Integrand size = 27, antiderivative size = 403 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{3 a c \left (a-c x^2\right )^{3/2}}-\frac {\sqrt {d+e x} (a A e-(4 A c d-3 a B e) x)}{6 a^2 c \sqrt {a-c x^2}}+\frac {(4 A c d-3 a B e) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {\left (4 A c d^2-3 a B d e-a A e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{6 a^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
1/3*(e*x+d)^(1/2)*(a*(A*e+B*d)+(A*c*d+B*a*e)*x)/a/c/(-c*x^2+a)^(3/2)-1/6*( e*x+d)^(1/2)*(a*A*e-(4*A*c*d-3*B*a*e)*x)/a^2/c/(-c*x^2+a)^(1/2)+1/6*(4*A*c *d-3*B*a*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^(3/2)/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)- 1/6*(-A*a*e^2+4*A*c*d^2-3*B*a*d*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 = 24.77 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-c x^2} \left (\frac {(d+e x) \left (-4 A c^2 d x^3+a^2 (2 B d+A e-B e x)+a c x \left (6 A d+A e x+3 B e x^2\right )\right )}{a^2 c \left (a-c x^2\right )^2}+\frac {e^2 (4 A c d-3 a B e) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )-i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) (4 A c d-3 a B e) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )-i \sqrt {a} \sqrt {c} e \left (4 A c d-3 a B e-\sqrt {a} A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{a^2 c^2 e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{6 \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(5/2),x]
Output:
(Sqrt[a - c*x^2]*(((d + e*x)*(-4*A*c^2*d*x^3 + a^2*(2*B*d + A*e - B*e*x) + a*c*x*(6*A*d + A*e*x + 3*B*e*x^2)))/(a^2*c*(a - c*x^2)^2) + (e^2*(4*A*c*d - 3*a*B*e)*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2) - I*Sqrt[c]*(Sqrt[ c]*d - Sqrt[a]*e)*(4*A*c*d - 3*a*B*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)*Ellip ticE[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]*Sqrt[c]*e*(4*A*c*d - 3*a* B*e - 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*ArcS inh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e) /(Sqrt[c]*d - Sqrt[a]*e)])/(a^2*c^2*e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(6*Sqrt[d + e*x])
Time = 0.66 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {684, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}-\frac {\int -\frac {4 A c d^2-3 a B e d-a A e^2+3 e (A c d-a B e) x}{2 \sqrt {d+e x} \left (a-c x^2\right )^{3/2}}dx}{3 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 A c d^2-3 a B e d-a A e^2+3 e (A c d-a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )^{3/2}}dx}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int \frac {c e \left (c d^2-a e^2\right ) (a A e+(4 A c d-3 a B e) x)}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {a A e+(4 A c d-3 a B e) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {e \left (\frac {(4 A c d-3 a B e) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (-a A e^2-3 a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\sqrt {1-\frac {c x^2}{a}} (4 A c d-3 a B e) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (-a A e^2-3 a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\left (-a A e^2-3 a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 A c d-3 a B e) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\left (-a A e^2-3 a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 A c d-3 a B e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (-a A e^2-3 a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 A c d-3 a B e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-a A e^2-3 a B d e+4 A c d^2\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 A c d-3 a B e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-a A e^2-3 a B d e+4 A c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (4 A c d-3 a B e) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a}-\frac {\sqrt {d+e x} \left (a A e \left (c d^2-a e^2\right )-x \left (c d^2-a e^2\right ) (4 A c d-3 a B e)\right )}{a \sqrt {a-c x^2} \left (c d^2-a e^2\right )}}{6 a c}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{3 a c \left (a-c x^2\right )^{3/2}}\) |
Input:
Int[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(5/2),x]
Output:
(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(3*a*c*(a - c*x^2)^(3/ 2)) + (-((Sqrt[d + e*x]*(a*A*e*(c*d^2 - a*e^2) - (4*A*c*d - 3*a*B*e)*(c*d^ 2 - a*e^2)*x))/(a*(c*d^2 - a*e^2)*Sqrt[a - c*x^2])) - (e*((-2*Sqrt[a]*(4*A *c*d - 3*a*B*e)*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]*(4*A*c*d^2 - 3*a*B*d*e - a*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])))/(2*a))/(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 - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 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. \(749\) vs. \(2(333)=666\).
