Integrand size = 31, antiderivative size = 562 \[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {2 B \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {b} d \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {\sqrt {c} (B c-A d) \sqrt {e} \arctan \left (\frac {\sqrt {b c^2+a d^2} \sqrt {e x}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{d^{3/2} \sqrt {b c^2+a d^2}}-\frac {2 \sqrt [4]{a} B \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{b^{3/4} d \sqrt {a+b x^2}}-\frac {\sqrt [4]{a} \left (\sqrt {a} B d-\sqrt {b} (2 B c-A d)\right ) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{b^{3/4} d \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {a+b x^2}}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) (B c-A d) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2}{4 \sqrt {a} \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d^2 \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {a+b x^2}} \] Output:
2*B*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/d/(a^(1/2)+b^(1/2)*x)+c^(1/2)*(-A* d+B*c)*e^(1/2)*arctan((a*d^2+b*c^2)^(1/2)*(e*x)^(1/2)/c^(1/2)/d^(1/2)/e^(1 /2)/(b*x^2+a)^(1/2))/d^(3/2)/(a*d^2+b*c^2)^(1/2)-2*a^(1/4)*B*e^(1/2)*(a^(1 /2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arc tan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))/b^(3/4)/d/(b*x^2+a) ^(1/2)-a^(1/4)*(a^(1/2)*B*d-b^(1/2)*(-A*d+2*B*c))*e^(1/2)*(a^(1/2)+b^(1/2) *x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/ 4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/b^(3/4)/d/(b^(1/2)*c-a^(1/2)* d)/(b*x^2+a)^(1/2)-1/2*(b^(1/2)*c+a^(1/2)*d)*(-A*d+B*c)*e^(1/2)*(a^(1/2)+b ^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticPi(sin(2*arctan( b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),-1/4*(b^(1/2)*c-a^(1/2)*d)^2/a^(1/2) /b^(1/2)/c/d,1/2*2^(1/2))/a^(1/4)/b^(1/4)/d^2/(b^(1/2)*c-a^(1/2)*d)/(b*x^2 +a)^(1/2)
Result contains complex when optimal does not.
Time = 22.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {2 \sqrt {e x} \sqrt {1+\frac {b x^2}{a}} \left (\sqrt {a} B d E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b} x}{\sqrt {a}}}\right )\right |-1\right )+i \left (\left (i \sqrt {a} B d+\sqrt {b} (B c-A d)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b} x}{\sqrt {a}}}\right ),-1\right )+\sqrt {b} (-B c+A d) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{\sqrt {b} c},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b} x}{\sqrt {a}}}\right ),-1\right )\right )\right )}{\sqrt {b} d^2 \sqrt {\frac {i \sqrt {b} x}{\sqrt {a}}} \sqrt {a+b x^2}} \] Input:
Integrate[(Sqrt[e*x]*(A + B*x))/((c + d*x)*Sqrt[a + b*x^2]),x]
Output:
(2*Sqrt[e*x]*Sqrt[1 + (b*x^2)/a]*(Sqrt[a]*B*d*EllipticE[I*ArcSinh[Sqrt[(I* Sqrt[b]*x)/Sqrt[a]]], -1] + I*((I*Sqrt[a]*B*d + Sqrt[b]*(B*c - A*d))*Ellip ticF[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1] + Sqrt[b]*(-(B*c) + A*d)* EllipticPi[((-I)*Sqrt[a]*d)/(Sqrt[b]*c), I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt [a]]], -1])))/(Sqrt[b]*d^2*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[a + b*x^2])
Time = 2.35 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2354, 2233, 27, 1510, 2227, 27, 761, 2221}
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.
