Integrand size = 27, antiderivative size = 437 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=-\frac {2 (B d-A e) \sqrt {a-c x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (B c d^2-4 A c d e+3 a B e^2\right ) \sqrt {a-c x^2}}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {c} \left (B c d^2-4 A c d e+3 a B e^2\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{3 e \left (c d^2-a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{3 e \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
-2/3*(-A*e+B*d)*(-c*x^2+a)^(1/2)/(-a*e^2+c*d^2)/(e*x+d)^(3/2)-2/3*(-4*A*c* d*e+3*B*a*e^2+B*c*d^2)*(-c*x^2+a)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)+2/3 *a^(1/2)*c^(1/2)*(-4*A*c*d*e+3*B*a*e^2+B*c*d^2)*(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))/e/(-a*e^2+c*d^2)^2/(c^(1/2)*(e*x+d)/(c^(1/2 )*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)-2/3*a^(1/2)*c^(1/2)*(-A*e+B*d)*(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))/e/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.75 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=\frac {2 \left (-e^2 (B d-A e) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (c d^2-a e^2\right ) \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (B c d^2-4 A c d e+3 a B e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {c} e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (3 A c d-3 a B e+\sqrt {a} \sqrt {c} (B d-A e)\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{3 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (c d^2 e-a e^3\right )^2 (d+e x)^{3/2} \sqrt {a-c x^2}} \] Input:
Integrate[(A + B*x)/((d + e*x)^(5/2)*Sqrt[a - c*x^2]),x]
Output:
(2*(-(e^2*(B*d - A*e)*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(c*d^2 - a*e^2)*(a - c*x^2)) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(B*c*d^2 - 4*A*c*d*e + 3*a*B*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)^(5/2)*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[c]*e*(Sqrt[c]*d - Sqrt[a]*e)*(3*A*c*d - 3*a*B*e + Sqrt[a]*Sqrt [c]*(B*d - A*e))*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a ]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(5/2)*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)]))/(3*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(c*d^2*e - a*e^3)^2*(d + e*x)^(3/2)*Sqrt[a - c*x^2])
Time = 0.64 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {688, 27, 688, 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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {a-c x^2} (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2 \int \frac {3 (A c d-a B e)+c (B d-A e) x}{2 (d+e x)^{3/2} \sqrt {a-c x^2}}dx}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 (A c d-a B e)+c (B d-A e) x}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {\frac {2 \int \frac {c \left (3 A c d^2-4 a B e d+a A e^2-\left (B c d^2-4 A c e d+3 a B e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \int \frac {3 A c d^2-4 a B e d+a A e^2-\left (B c d^2-4 A c e d+3 a B e^2\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {c \left (\frac {\left (c d^2-a e^2\right ) (B d-A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {\left (3 a B e^2-4 A c d e+B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {c \left (\frac {\left (c d^2-a e^2\right ) (B d-A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {\sqrt {1-\frac {c x^2}{a}} \left (3 a B e^2-4 A c d e+B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {c \left (\frac {\left (c d^2-a e^2\right ) (B d-A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a B e^2-4 A c d e+B c d^2\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {c \left (\frac {\left (c d^2-a e^2\right ) (B d-A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a B e^2-4 A c d e+B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {c \left (\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (B d-A e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a B e^2-4 A c d e+B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a B e^2-4 A c d e+B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a B e^2-4 A c d e+B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}\right )}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a B e^2-4 A c d e+B c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
Input:
Int[(A + B*x)/((d + e*x)^(5/2)*Sqrt[a - c*x^2]),x]
Output:
(-2*(B*d - A*e)*Sqrt[a - c*x^2])/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + ((- 2*(B*c*d^2 - 4*A*c*d*e + 3*a*B*e^2)*Sqrt[a - c*x^2])/((c*d^2 - a*e^2)*Sqrt [d + e*x]) + (c*((2*Sqrt[a]*(B*c*d^2 - 4*A*c*d*e + 3*a*B*e^2)*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)])/(Sqrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x)) /(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) - (2*Sqrt[a]*(B*d - A*e)*(c*d^2 - a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^ 2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqr t[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2])))/(c*d^2 - a*e^2))/(3*(c*d^2 - a*e^2))
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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(367)=734\).
