Integrand size = 27, antiderivative size = 433 \[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\frac {4 \left (4 B c d^2-7 A c d e-5 a B e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}}{105 c e^2}-\frac {2 (d+e x)^{3/2} (4 B d-7 A e-5 B e x) \sqrt {a-c x^2}}{35 e^2}+\frac {4 \sqrt {a} \left (4 B c d^3-7 A c d^2 e-8 a B d e^2-21 a A e^3\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{105 \sqrt {c} e^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {4 \sqrt {a} \left (c d^2-a e^2\right ) \left (4 B c d^2-7 A c d e-5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{105 c^{3/2} e^3 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
4/105*(-7*A*c*d*e-5*B*a*e^2+4*B*c*d^2)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2)/c/e^ 2-2/35*(e*x+d)^(3/2)*(-5*B*e*x-7*A*e+4*B*d)*(-c*x^2+a)^(1/2)/e^2+4/105*a^( 1/2)*(-21*A*a*e^3-7*A*c*d^2*e-8*B*a*d*e^2+4*B*c*d^3)*(e*x+d)^(1/2)*(1-c*x^ 2/a)^(1/2)*EllipticE(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1 /2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/c^(1/2)/e^3/(c^(1/2)*(e*x+d)/(c^(1/2)* d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)-4/105*a^(1/2)*(-a*e^2+c*d^2)*(-7*A*c* d*e-5*B*a*e^2+4*B*c*d^2)*(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1- c*x^2/a)^(1/2)*EllipticF(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*( a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/c^(3/2)/e^3/(e*x+d)^(1/2)/(-c*x^2+ a)^(1/2)
Result contains complex when optimal does not.
Time = 26.74 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.35 \[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\frac {2 \sqrt {a-c x^2} \left (-\left ((d+e x) \left (10 a B e^2-7 A c e (d+3 e x)+B c \left (4 d^2-3 d e x-15 e^2 x^2\right )\right )\right )-\frac {2 \left (e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-7 A e \left (c d^2+3 a e^2\right )+4 B \left (c d^3-2 a d e^2\right )\right ) \left (-a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-4 B c d^3+7 A c d^2 e+8 a B d e^2+21 a A e^3\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} e \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (4 B c d^2+3 \sqrt {a} B \sqrt {c} d e-7 A c d e-5 a B e^2+21 \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )}\right )}{105 c e^2 \sqrt {d+e x}} \] Input:
Integrate[(A + B*x)*Sqrt[d + e*x]*Sqrt[a - c*x^2],x]
Output:
(2*Sqrt[a - c*x^2]*(-((d + e*x)*(10*a*B*e^2 - 7*A*c*e*(d + 3*e*x) + B*c*(4 *d^2 - 3*d*e*x - 15*e^2*x^2))) - (2*(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(- 7*A*e*(c*d^2 + 3*a*e^2) + 4*B*(c*d^3 - 2*a*d*e^2))*(-a + c*x^2) + I*Sqrt[c ]*(Sqrt[c]*d - Sqrt[a]*e)*(-4*B*c*d^3 + 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A *e^3)*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c ] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d + (Sqrt[a ]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a] *e)] + I*Sqrt[a]*e*(-(Sqrt[c]*d) + Sqrt[a]*e)*(4*B*c*d^2 + 3*Sqrt[a]*B*Sqr t[c]*d*e - 7*A*c*d*e - 5*a*B*e^2 + 21*Sqrt[a]*A*Sqrt[c]*e^2)*Sqrt[(e*(Sqrt [a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x)) ]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[ d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)]))/(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2))))/(105*c*e^2*Sqrt[d + e*x])
Time = 0.66 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {687, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \sqrt {a-c x^2} (A+B x) \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {2 \int -\frac {(7 A c d+a B e+c (B d+7 A e) x) \sqrt {a-c x^2}}{2 \sqrt {d+e x}}dx}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(7 A c d+a B e+c (B d+7 A e) x) \sqrt {a-c x^2}}{\sqrt {d+e x}}dx}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {-\frac {4 \int \frac {c \left (a e \left (B c d^2-28 A c e d-5 a B e^2\right )+c \left (4 B c d^3-7 A c e d^2-8 a B e^2 d-21 a A e^3\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 c e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a e \left (B c d^2-28 A c e d-5 a B e^2\right )+c \left (4 B c d^3-7 A c e d^2-8 a B e^2 d-21 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {c \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {c \sqrt {1-\frac {c x^2}{a}} \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2-7 A c d e+4 B c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-21 a A e^3-8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{15 e^2}}{7 c}-\frac {2 B \left (a-c x^2\right )^{3/2} \sqrt {d+e x}}{7 c}\) |