| \(\displaystyle \int \frac {\sqrt {e x} (A+B x)}{\sqrt {a+b x^2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2354 |
| \(\displaystyle \frac {2 \int \frac {e x (A e+B x e)}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 2233 |
| \(\displaystyle \frac {2 \left (\frac {e^2 \int \frac {\sqrt {b} \left (\sqrt {a} B c e+\left (\sqrt {a} B d-\sqrt {b} (B c-A d)\right ) x e\right )}{e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{b d}-\frac {\sqrt {a} B e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} d}\right )}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {e \int \frac {\sqrt {a} B c e+\left (\sqrt {a} B d-\sqrt {b} (B c-A d)\right ) x e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} d}-\frac {B \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} d}\right )}{e}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 \left (\frac {e \int \frac {\sqrt {a} B c e+\left (\sqrt {a} B d-\sqrt {b} (B c-A d)\right ) x e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} d}-\frac {B \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {b} d}\right )}{e}\) |
| \(\Big \downarrow \) 2227 |
| \(\displaystyle \frac {2 \left (\frac {e \left (-\frac {\sqrt {a} \left (\sqrt {a} B d-\sqrt {b} (2 B c-A d)\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} c-\sqrt {a} d}-\frac {\sqrt {a} \sqrt {b} c e (B c-A d) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} d}-\frac {B \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {b} d}\right )}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {e \left (-\frac {\sqrt {a} \left (\sqrt {a} B d-\sqrt {b} (2 B c-A d)\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} c-\sqrt {a} d}-\frac {\sqrt {b} c (B c-A d) \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} d}-\frac {B \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {b} d}\right )}{e}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 \left (\frac {e \left (-\frac {\sqrt {b} c (B c-A d) \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b} c-\sqrt {a} d}-\frac {\sqrt [4]{a} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \left (\sqrt {a} B d-\sqrt {b} (2 B c-A d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {e} \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right )}\right )}{\sqrt {b} d}-\frac {B \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {b} d}\right )}{e}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {2 \left (\frac {e \left (-\frac {\sqrt {b} c (B c-A d) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {b} c-\sqrt {a} d}-\frac {\sqrt [4]{a} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \left (\sqrt {a} B d-\sqrt {b} (2 B c-A d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {e} \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right )}\right )}{\sqrt {b} d}-\frac {B \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {b} d}\right )}{e}\) |
Input:
Int[(Sqrt[e*x]*(A + B*x))/((c + d*x)*Sqrt[a + b*x^2]),x]
Output:
(2*(-((B*(-((e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(Sqrt[a]*e + Sqrt[b]*e*x)) + ( a^(1/4)*Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a ]*e + Sqrt[b]*e*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt [e])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])))/(Sqrt[b]*d)) + (e*(-1/2*(a^(1/4)* (Sqrt[a]*B*d - Sqrt[b]*(2*B*c - A*d))*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^ 2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sq rt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(b^(1/4)*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[e ]*Sqrt[a + b*x^2]) - (Sqrt[b]*c*(B*c - A*d)*(-1/2*((Sqrt[b]*c - Sqrt[a]*d) *Sqrt[e]*ArcTan[(Sqrt[b*c^2 + a*d^2]*Sqrt[e*x])/(Sqrt[c]*Sqrt[d]*Sqrt[e]*S qrt[a + b*x^2])])/(Sqrt[c]*Sqrt[d]*Sqrt[b*c^2 + a*d^2]) + ((Sqrt[b]*c + Sq rt[a]*d)*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + S qrt[b]*e*x)^2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[b]*c)/Sqrt[a] - d)^2)/(Sqrt [b]*c*d), 2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(4*a^(1/4 )*b^(1/4)*c*d*Sqrt[e]*Sqrt[a + b*x^2])))/(Sqrt[b]*c - Sqrt[a]*d)))/(Sqrt[b ]*d)))/e
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_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e ) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x ], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) - a*B*(e + d*q) )/(c*d^2 - a*e^2) Int[1/Sqrt[a + c*x^4], x], x] + Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2)) Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x] , x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff [P4x, x, 4]}, Simp[-C/(e*q) Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim p[1/(c*e) Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x ^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(Px_)*((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2) ^(p_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[(Px /. x -> x^k/e)*x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x ], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolyQ[Px, x] && FractionQ[m]
