Integrand size = 27, antiderivative size = 311 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\frac {(a B+A c x) \sqrt {d+e x}}{a c \sqrt {a-c x^2}}+\frac {A \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {(A c d-a B e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} c^{3/2} \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
(A*c*x+B*a)*(e*x+d)^(1/2)/a/c/(-c*x^2+a)^(1/2)+A*(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)/c^(1/2)/(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)*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^(1/2)/ c^(3/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.55 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-c x^2} \left (A e+\frac {(a B+A c x) (d+e x)}{a-c x^2}-\frac {i A \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}+\frac {i \sqrt {a} \left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{a c \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^(3/2),x]
Output:
(Sqrt[a - c*x^2]*(A*e + ((a*B + A*c*x)*(d + e*x))/(a - c*x^2) - (I*A*Sqrt[ c]*(Sqrt[c]*d - Sqrt[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)*EllipticE[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)])/(e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) + (I*Sqrt[a]*(Sqrt[a]*B - A*Sqrt[c])*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 ticF[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*Sqrt[d + e*x])
Time = 0.45 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {685, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 685 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {\int \frac {e (a B+A c x)}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \int \frac {a B+A c x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{2 a c}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {A c \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {(A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a c}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {A c \sqrt {1-\frac {c x^2}{a}} \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {(A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a c}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (-\frac {(A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} A \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (-\frac {(A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} A \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (-\frac {\sqrt {1-\frac {c x^2}{a}} (A c d-a B e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} A \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} (A c d-a B e) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} A \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} (A c d-a B e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} A \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
Input:
Int[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^(3/2),x]
Output:
((a*B + A*c*x)*Sqrt[d + e*x])/(a*c*Sqrt[a - c*x^2]) - (e*((-2*Sqrt[a]*A*Sq rt[c]*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]*(A*c*d - a*B*e)*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*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])
Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(253)=506\).
Time = 1.84 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.01
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (-c e x -c d \right ) \left (\frac {A x}{2 a c}+\frac {B}{2 c^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (-\frac {3 B e}{2 c}+\frac {A c d +B a e}{a c}-\frac {A d}{a}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}-\frac {A e \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{a \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(625\) |
| default | \(\frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, \left (A \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) a c \,e^{2}-A \sqrt {a c}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) c d e -A \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) a c \,e^{2}+A \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) c^{2} d^{2}-B \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) a c d e +B \sqrt {a c}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) a \,e^{2}+A \,c^{2} e^{2} x^{2}+A \,c^{2} d e x +B a c \,e^{2} x +B a c d e \right )}{c^{2} a \left (-c e \,x^{3}-c d \,x^{2}+a e x +a d \right ) e}\) | \(930\) |
Input:
int((B*x+A)*(e*x+d)^(1/2)/(-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*x-c*d) *(1/2*A/a/c*x+1/2*B/c^2)/((x^2-a/c)*(-c*e*x-c*d))^(1/2)+2*(-3/2*B*e/c+(A*c *d+B*a*e)/a/c-A/a*d)*(d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2))) ^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1 /2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*Elli pticF(((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))-A*e/a*(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 = 0.12 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\frac {{\left (A a c d - 3 \, B a^{2} e - {\left (A c^{2} d - 3 \, B a c e\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) - 3 \, {\left (A c^{2} e x^{2} - A a c e\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (A c^{2} e x + B a c e\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a c^{3} e x^{2} - a^{2} c^{2} e\right )}} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/3*((A*a*c*d - 3*B*a^2*e - (A*c^2*d - 3*B*a*c*e)*x^2)*sqrt(-c*e)*weierstr assPInverse(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*c^2*e*x^2 - A*a*c*e)*sqrt(-c*e)*weierstrassZ eta(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), weie rstrassPInverse(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*c^2*e*x + B*a*c*e)*sqrt(-c*x^2 + a)*sqr t(e*x + d))/(a*c^3*e*x^2 - a^2*c^2*e)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a - c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(1/2)/(-c*x**2+a)**(3/2),x)
Output:
Integral((A + B*x)*sqrt(d + e*x)/(a - c*x**2)**(3/2), x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (-c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*sqrt(e*x + d)/(-c*x^2 + a)^(3/2), x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (-c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x + A)*sqrt(e*x + d)/(-c*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (a-c\,x^2\right )}^{3/2}} \,d x \] Input:
int(((A + B*x)*(d + e*x)^(1/2))/(a - c*x^2)^(3/2),x)
Output:
int(((A + B*x)*(d + e*x)^(1/2))/(a - c*x^2)^(3/2), x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(3/2),x)
Output:
(2*sqrt(d + e*x)*sqrt(a - c*x**2)*a*e + 2*sqrt(d + e*x)*sqrt(a - c*x**2)*b *d - int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a*d**2 - sqrt(a - c*x**2)*a*e**2* x**2 - sqrt(a - c*x**2)*c*d**2*x**2 + sqrt(a - c*x**2)*c*e**2*x**4),x)*a** 3*d*e**2 - int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a*d**2 - sqrt(a - c*x**2)*a *e**2*x**2 - sqrt(a - c*x**2)*c*d**2*x**2 + sqrt(a - c*x**2)*c*e**2*x**4), x)*a**2*b*d**2*e + 2*int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a*d**2 - sqrt(a - c*x**2)*a*e**2*x**2 - sqrt(a - c*x**2)*c*d**2*x**2 + sqrt(a - c*x**2)*c*e **2*x**4),x)*a**2*c*d**3 + int(sqrt(d + e*x)/(sqrt(a - c*x**2)*a*d**2 - sq rt(a - c*x**2)*a*e**2*x**2 - sqrt(a - c*x**2)*c*d**2*x**2 + sqrt(a - c*x** 2)*c*e**2*x**4),x)*a**2*c*d*e**2*x**2 + int(sqrt(d + e*x)/(sqrt(a - c*x**2 )*a*d**2 - sqrt(a - c*x**2)*a*e**2*x**2 - sqrt(a - c*x**2)*c*d**2*x**2 + s qrt(a - c*x**2)*c*e**2*x**4),x)*a*b*c*d**2*e*x**2 - 2*int(sqrt(d + e*x)/(s qrt(a - c*x**2)*a*d**2 - sqrt(a - c*x**2)*a*e**2*x**2 - sqrt(a - c*x**2)*c *d**2*x**2 + sqrt(a - c*x**2)*c*e**2*x**4),x)*a*c**2*d**3*x**2 - int((sqrt (d + e*x)*sqrt(a - c*x**2)*x**2)/(a**2*d + a**2*e*x - 2*a*c*d*x**2 - 2*a*c *e*x**3 + c**2*d*x**4 + c**2*e*x**5),x)*a**2*c*e**2 + int((sqrt(d + e*x)*s qrt(a - c*x**2)*x**2)/(a**2*d + a**2*e*x - 2*a*c*d*x**2 - 2*a*c*e*x**3 + c **2*d*x**4 + c**2*e*x**5),x)*a*b*c*d*e + int((sqrt(d + e*x)*sqrt(a - c*x** 2)*x**2)/(a**2*d + a**2*e*x - 2*a*c*d*x**2 - 2*a*c*e*x**3 + c**2*d*x**4 + c**2*e*x**5),x)*a*c**2*e**2*x**2 - int((sqrt(d + e*x)*sqrt(a - c*x**2)*...