Input:
Int[(A + B*x)*Sqrt[d + e*x]*Sqrt[a - c*x^2],x]
Output:
(-2*B*Sqrt[d + e*x]*(a - c*x^2)^(3/2))/(7*c) + ((-2*Sqrt[d + e*x]*(4*B*c*d ^2 - 7*A*c*d*e - 5*a*B*e^2 - 3*c*e*(B*d + 7*A*e)*x)*Sqrt[a - c*x^2])/(15*e ^2) - (2*((-2*Sqrt[a]*Sqrt[c]*(4*B*c*d^3 - 7*A*c*d^2*e - 8*a*B*d*e^2 - 21* a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt [c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(e*Sqrt[(Sqrt[ c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]*(c*d^ 2 - a*e^2)*(4*B*c*d^2 - 7*A*c*d*e - 5*a*B*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(S qrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt [c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqr t[d + e*x]*Sqrt[a - c*x^2])))/(15*e^2))/(7*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(791\) vs. \(2(363)=726\).
Time = 4.13 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.83
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {2 B \,x^{2} \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{7}-\frac {2 \left (-A c e -\frac {1}{7} B c d \right ) x \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{5 c e}-\frac {2 \left (-A c d +\frac {2 B a e}{7}-\frac {4 d \left (-A c e -\frac {1}{7} B c d \right )}{5 e}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (A a d +\frac {2 a d \left (-A c e -\frac {1}{7} B c d \right )}{5 c e}+\frac {a \left (-A c d +\frac {2 B a e}{7}-\frac {4 d \left (-A c e -\frac {1}{7} B c d \right )}{5 e}\right )}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (a A e +\frac {3 B a d}{7}+\frac {3 a \left (-A c e -\frac {1}{7} B c d \right )}{5 c}-\frac {2 d \left (-A c d +\frac {2 B a e}{7}-\frac {4 d \left (-A c e -\frac {1}{7} B c d \right )}{5 e}\right )}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(792\) |
| risch | \(\frac {2 \left (15 e^{2} B c \,x^{2}+21 A c \,e^{2} x +3 B c d e x +7 A c d e -10 B a \,e^{2}-4 B c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{105 c \,e^{2}}+\frac {2 \left (\frac {\left (21 A a \,e^{3}+7 A c \,d^{2} e +8 B a d \,e^{2}-4 B c \,d^{3}\right ) \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \left (\left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )-\frac {d \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{e}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {5 B \,e^{3} a^{2} \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {28 A a d \,e^{2} \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}-\frac {B a \,d^{2} e \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right ) \sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}}{105 e^{2} c \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(887\) |
| default | \(\text {Expression too large to display}\) | \(2428\) |
Input:
int((B*x+A)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(2/7*B*x^2*(-c*e *x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/5*(-A*c*e-1/7*B*c*d)/c/e*x*(-c*e*x^3-c*d*x ^2+a*e*x+a*d)^(1/2)-2/3*(-A*c*d+2/7*B*a*e-4/5*d/e*(-A*c*e-1/7*B*c*d))/c/e* (-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)+2*(A*a*d+2/5*a/c*d/e*(-A*c*e-1/7*B*c*d) +1/3*a/c*(-A*c*d+2/7*B*a*e-4/5*d/e*(-A*c*e-1/7*B*c*d)))*(d/e-1/c*(a*c)^(1/ 2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*( a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c* e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^( 1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+2*(a*A*e+3/7*B *a*d+3/5*a/c*(-A*c*e-1/7*B*c*d)-2/3*d/e*(-A*c*d+2/7*B*a*e-4/5*d/e*(-A*c*e- 1/7*B*c*d)))*(d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*( (x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d /e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*( a*c)^(1/2))*EllipticE(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a* c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+1/c*(a*c)^(1/2)*EllipticF(((x+d/e )/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/ 2)))^(1/2))))
