Integrand size = 27, antiderivative size = 336 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{a c \sqrt {a-c x^2}}+\frac {(A c d+3 a B e) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {A \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} c^{3/2} \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
(e*x+d)^(1/2)*(a*(A*e+B*d)+(A*c*d+B*a*e)*x)/a/c/(-c*x^2+a)^(1/2)+(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^(1/2)/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)-A*(-a *e^2+c*d^2)*(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 = 24.56 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-c x^2} \left (-\frac {c (d+e x) (a A e+A c d x+a B (d+e x))}{-a+c x^2}-\frac {e^2 (A c d+3 a B e) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) (A c d+3 a B e) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} \left (-3 \sqrt {a} B+A \sqrt {c}\right ) \sqrt {c} e \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{a c^2 \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(3/2),x]
Output:
(Sqrt[a - c*x^2]*(-((c*(d + e*x)*(a*A*e + A*c*d*x + a*B*(d + e*x)))/(-a + c*x^2)) - (e^2*(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)*(A*c*d + 3*a*B*e)*Sqrt[(e*(Sqrt[a]/S qrt[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 - Sqrt[a]*e)] + I*Sqrt[a]*(-3*Sqr t[a]*B + A*Sqrt[c])*Sqrt[c]*e*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[(e*(Sqrt[a]/Sqr t[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))))/(a*c^2*Sqrt[d + e*x])
Time = 0.49 (sec) , antiderivative size = 343, 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 = {684, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {\int \frac {e (a (3 B d+A e)+(A c d+3 a B e) x)}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \int \frac {a (3 B d+A e)+(A c d+3 a B e) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{2 a c}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {(3 a B e+A c d) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {A \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a c}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {\sqrt {1-\frac {c x^2}{a}} (3 a B e+A c d) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {A \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a c}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 a B e+A c d) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {A \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{2 a c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (-\frac {A \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 a B e+A c d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (-\frac {A \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 a B e+A c d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {2 \sqrt {a} A \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 a B e+A c d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{a c \sqrt {a-c x^2}}-\frac {e \left (\frac {2 \sqrt {a} A \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 a B e+A c d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{2 a c}\) |
Input:
Int[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(3/2),x]
Output:
(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(a*c*Sqrt[a - c*x^2]) - (e*((-2*Sqrt[a]*(A*c*d + 3*a*B*e)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*Elli pticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sq rt[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]*A*(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 - (Sqr t[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sq rt[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 - 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])
Leaf count of result is larger than twice the leaf count of optimal. \(702\) vs. \(2(278)=556\).
Time = 4.90 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.09
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (-c e x -c d \right ) \left (\frac {\left (A c d +B a e \right ) x}{2 a \,c^{2}}+\frac {A e +B d}{2 c^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{c}\right ) \left (-c e x -c d \right )}}+\frac {2 \left (-\frac {e \left (A e +2 B d \right )}{c}+\frac {A a \,e^{2}+A c \,d^{2}+2 B a d e}{a c}-\frac {e \left (A e +B d \right )}{2 c}-\frac {d \left (A c d +B a e \right )}{c a}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (-\frac {B \,e^{2}}{c}-\frac {\left (A c d +B a e \right ) e}{2 a c}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(703\) |
| default | \(\text {Expression too large to display}\) | \(1252\) |
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*x-c*d) *(1/2*(A*c*d+B*a*e)/a/c^2*x+1/2*(A*e+B*d)/c^2)/((x^2-a/c)*(-c*e*x-c*d))^(1 /2)+2*(-e*(A*e+2*B*d)/c+(A*a*e^2+A*c*d^2+2*B*a*d*e)/a/c-1/2/c*e*(A*e+B*d)- 1/c*d*(A*c*d+B*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)*El lipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/ e-1/c*(a*c)^(1/2)))^(1/2))+2*(-B*e^2/c-1/2*(A*c*d+B*a*e)*e/a/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 = 328, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\frac {{\left (A a c d^{2} - 6 \, B a^{2} d e - 3 \, A a^{2} e^{2} - {\left (A c^{2} d^{2} - 6 \, B a c d e - 3 \, A a c e^{2}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (A a c d e + 3 \, B a^{2} e^{2} - {\left (A c^{2} d e + 3 \, B a c e^{2}\right )} x^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (B a c d e + A a c e^{2} + {\left (A c^{2} d e + B a c e^{2}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a c^{3} e x^{2} - a^{2} c^{2} e\right )}} \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/3*((A*a*c*d^2 - 6*B*a^2*d*e - 3*A*a^2*e^2 - (A*c^2*d^2 - 6*B*a*c*d*e - 3 *A*a*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*(A*a*c*d*e + 3*B*a^2*e^2 - (A*c^2*d*e + 3*B*a*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), 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*(B*a*c*d*e + A*a*c*e^2 + (A*c^2*d*e + B*a*c*e^ 2)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(a*c^3*e*x^2 - a^2*c^2*e)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\left (a - c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(3/2)/(-c*x**2+a)**(3/2),x)
Output:
Integral((A + B*x)*(d + e*x)**(3/2)/(a - c*x**2)**(3/2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (-c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(e*x + d)^(3/2)/(-c*x^2 + a)^(3/2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (-c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(e*x + d)^(3/2)/(-c*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (a-c\,x^2\right )}^{3/2}} \,d x \] Input:
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(3/2),x)
Output:
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(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:
(6*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*e**2 + 4*sqrt(d + e*x)*sqrt(a - c*x* *2)*a*c*d*e + 2*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c*d**2 - 4*sqrt(d + e*x)* sqrt(a - c*x**2)*b*c*d*e*x - 3*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*b*d*e**3 - 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**3*c*d**2*e**2 + 3*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**2*b*c*d**3*e + 3*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*c*d*e**3* x**2 + 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**2*d**4 + 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**2*d**2*e**2*x**2 - 3*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*b*c**2*d**3*e*x**2 - 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*c**3*d**4*x**2 - 3*...