Time = 1.24 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(\frac {\left (A \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b c d -A \sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) d^{2}-A \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {-a b}\, d}{d \sqrt {-a b}-b c}, \frac {\sqrt {2}}{2}\right ) b c d +B \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}-B \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}+2 B \sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c d +B \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {-a b}\, d}{d \sqrt {-a b}-b c}, \frac {\sqrt {2}}{2}\right ) b \,c^{2}-2 B \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}-2 B \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c d \right ) \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-a b}\, \sqrt {e x}}{b \sqrt {b \,x^{2}+a}\, d^{2} \left (b c -d \sqrt {-a b}\right ) x}\) | \(443\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {x e \left (b \,x^{2}+a \right )}\, \left (\frac {\left (A d -B c \right ) e \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} b \sqrt {b e \,x^{3}+a e x}}+\frac {B e \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{d b \sqrt {b e \,x^{3}+a e x}}-\frac {c \left (A d -B c \right ) e \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{b \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{3} b \sqrt {b e \,x^{3}+a e x}\, \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(492\) |
Input:
int((e*x)^(1/2)*(B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(A*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*b*c*d-A* (-a*b)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2) )*d^2-A*EllipticPi(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),(-a*b)^(1/2)*d/ (d*(-a*b)^(1/2)-b*c),1/2*2^(1/2))*b*c*d+B*EllipticF(((b*x+(-a*b)^(1/2))/(- a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2-B*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b )^(1/2))^(1/2),1/2*2^(1/2))*b*c^2+2*B*(-a*b)^(1/2)*EllipticF(((b*x+(-a*b)^ (1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*c*d+B*EllipticPi(((b*x+(-a*b)^(1/2 ))/(-a*b)^(1/2))^(1/2),(-a*b)^(1/2)*d/(d*(-a*b)^(1/2)-b*c),1/2*2^(1/2))*b* c^2-2*B*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d ^2-2*B*(-a*b)^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2* 2^(1/2))*c*d)*(-b/(-a*b)^(1/2)*x)^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2)) ^(1/2)*2^(1/2)*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-a*b)^(1/2)*(e*x)^ (1/2)/b/(b*x^2+a)^(1/2)/d^2/(b*c-d*(-a*b)^(1/2))/x
Timed out. \[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(1/2)*(B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {e x} \left (A + B x\right )}{\sqrt {a + b x^{2}} \left (c + d x\right )}\, dx \] Input:
integrate((e*x)**(1/2)*(B*x+A)/(d*x+c)/(b*x**2+a)**(1/2),x)
Output:
Integral(sqrt(e*x)*(A + B*x)/(sqrt(a + b*x**2)*(c + d*x)), x)
\[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x}}{\sqrt {b x^{2} + a} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(1/2)*(B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="maxima ")
Output:
integrate((B*x + A)*sqrt(e*x)/(sqrt(b*x^2 + a)*(d*x + c)), x)
\[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x}}{\sqrt {b x^{2} + a} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(1/2)*(B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x + A)*sqrt(e*x)/(sqrt(b*x^2 + a)*(d*x + c)), x)
Timed out. \[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {e\,x}\,\left (A+B\,x\right )}{\sqrt {b\,x^2+a}\,\left (c+d\,x\right )} \,d x \] Input:
int(((e*x)^(1/2)*(A + B*x))/((a + b*x^2)^(1/2)*(c + d*x)),x)
Output:
int(((e*x)^(1/2)*(A + B*x))/((a + b*x^2)^(1/2)*(c + d*x)), x)
\[ \int \frac {\sqrt {e x} (A+B x)}{(c+d x) \sqrt {a+b x^2}} \, dx=\sqrt {e}\, \left (\left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}\, x}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) b +\left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b d \,x^{3}+b c \,x^{2}+a d x +a c}d x \right ) a \right ) \] Input:
int((e*x)^(1/2)*(B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x)
Output:
sqrt(e)*(int((sqrt(x)*sqrt(a + b*x**2)*x)/(a*c + a*d*x + b*c*x**2 + b*d*x* *3),x)*b + int((sqrt(x)*sqrt(a + b*x**2))/(a*c + a*d*x + b*c*x**2 + b*d*x* *3),x)*a)