Time = 8.34 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 e^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (-c e \,x^{2}+a e \right ) \left (4 A c d e -3 B a \,e^{2}-B c \,d^{2}\right )}{3 e \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {c \left (A e -B d \right )}{3 e \left (a \,e^{2}-c \,d^{2}\right )}+\frac {c d \left (4 A c d e -3 B a \,e^{2}-B c \,d^{2}\right )}{3 e \left (a \,e^{2}-c \,d^{2}\right )^{2}}\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 c \left (4 A c d e -3 B a \,e^{2}-B c \,d^{2}\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 )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(776\) |
| default | \(\text {Expression too large to display}\) | \(3938\) |
Input:
int((B*x+A)/(e*x+d)^(5/2)/(-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)*(-2/3/e^2/(a*e^2 -c*d^2)*(A*e-B*d)*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2+2/3*(-c*e*x ^2+a*e)/e/(a*e^2-c*d^2)^2*(4*A*c*d*e-3*B*a*e^2-B*c*d^2)/((x+d/e)*(-c*e*x^2 +a*e))^(1/2)+2*(1/3*c/e*(A*e-B*d)/(a*e^2-c*d^2)+1/3*c*d/e*(4*A*c*d*e-3*B*a *e^2-B*c*d^2)/(a*e^2-c*d^2)^2)*(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))+2/3*c*(4*A*c*d*e-3*B*a*e^2-B*c*d^2)/(a* e^2-c*d^2)^2*(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 = 545, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=-\frac {2 \, {\left ({\left (B c d^{5} + 5 \, A c d^{4} e - 9 \, B a d^{3} e^{2} + 3 \, A a d^{2} e^{3} + {\left (B c d^{3} e^{2} + 5 \, A c d^{2} e^{3} - 9 \, B a d e^{4} + 3 \, A a e^{5}\right )} x^{2} + 2 \, {\left (B c d^{4} e + 5 \, A c d^{3} e^{2} - 9 \, B a d^{2} e^{3} + 3 \, A a d e^{4}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (B c d^{4} e - 4 \, A c d^{3} e^{2} + 3 \, B a d^{2} e^{3} + {\left (B c d^{2} e^{3} - 4 \, A c d e^{4} + 3 \, B a e^{5}\right )} x^{2} + 2 \, {\left (B c d^{3} e^{2} - 4 \, A c d^{2} e^{3} + 3 \, B a d e^{4}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (2 \, B c d^{3} e^{2} - 5 \, A c d^{2} e^{3} + 2 \, B a d e^{4} + A a e^{5} + {\left (B c d^{2} e^{3} - 4 \, A c d e^{4} + 3 \, B a e^{5}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (c^{2} d^{6} e^{2} - 2 \, a c d^{4} e^{4} + a^{2} d^{2} e^{6} + {\left (c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{3} - 2 \, a c d^{3} e^{5} + a^{2} d e^{7}\right )} x\right )}} \] Input:
integrate((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/9*((B*c*d^5 + 5*A*c*d^4*e - 9*B*a*d^3*e^2 + 3*A*a*d^2*e^3 + (B*c*d^3*e^ 2 + 5*A*c*d^2*e^3 - 9*B*a*d*e^4 + 3*A*a*e^5)*x^2 + 2*(B*c*d^4*e + 5*A*c*d^ 3*e^2 - 9*B*a*d^2*e^3 + 3*A*a*d*e^4)*x)*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) + 3*(B*c*d^4*e - 4*A*c*d^3*e^2 + 3*B*a*d^2*e^3 + (B*c*d^2*e^3 - 4* A*c*d*e^4 + 3*B*a*e^5)*x^2 + 2*(B*c*d^3*e^2 - 4*A*c*d^2*e^3 + 3*B*a*d*e^4) *x)*sqrt(-c*e)*weierstrassZeta(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), 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)) + 3*(2*B*c*d^3*e^2 - 5*A*c*d^2*e^3 + 2*B*a*d*e^4 + A*a*e^5 + (B*c*d^2*e^3 - 4*A*c*d*e^4 + 3*B *a*e^5)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c^2*d^6*e^2 - 2*a*c*d^4*e^4 + a^2*d^2*e^6 + (c^2*d^4*e^4 - 2*a*c*d^2*e^6 + a^2*e^8)*x^2 + 2*(c^2*d^5*e^3 - 2*a*c*d^3*e^5 + a^2*d*e^7)*x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=\int \frac {A + B x}{\sqrt {a - c x^{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((B*x+A)/(e*x+d)**(5/2)/(-c*x**2+a)**(1/2),x)
Output:
Integral((A + B*x)/(sqrt(a - c*x**2)*(d + e*x)**(5/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x + A)/(sqrt(-c*x^2 + a)*(e*x + d)^(5/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x + A)/(sqrt(-c*x^2 + a)*(e*x + d)^(5/2)), x)
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {a-c\,x^2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((A + B*x)/((a - c*x^2)^(1/2)*(d + e*x)^(5/2)),x)
Output:
int((A + B*x)/((a - c*x^2)^(1/2)*(d + e*x)^(5/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a-c x^2}} \, dx=\left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x}{-c \,e^{3} x^{5}-3 c d \,e^{2} x^{4}+a \,e^{3} x^{3}-3 c \,d^{2} e \,x^{3}+3 a d \,e^{2} x^{2}-c \,d^{3} x^{2}+3 a \,d^{2} e x +a \,d^{3}}d x \right ) b +\left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c \,e^{3} x^{5}-3 c d \,e^{2} x^{4}+a \,e^{3} x^{3}-3 c \,d^{2} e \,x^{3}+3 a d \,e^{2} x^{2}-c \,d^{3} x^{2}+3 a \,d^{2} e x +a \,d^{3}}d x \right ) a \] Input:
int((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+a)^(1/2),x)
Output:
int((sqrt(d + e*x)*sqrt(a - c*x**2)*x)/(a*d**3 + 3*a*d**2*e*x + 3*a*d*e**2 *x**2 + a*e**3*x**3 - c*d**3*x**2 - 3*c*d**2*e*x**3 - 3*c*d*e**2*x**4 - c* e**3*x**5),x)*b + int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d**3 + 3*a*d**2* e*x + 3*a*d*e**2*x**2 + a*e**3*x**3 - c*d**3*x**2 - 3*c*d**2*e*x**3 - 3*c* d*e**2*x**4 - c*e**3*x**5),x)*a