \sqrt {e x + d} A c^{3} d^{5} e - 5 \, {\left (e x + d\right )}^{\frac {7}{2}} B a c^{2} d e^{2} + 15 \, {\left (e x + d\right )}^{\frac {5}{2}} B a c^{2} d^{2} e^{2} - 15 \, {\left (e x + d\right )}^{\frac {3}{2}} B a c^{2} d^{3} e^{2} + 5 \, \sqrt {e x + d} B a c^{2} d^{4} e^{2} - 3 \, {\left (e x + d\right )}^{\frac {7}{2}} A a c^{2} e^{3} + 8 \, {\left (e x + d\right )}^{\frac {5}{2}} A a c^{2} d e^{3} - 17 \, {\left (e x + d\right )}^{\frac {3}{2}} A a c^{2} d^{2} e^{3} + 12 \, \sqrt {e x + d} A a c^{2} d^{3} e^{3} - 9 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} c e^{4} + 15 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} c d e^{4} - 10 \, \sqrt {e x + d} B a^{2} c d^{2} e^{4} - {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} c e^{5} - 6 \, \sqrt {e x + d} A a^{2} c d e^{5} + 5 \, \sqrt {e x + d} B a^{3} e^{6}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c^{2}} \]
1/32*(5*B*a^2*c*d*e^4*abs(c) - 3*(2*a*c^2*d^2*e - a^2*c*e^3)*A*e^2*abs(c) - 6*(sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c*d*e^3)*A*abs(c)*abs(e) + 5*(sqrt( a*c)*a*c*d^2*e^2 - sqrt(a*c)*a^2*e^4)*B*abs(c)*abs(e) + 3*(4*c^3*d^4*e - 3 *a*c^2*d^2*e^3)*A*abs(c) - 5*(2*a*c^2*d^3*e^2 - a^2*c*d*e^4)*B*abs(c))*arc tan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3 *c^2*e^2)*a^2*c^3))/(a^2*c^3)))/((a^3*c^3*e - sqrt(a*c)*a^2*c^3*d)*sqrt(-c ^2*d - sqrt(a*c)*c*e)*abs(e)) + 1/32*(5*B*a^2*c*d*e^4*abs(c) - 3*(2*a*c^2* d^2*e - a^2*c*e^3)*A*e^2*abs(c) + 6*(sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c*d *e^3)*A*abs(c)*abs(e) - 5*(sqrt(a*c)*a*c*d^2*e^2 - sqrt(a*c)*a^2*e^4)*B*ab s(c)*abs(e) + 3*(4*c^3*d^4*e - 3*a*c^2*d^2*e^3)*A*abs(c) - 5*(2*a*c^2*d^3* e^2 - a^2*c*d*e^4)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d - sqrt( a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))/(a^2*c^3)))/((a^3*c^3* e + sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 1/16*(6*(e *x + d)^(7/2)*A*c^3*d^2*e - 18*(e*x + d)^(5/2)*A*c^3*d^3*e + 18*(e*x + d)^ (3/2)*A*c^3*d^4*e - 6*sqrt(e*x + d)*A*c^3*d^5*e - 5*(e*x + d)^(7/2)*B*a*c^ 2*d*e^2 + 15*(e*x + d)^(5/2)*B*a*c^2*d^2*e^2 - 15*(e*x + d)^(3/2)*B*a*c^2* d^3*e^2 + 5*sqrt(e*x + d)*B*a*c^2*d^4*e^2 - 3*(e*x + d)^(7/2)*A*a*c^2*e^3 + 8*(e*x + d)^(5/2)*A*a*c^2*d*e^3 - 17*(e*x + d)^(3/2)*A*a*c^2*d^2*e^3 + 1 2*sqrt(e*x + d)*A*a*c^2*d^3*e^3 - 9*(e*x + d)^(5/2)*B*a^2*c*e^4 + 15*(e*x + d)^(3/2)*B*a^2*c*d*e^4 - 10*sqrt(e*x + d)*B*a^2*c*d^2*e^4 - (e*x + d)...
Time = 11.67 (sec) , antiderivative size = 7702, normalized size of antiderivative = 20.70 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
((e*(d + e*x)^(7/2)*(3*A*a*e^2 - 6*A*c*d^2 + 5*B*a*d*e))/(16*a^2) - ((d + e*x)^(1/2)*(5*B*a^3*e^6 - 6*A*c^3*d^5*e + 12*A*a*c^2*d^3*e^3 + 5*B*a*c^2*d ^4*e^2 - 10*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5))/(16*a^2*c^2) + ((d + e*x)^ (3/2)*(A*a^2*e^5 - 15*B*a^2*d*e^4 - 18*A*c^2*d^4*e + 17*A*a*c*d^2*e^3 + 15 *B*a*c*d^3*e^2))/(16*a^2*c) + (e*(d + e*x)^(5/2)*(18*A*c^2*d^3 + 9*B*a^2*e ^3 - 8*A*a*c*d*e^2 - 15*B*a*c*d^2*e))/(16*a^2*c))/(c^2*(d + e*x)^4 + a^2*e ^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d* e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2) - atan(((((20480*B*a ^7*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6 *c^5*d^2*e^4)/(4096*a^6*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(25*B^2*a* e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30* A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^( 1/2))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5 *(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30 *A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a ^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/ (4096*a^10*c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^ 6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4*...