Integrand size = 27, antiderivative size = 452 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx=-\frac {2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}}+\frac {\sqrt {d+e x} \left (a \left (3 B c d^2-4 A c d e+a B e^2\right )+c \left (A c d^2-4 a B d e+3 a A e^2\right ) x\right )}{a \left (c d^2-a e^2\right )^2 \sqrt {a-c x^2}}+\frac {\sqrt {c} \left (A c d^2-4 a B d e+3 a A e^2\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \left (c d^2-a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {(A c d-a B e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {c} \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
(2*A*e-2*B*d)/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)+(e*x+d)^(1/2)* (a*(-4*A*c*d*e+B*a*e^2+3*B*c*d^2)+c*(3*A*a*e^2+A*c*d^2-4*B*a*d*e)*x)/a/(-a *e^2+c*d^2)^2/(-c*x^2+a)^(1/2)+c^(1/2)*(3*A*a*e^2+A*c*d^2-4*B*a*d*e)*(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))/a^(1/2)/(-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)-(A*c*d-B *a*e)*(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c*x^2/a)^(1/2)*Elli pticF(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))/a^(1/2)/c^(1/2)/(-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 = 24.92 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-c x^2} \left (2 a e^2 (B d-A e)+e \left (A c d^2-4 a B d e+3 a A e^2\right )-\frac {(d+e x) \left (a^2 B e^2+A c^2 d^2 x+a c (B d (d-2 e x)+A e (-2 d+e x))\right )}{-a+c x^2}-\frac {i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (A c d^2-4 a B d e+3 a A e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}-\frac {i \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (A c d-a B e+3 \sqrt {a} \sqrt {c} (B d-A e)\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{a \left (c d^2-a e^2\right )^2 \sqrt {d+e x}} \] Input:
Integrate[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)^(3/2)),x]
Output:
(Sqrt[a - c*x^2]*(2*a*e^2*(B*d - A*e) + e*(A*c*d^2 - 4*a*B*d*e + 3*a*A*e^2 ) - ((d + e*x)*(a^2*B*e^2 + A*c^2*d^2*x + a*c*(B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(-a + c*x^2) - (I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(A*c*d^2 - 4* a*B*d*e + 3*a*A*e^2)*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sq rt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sq rt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt [c]*d - Sqrt[a]*e)])/(e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) - (I* Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)*(A*c*d - a*B*e + 3*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)^(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)])/(Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(a*(c*d^2 - a*e^2)^2 *Sqrt[d + e*x])
Time = 0.68 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {686, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right )^{3/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {\int -\frac {c e (3 a (B d-A e)+(A c d-a B e) x)}{2 (d+e x)^{3/2} \sqrt {a-c x^2}}dx}{a c \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {3 a (B d-A e)+(A c d-a B e) x}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {e \left (\frac {2 \int \frac {a \left (3 B c d^2-4 A c e d+a B e^2\right )-c \left (A c d^2-4 a B e d+3 a A e^2\right ) x}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {\int \frac {a \left (3 B c d^2-4 A c e d+a B e^2\right )-c \left (A c d^2-4 a B e d+3 a A e^2\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {e \left (\frac {\frac {\left (c d^2-a e^2\right ) (A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {c \left (3 a A e^2-4 a B d e+A c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {e \left (\frac {\frac {\left (c d^2-a e^2\right ) (A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {c \sqrt {1-\frac {c x^2}{a}} \left (3 a A e^2-4 a B d e+A c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {e \left (\frac {\frac {\left (c d^2-a e^2\right ) (A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^2-4 a B d e+A c d^2\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {e \left (\frac {\frac {\left (c d^2-a e^2\right ) (A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^2-4 a B d e+A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {e \left (\frac {\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^2-4 a B d e+A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {e \left (\frac {\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^2-4 a B d e+A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} (A c d-a B e) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {e \left (\frac {\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^2-4 a B d e+A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} (A c d-a B e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^2-4 a B d e+A c d^2\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{2 a \left (c d^2-a e^2\right )}+\frac {x (A c d-a B e)+a (B d-A e)}{a \sqrt {a-c x^2} \sqrt {d+e x} \left (c d^2-a e^2\right )}\) |
Input:
Int[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)^(3/2)),x]
Output:
(a*(B*d - A*e) + (A*c*d - a*B*e)*x)/(a*(c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[ a - c*x^2]) + (e*((-2*(A*c*d^2 - 4*a*B*d*e + 3*a*A*e^2)*Sqrt[a - c*x^2])/( (c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((2*Sqrt[a]*Sqrt[c]*(A*c*d^2 - 4*a*B*d*e + 3*a*A*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)])/(e*Sqrt[(S qrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) - (2*Sqrt[a]*( A*c*d - a*B*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)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[ a - c*x^2]))/(c*d^2 - a*e^2)))/(2*a*(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[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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. \(883\) vs. \(2(390)=780\).