Time = 0.11 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.80 \[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, B c^{2} d^{4} - 7 \, A c^{2} d^{3} e - 11 \, B a c d^{2} e^{2} + 63 \, A a c d e^{3} + 15 \, B a^{2} e^{4}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (4 \, B c^{2} d^{3} e - 7 \, A c^{2} d^{2} e^{2} - 8 \, B a c d e^{3} - 21 \, A a c e^{4}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (15 \, B c^{2} e^{4} x^{2} - 4 \, B c^{2} d^{2} e^{2} + 7 \, A c^{2} d e^{3} - 10 \, B a c e^{4} + 3 \, {\left (B c^{2} d e^{3} + 7 \, A c^{2} e^{4}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{315 \, c^{2} e^{4}} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/315*(2*(4*B*c^2*d^4 - 7*A*c^2*d^3*e - 11*B*a*c*d^2*e^2 + 63*A*a*c*d*e^3 + 15*B*a^2*e^4)*sqrt(-c*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e ^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 6*(4*B*c^2*d^ 3*e - 7*A*c^2*d^2*e^2 - 8*B*a*c*d*e^3 - 21*A*a*c*e^4)*sqrt(-c*e)*weierstra ssZeta(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), w eierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2 )/(c*e^3), 1/3*(3*e*x + d)/e)) - 3*(15*B*c^2*e^4*x^2 - 4*B*c^2*d^2*e^2 + 7 *A*c^2*d*e^3 - 10*B*a*c*e^4 + 3*(B*c^2*d*e^3 + 7*A*c^2*e^4)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c^2*e^4)
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\int \left (A + B x\right ) \sqrt {a - c x^{2}} \sqrt {d + e x}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(1/2)*(-c*x**2+a)**(1/2),x)
Output:
Integral((A + B*x)*sqrt(a - c*x**2)*sqrt(d + e*x), x)
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\int { \sqrt {-c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-c*x^2 + a)*(B*x + A)*sqrt(e*x + d), x)
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\int { \sqrt {-c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^2 + a)*(B*x + A)*sqrt(e*x + d), x)
Timed out. \[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\int \sqrt {a-c\,x^2}\,\left (A+B\,x\right )\,\sqrt {d+e\,x} \,d x \] Input:
int((a - c*x^2)^(1/2)*(A + B*x)*(d + e*x)^(1/2),x)
Output:
int((a - c*x^2)^(1/2)*(A + B*x)*(d + e*x)^(1/2), x)
\[ \int (A+B x) \sqrt {d+e x} \sqrt {a-c x^2} \, dx=\frac {-14 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, a^{2} e^{2}-12 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, a b d e +14 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, a c d e x +2 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, b c \,d^{2} x +10 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, b c d e \,x^{2}-21 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} c \,e^{3}-8 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a b c d \,e^{2}-7 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a \,c^{2} d^{2} e +4 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) b \,c^{2} d^{3}+7 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{3} e^{3}+6 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} b d \,e^{2}+21 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} c \,d^{2} e -2 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a b c \,d^{3}}{35 c d e} \] Input:
int((B*x+A)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2),x)
Output:
( - 14*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*e**2 - 12*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*d*e + 14*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c*d*e*x + 2*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c*d**2*x + 10*sqrt(d + e*x)*sqrt(a - c*x**2)*b* c*d*e*x**2 - 21*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c *d*x**2 - c*e*x**3),x)*a**2*c*e**3 - 8*int((sqrt(d + e*x)*sqrt(a - c*x**2) *x**2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*b*c*d*e**2 - 7*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*c* *2*d**2*e + 4*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d *x**2 - c*e*x**3),x)*b*c**2*d**3 + 7*int((sqrt(d + e*x)*sqrt(a - c*x**2))/ (a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**3*e**3 + 6*int((sqrt(d + e*x)*sq rt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*b*d*e**2 + 21* int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x )*a**2*c*d**2*e - 2*int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c* d*x**2 - c*e*x**3),x)*a*b*c*d**3)/(35*c*d*e)