Time = 6.09 (sec) , antiderivative size = 750, normalized size of antiderivative = 1.86
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {\left (A c d +B a e \right ) x}{3 c^{3} a}+\frac {A e +B d}{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 (4 A c d -3 B a e \right ) x}{12 a^{2} c^{2}}-\frac {A e}{12 a \,c^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (-\frac {A a \,e^{2}-4 A c \,d^{2}+3 B a d e}{6 c \,a^{2}}+\frac {A \,e^{2}}{12 a c}-\frac {d \left (4 A c d -3 B a e \right )}{6 c \,a^{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 {\left (4 A c d -3 B a e \right ) e \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 a^{2} c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(750\) |
| default | \(\text {Expression too large to display}\) | \(2786\) |
Input:
int((B*x+A)*(e*x+d)^(3/2)/(-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*d+B*a *e)/c^3/a*x+1/3*(A*e+B*d)/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*(4*A*c*d-3*B*a*e)/a^2/c^2*x-1/12*A*e/a/c^2)/((x^2 -a/c)*(-c*e*x-c*d))^(1/2)+2*(-1/6/c*(A*a*e^2-4*A*c*d^2+3*B*a*d*e)/a^2+1/12 /a/c*A*e^2-1/6/c*d*(4*A*c*d-3*B*a*e)/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)*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*(4*A*c*d-3*B*a*e)*e/a^2/ c*(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 = 0.10 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=-\frac {{\left (4 \, A a^{2} c d^{2} - 3 \, B a^{3} d e - 3 \, A a^{3} e^{2} + {\left (4 \, A c^{3} d^{2} - 3 \, B a c^{2} d e - 3 \, A a c^{2} e^{2}\right )} x^{4} - 2 \, {\left (4 \, A a c^{2} d^{2} - 3 \, B a^{2} c d e - 3 \, A a^{2} c e^{2}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (4 \, A a^{2} c d e - 3 \, B a^{3} e^{2} + {\left (4 \, A c^{3} d e - 3 \, B a c^{2} e^{2}\right )} x^{4} - 2 \, {\left (4 \, A a c^{2} d e - 3 \, B a^{2} c e^{2}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (A a c^{2} e^{2} x^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2} - {\left (4 \, A c^{3} d e - 3 \, B a c^{2} e^{2}\right )} x^{3} + {\left (6 \, A a c^{2} d e - B a^{2} c e^{2}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{2} c^{4} e x^{4} - 2 \, a^{3} c^{3} e x^{2} + a^{4} c^{2} e\right )}} \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/18*((4*A*a^2*c*d^2 - 3*B*a^3*d*e - 3*A*a^3*e^2 + (4*A*c^3*d^2 - 3*B*a*c ^2*d*e - 3*A*a*c^2*e^2)*x^4 - 2*(4*A*a*c^2*d^2 - 3*B*a^2*c*d*e - 3*A*a^2*c *e^2)*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*d*e - 3*B*a^3*e^2 + (4*A*c^3*d*e - 3*B*a*c^2*e^2)*x^4 - 2*(4*A*a*c^2*d*e - 3*B*a ^2*c*e^2)*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*(A*a *c^2*e^2*x^2 + 2*B*a^2*c*d*e + A*a^2*c*e^2 - (4*A*c^3*d*e - 3*B*a*c^2*e^2) *x^3 + (6*A*a*c^2*d*e - B*a^2*c*e^2)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(a ^2*c^4*e*x^4 - 2*a^3*c^3*e*x^2 + a^4*c^2*e)
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)*(e*x+d)**(3/2)/(-c*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (-c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(e*x + d)^(3/2)/(-c*x^2 + a)^(5/2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (-c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(e*x + d)^(3/2)/(-c*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (a-c\,x^2\right )}^{5/2}} \,d x \] Input:
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(5/2),x)
Output:
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(5/2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(5/2),x)
Output:
(3*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*b*d*e**3 - 3*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*c*d**2*e**2 - 3*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*b*c*d**3*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*b*c*d*e**3*x* *2 + 3*sqrt(a - c*x**2)*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a**2*d**2 - sq rt(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**2*d**4 + 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 ...