Time = 8.42 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.96
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {2 c e \left (\frac {\left (3 A a \,e^{2}+A c \,d^{2}-4 B a d e \right ) x^{2}}{2 a \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}-\frac {\left (A c d -B a e \right ) x}{2 a c e \left (a \,e^{2}-c \,d^{2}\right )}-\frac {2 A a \,e^{3}+2 A c \,d^{2} e -3 B a d \,e^{2}-B c \,d^{3}}{2 c e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\right )}{\sqrt {-\left (x^{3}+\frac {d \,x^{2}}{e}-\frac {a x}{c}-\frac {a d}{c e}\right ) c e}}+\frac {2 \left (-\frac {6 A a c d \,e^{2}-2 A \,c^{2} d^{3}-3 B \,e^{3} a^{2}-B a c \,d^{2} e}{2 a \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {A c d -B a e}{a \left (a \,e^{2}-c \,d^{2}\right )}\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 {c e \left (3 A a \,e^{2}+A c \,d^{2}-4 B a d 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 )}{a \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\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}}\) | \(884\) |
| default | \(\text {Expression too large to display}\) | \(2069\) |
Input:
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),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*c*e*(1/2*(3*A *a*e^2+A*c*d^2-4*B*a*d*e)/a/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)*x^2-1/2*(A*c*d -B*a*e)/a/c/e/(a*e^2-c*d^2)*x-1/2*(2*A*a*e^3+2*A*c*d^2*e-3*B*a*d*e^2-B*c*d ^3)/c/e/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4))/(-(x^3+d/e*x^2-a/c*x-a/c*d/e)*c*e )^(1/2)+2*(-1/2*(6*A*a*c*d*e^2-2*A*c^2*d^3-3*B*a^2*e^3-B*a*c*d^2*e)/a/(a^2 *e^4-2*a*c*d^2*e^2+c^2*d^4)+(A*c*d-B*a*e)/a/(a*e^2-c*d^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))-c*e*(3*A*a* e^2+A*c*d^2-4*B*a*d*e)/a/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)*(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))))
Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (390) = 780\).
Time = 0.12 (sec) , antiderivative size = 804, normalized size of antiderivative = 1.78 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/3*((A*a*c^2*d^4 + 5*B*a^2*c*d^3*e - 9*A*a^2*c*d^2*e^2 + 3*B*a^3*d*e^3 - (A*c^3*d^3*e + 5*B*a*c^2*d^2*e^2 - 9*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*x^3 - (A*c^3*d^4 + 5*B*a*c^2*d^3*e - 9*A*a*c^2*d^2*e^2 + 3*B*a^2*c*d*e^3)*x^2 + (A*a*c^2*d^3*e + 5*B*a^2*c*d^2*e^2 - 9*A*a^2*c*d*e^3 + 3*B*a^3*e^4)*x)*sq rt(-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*(A*a*c^2*d^3*e - 4*B*a^2*c*d^2 *e^2 + 3*A*a^2*c*d*e^3 - (A*c^3*d^2*e^2 - 4*B*a*c^2*d*e^3 + 3*A*a*c^2*e^4) *x^3 - (A*c^3*d^3*e - 4*B*a*c^2*d^2*e^2 + 3*A*a*c^2*d*e^3)*x^2 + (A*a*c^2* d^2*e^2 - 4*B*a^2*c*d*e^3 + 3*A*a^2*c*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), weierstra ssPInverse(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*a*c^2*d^3*e - 2*A*a*c^2*d^2*e^2 + 3*B*a^2*c* d*e^3 - 2*A*a^2*c*e^4 + (A*c^3*d^2*e^2 - 4*B*a*c^2*d*e^3 + 3*A*a*c^2*e^4)* x^2 + (A*c^3*d^3*e - B*a*c^2*d^2*e^2 - A*a*c^2*d*e^3 + B*a^2*c*e^4)*x)*sqr t(-c*x^2 + a)*sqrt(e*x + d))/(a^2*c^3*d^5*e - 2*a^3*c^2*d^3*e^3 + a^4*c*d* e^5 - (a*c^4*d^4*e^2 - 2*a^2*c^3*d^2*e^4 + a^3*c^2*e^6)*x^3 - (a*c^4*d^5*e - 2*a^2*c^3*d^3*e^3 + a^3*c^2*d*e^5)*x^2 + (a^2*c^3*d^4*e^2 - 2*a^3*c^2*d ^2*e^4 + a^4*c*e^6)*x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (a - c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a)**(3/2),x)
Output:
Integral((A + B*x)/((a - c*x**2)**(3/2)*(d + e*x)**(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x + A)/((-c*x^2 + a)^(3/2)*(e*x + d)^(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x + A)/((-c*x^2 + a)^(3/2)*(e*x + d)^(3/2)), x)
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (a-c\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:
int((A + B*x)/((a - c*x^2)^(3/2)*(d + e*x)^(3/2)),x)
Output:
int((A + B*x)/((a - c*x^2)^(3/2)*(d + e*x)^(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x)
Output:
( - sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**3)/(a**2*d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*e*x**3 - 2*a*c *e**2*x**4 + c**2*d**2*x**4 + 2*c**2*d*e*x**5 + c**2*e**2*x**6),x)*a*c*d*e **2 - sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**3)/(a**2*d** 2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*e*x**3 - 2*a *c*e**2*x**4 + c**2*d**2*x**4 + 2*c**2*d*e*x**5 + c**2*e**2*x**6),x)*a*c*e **3*x - 2*sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**3)/(a**2 *d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*e*x**3 - 2*a*c*e**2*x**4 + c**2*d**2*x**4 + 2*c**2*d*e*x**5 + c**2*e**2*x**6),x)*b *c*d**2*e - 2*sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**3)/( a**2*d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*e*x* *3 - 2*a*c*e**2*x**4 + c**2*d**2*x**4 + 2*c**2*d*e*x**5 + c**2*e**2*x**6), x)*b*c*d*e**2*x - sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x)/ (a**2*d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*e*x **3 - 2*a*c*e**2*x**4 + c**2*d**2*x**4 + 2*c**2*d*e*x**5 + c**2*e**2*x**6) ,x)*a**2*d*e**2 - sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x)/ (a**2*d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*e*x **3 - 2*a*c*e**2*x**4 + c**2*d**2*x**4 + 2*c**2*d*e*x**5 + c**2*e**2*x**6) ,x)*a**2*e**3*x - 4*sqrt(a - c*x**2)*int((sqrt(d + e*x)*sqrt(a - c*x**2))/ (a**2*d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 - 2*a*c*d**2*x**2 - 4*a*c*d*...