Integrand size = 27, antiderivative size = 438 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\frac {(a B+A c x) \sqrt {d+e x}}{3 a c \left (a-c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c \left (4 A c d^2-a B d e-3 a A e^2\right ) x\right )}{6 a^2 c \left (c d^2-a e^2\right ) \sqrt {a-c x^2}}+\frac {\left (4 A c d^2-a B d e-3 a A e^2\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} \sqrt {c} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {(4 A c d-a B e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
1/3*(A*c*x+B*a)*(e*x+d)^(1/2)/a/c/(-c*x^2+a)^(3/2)-1/6*(e*x+d)^(1/2)*(a*e* (A*c*d-B*a*e)-c*(-3*A*a*e^2+4*A*c*d^2-B*a*d*e)*x)/a^2/c/(-a*e^2+c*d^2)/(-c *x^2+a)^(1/2)+1/6*(-3*A*a*e^2+4*A*c*d^2-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^(3/2)/c^(1/2)/(-a*e^2+c*d^2)/(c^(1/2)*(e *x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)-1/6*(4*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)*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))/a^(3/2)/c^(3/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.79 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-c x^2} \left (\frac {(d+e x) \left (2 a \left (c d^2-a e^2\right ) (a B+A c x)+\left (a-c x^2\right ) \left (a^2 B e^2+4 A c^2 d^2 x-a c e (B d x+A (d+3 e x))\right )\right )}{\left (a-c x^2\right )^2}+\frac {e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (4 A c d^2-a B d e-3 a A e^2\right ) \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 A c d^2-a B d e-3 a A e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 A c d-a B e+3 \sqrt {a} A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )}\right )}{6 a^2 c \left (c d^2-a e^2\right ) \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^(5/2),x]
Output:
(Sqrt[a - c*x^2]*(((d + e*x)*(2*a*(c*d^2 - a*e^2)*(a*B + A*c*x) + (a - c*x ^2)*(a^2*B*e^2 + 4*A*c^2*d^2*x - a*c*e*(B*d*x + A*(d + 3*e*x)))))/(a - c*x ^2)^2 + (e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(4*A*c*d^2 - a*B*d*e - 3*a*A*e ^2)*(a - c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(4*A*c*d^2 - a*B*d*e - 3*a*A*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)*EllipticE[I*ArcSinh[Sqrt[-d + ( Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - S qrt[a]*e)] + I*Sqrt[a]*e*(Sqrt[c]*d - Sqrt[a]*e)*(4*A*c*d - a*B*e + 3*Sqrt [a]*A*Sqrt[c]*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)])/(e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2))))/(6*a^2*c *(c*d^2 - a*e^2)*Sqrt[d + e*x])
Time = 0.66 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {685, 27, 686, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 685 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}-\frac {\int -\frac {4 A c d-a B e+3 A c e x}{2 \sqrt {d+e x} \left (a-c x^2\right )^{3/2}}dx}{3 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 A c d-a B e+3 A c e x}{\sqrt {d+e x} \left (a-c x^2\right )^{3/2}}dx}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int \frac {c e \left (a e (A c d-a B e)+c \left (4 A c d^2-a B e d-3 a A e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {a e (A c d-a B e)+c \left (4 A c d^2-a B e d-3 a A e^2\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {e \left (\frac {c \left (-3 a A e^2-a B d e+4 A c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) (4 A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {e \left (\frac {c \sqrt {1-\frac {c x^2}{a}} \left (-3 a A e^2-a B d e+4 A c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) (4 A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\left (c d^2-a e^2\right ) (4 A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a A e^2-a B d e+4 A c d^2\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\left (c d^2-a e^2\right ) (4 A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a A e^2-a B d e+4 A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (4 A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a A e^2-a B d e+4 A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} (4 A c d-a B e) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a A e^2-a B d e+4 A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} (4 A c d-a B e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a A e^2-a B d e+4 A c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-3 a A e^2-a B d e+4 A c d^2\right )\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{3 a c \left (a-c x^2\right )^{3/2}}\) |
Input:
Int[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^(5/2),x]
Output:
((a*B + A*c*x)*Sqrt[d + e*x])/(3*a*c*(a - c*x^2)^(3/2)) + (-((Sqrt[d + e*x ]*(a*e*(A*c*d - a*B*e) - c*(4*A*c*d^2 - a*B*d*e - 3*a*A*e^2)*x))/(a*(c*d^2 - a*e^2)*Sqrt[a - c*x^2])) - (e*((-2*Sqrt[a]*Sqrt[c]*(4*A*c*d^2 - 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[( Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]* (4*A*c*d - a*B*e)*(c*d^2 - a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sq rt[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]*Sq rt[a - c*x^2])))/(2*a*(c*d^2 - a*e^2)))/(6*a*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*(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_))^(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])
Leaf count of result is larger than twice the leaf count of optimal. \(835\) vs. \(2(368)=736\).
Time = 1.87 (sec) , antiderivative size = 836, normalized size of antiderivative = 1.91
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {A x}{3 c^{2} a}+\frac {B}{3 c^{3}}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{\left (x^{2}-\frac {a}{c}\right )^{2}}-\frac {2 \left (-c e x -c d \right ) \left (\frac {\left (3 A a \,e^{2}-4 A c \,d^{2}+B a d e \right ) x}{12 c \,a^{2} \left (a \,e^{2}-c \,d^{2}\right )}+\frac {e \left (A c d -B a e \right )}{12 c^{2} a \left (a \,e^{2}-c \,d^{2}\right )}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (\frac {4 A c d -B a e}{6 c \,a^{2}}-\frac {e^{2} \left (A c d -B a e \right )}{12 c a \left (a \,e^{2}-c \,d^{2}\right )}-\frac {d \left (3 A a \,e^{2}-4 A c \,d^{2}+B a d e \right )}{6 a^{2} \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 {e \left (3 A a \,e^{2}-4 A c \,d^{2}+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 )}{6 \left (a \,e^{2}-c \,d^{2}\right ) a^{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}}\) | \(836\) |
| default | \(\text {Expression too large to display}\) | \(3507\) |
Input:
int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*((1/3*A/c^2/a*x+ 1/3*B/c^3)*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x^2-a/c)^2-2*(-c*e*x-c*d)*( 1/12/c*(3*A*a*e^2-4*A*c*d^2+B*a*d*e)/a^2/(a*e^2-c*d^2)*x+1/12*e*(A*c*d-B*a *e)/c^2/a/(a*e^2-c*d^2))/((x^2-a/c)*(-c*e*x-c*d))^(1/2)+2*(1/6/c*(4*A*c*d- B*a*e)/a^2-1/12/c*e^2*(A*c*d-B*a*e)/a/(a*e^2-c*d^2)-1/6*d*(3*A*a*e^2-4*A*c *d^2+B*a*d*e)/a^2/(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))-1/6*e*(3*A*a*e^2-4*A*c*d^2+B*a*d*e)/( a*e^2-c*d^2)/a^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 = 662, normalized size of antiderivative = 1.51 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=-\frac {{\left (4 \, A a^{2} c^{2} d^{3} - B a^{3} c d^{2} e - 6 \, A a^{3} c d e^{2} + 3 \, B a^{4} e^{3} + {\left (4 \, A c^{4} d^{3} - B a c^{3} d^{2} e - 6 \, A a c^{3} d e^{2} + 3 \, B a^{2} c^{2} e^{3}\right )} x^{4} - 2 \, {\left (4 \, A a c^{3} d^{3} - B a^{2} c^{2} d^{2} e - 6 \, A a^{2} c^{2} d e^{2} + 3 \, B a^{3} c e^{3}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (4 \, A a^{2} c^{2} d^{2} e - B a^{3} c d e^{2} - 3 \, A a^{3} c e^{3} + {\left (4 \, A c^{4} d^{2} e - B a c^{3} d e^{2} - 3 \, A a c^{3} e^{3}\right )} x^{4} - 2 \, {\left (4 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - 3 \, A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (2 \, B a^{2} c^{2} d^{2} e - A a^{2} c^{2} d e^{2} - B a^{3} c e^{3} - {\left (4 \, A c^{4} d^{2} e - B a c^{3} d e^{2} - 3 \, A a c^{3} e^{3}\right )} x^{3} + {\left (A a c^{3} d e^{2} - B a^{2} c^{2} e^{3}\right )} x^{2} + {\left (6 \, A a c^{3} d^{2} e - B a^{2} c^{2} d e^{2} - 5 \, A a^{2} c^{2} e^{3}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{4} c^{3} d^{2} e - a^{5} c^{2} e^{3} + {\left (a^{2} c^{5} d^{2} e - a^{3} c^{4} e^{3}\right )} x^{4} - 2 \, {\left (a^{3} c^{4} d^{2} e - a^{4} c^{3} e^{3}\right )} x^{2}\right )}} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/18*((4*A*a^2*c^2*d^3 - B*a^3*c*d^2*e - 6*A*a^3*c*d*e^2 + 3*B*a^4*e^3 + (4*A*c^4*d^3 - B*a*c^3*d^2*e - 6*A*a*c^3*d*e^2 + 3*B*a^2*c^2*e^3)*x^4 - 2* (4*A*a*c^3*d^3 - B*a^2*c^2*d^2*e - 6*A*a^2*c^2*d*e^2 + 3*B*a^3*c*e^3)*x^2) *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*(4*A*a^2*c^2*d^2*e - B*a^3* c*d*e^2 - 3*A*a^3*c*e^3 + (4*A*c^4*d^2*e - B*a*c^3*d*e^2 - 3*A*a*c^3*e^3)* x^4 - 2*(4*A*a*c^3*d^2*e - B*a^2*c^2*d*e^2 - 3*A*a^2*c^2*e^3)*x^2)*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*a^2*c^2*d^2*e - A*a^2 *c^2*d*e^2 - B*a^3*c*e^3 - (4*A*c^4*d^2*e - B*a*c^3*d*e^2 - 3*A*a*c^3*e^3) *x^3 + (A*a*c^3*d*e^2 - B*a^2*c^2*e^3)*x^2 + (6*A*a*c^3*d^2*e - B*a^2*c^2* d*e^2 - 5*A*a^2*c^2*e^3)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(a^4*c^3*d^2*e - a^5*c^2*e^3 + (a^2*c^5*d^2*e - a^3*c^4*e^3)*x^4 - 2*(a^3*c^4*d^2*e - a^ 4*c^3*e^3)*x^2)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a - c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(1/2)/(-c*x**2+a)**(5/2),x)
Output:
Integral((A + B*x)*sqrt(d + e*x)/(a - c*x**2)**(5/2), x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (-c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*sqrt(e*x + d)/(-c*x^2 + a)^(5/2), x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (-c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*sqrt(e*x + d)/(-c*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (a-c\,x^2\right )}^{5/2}} \,d x \] Input:
int(((A + B*x)*(d + e*x)^(1/2))/(a - c*x^2)^(5/2),x)
Output:
int(((A + B*x)*(d + e*x)^(1/2))/(a - c*x^2)^(5/2), x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(5/2),x)
Output:
( - sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**2 - sqrt( a - c*x**2)*a**2*e**2*x**2 - 2*sqrt(a - c*x**2)*a*c*d**2*x**2 + 2*sqrt(a - c*x**2)*a*c*e**2*x**4 + sqrt(a - c*x**2)*c**2*d**2*x**4 - sqrt(a - c*x**2 )*c**2*e**2*x**6),x)*a**4*d*e**2 - sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqr t(a - c*x**2)*a**2*d**2 - sqrt(a - c*x**2)*a**2*e**2*x**2 - 2*sqrt(a - c*x **2)*a*c*d**2*x**2 + 2*sqrt(a - c*x**2)*a*c*e**2*x**4 + sqrt(a - c*x**2)*c **2*d**2*x**4 - sqrt(a - c*x**2)*c**2*e**2*x**6),x)*a**3*b*d**2*e + 6*sqrt (a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**2 - sqrt(a - c*x* *2)*a**2*e**2*x**2 - 2*sqrt(a - c*x**2)*a*c*d**2*x**2 + 2*sqrt(a - c*x**2) *a*c*e**2*x**4 + sqrt(a - c*x**2)*c**2*d**2*x**4 - sqrt(a - c*x**2)*c**2*e **2*x**6),x)*a**3*c*d**3 + 2*sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**2 - sqrt(a - c*x**2)*a**2*e**2*x**2 - 2*sqrt(a - c*x**2)*a *c*d**2*x**2 + 2*sqrt(a - c*x**2)*a*c*e**2*x**4 + sqrt(a - c*x**2)*c**2*d* *2*x**4 - sqrt(a - c*x**2)*c**2*e**2*x**6),x)*a**3*c*d*e**2*x**2 + 2*sqrt( a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**2 - sqrt(a - c*x** 2)*a**2*e**2*x**2 - 2*sqrt(a - c*x**2)*a*c*d**2*x**2 + 2*sqrt(a - c*x**2)* a*c*e**2*x**4 + sqrt(a - c*x**2)*c**2*d**2*x**4 - sqrt(a - c*x**2)*c**2*e* *2*x**6),x)*a**2*b*c*d**2*e*x**2 - 12*sqrt(a - c*x**2)*int(sqrt(d + e*x)/( sqrt(a - c*x**2)*a**2*d**2 - sqrt(a - c*x**2)*a**2*e**2*x**2 - 2*sqrt(a - c*x**2)*a*c*d**2*x**2 + 2*sqrt(a - c*x**2)*a*c*e**2*x**4 + sqrt(a - c*x...