\(\int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx\) [680]

Optimal result

Integrand size = 21, antiderivative size = 136 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}} \]

[Out]

-2*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)
*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/c^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {733, 435} \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}} \]

[In]

Int[Sqrt[d + e*x]/Sqrt[a + c*x^2],x]

[Out]

(-2*Sqrt[-a]*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a
*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {-a} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 20.54 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\frac {2 i \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} d+\sqrt {a} e}} \sqrt {d+e x} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-i \sqrt {a} e}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-i \sqrt {a} e}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (d+e x)}{e \left (i \sqrt {a}+\sqrt {c} x\right )}} \sqrt {a+c x^2}} \]

[In]

Integrate[Sqrt[d + e*x]/Sqrt[a + c*x^2],x]

[Out]

((2*I)*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[d + e*x]*
(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[
c]*d + I*Sqrt[a]*e)] - EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d
- I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(Sqrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x))/(e*(I*Sqrt[a] + Sqrt[c]*x))]*
Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(108)=216\).

Time = 2.14 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.91

method	result	size
default	\(\frac {2 \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}\, \left (-\sqrt {-a c}\, e +c d \right ) \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, \left (\sqrt {-a c}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) e -\sqrt {-a c}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) e +d F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c -E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c d \right )}{e \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) c^{2}}\)	\(396\)
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 d \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 e \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(556\)

[In]

int((e*x+d)^(1/2)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(-(-a*c)^(1/2)*e+c*d)*(-c*(e*x+d)/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1
/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*((-a*c)^(1/2)*EllipticF((
-c*(e*x+d)/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*e-(-a*c)^(1/2)*Elli
pticE((-c*(e*x+d)/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*e+d*Elliptic
F((-c*(e*x+d)/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c-EllipticE((-c*
(e*x+d)/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*d)/e/(c*e*x^3+c*d*x^
2+a*e*x+a*d)/c^2

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {c e} d {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) - 3 \, \sqrt {c e} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )\right )}}{3 \, c e} \]

[In]

integrate((e*x+d)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

2/3*(2*sqrt(c*e)*d*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*sqrt(c*e)*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)))/(
c*e)

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {a + c x^{2}}}\, dx \]

[In]

integrate((e*x+d)**(1/2)/(c*x**2+a)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/sqrt(a + c*x**2), x)

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + a}} \,d x } \]

[In]

integrate((e*x+d)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/sqrt(c*x^2 + a), x)

Giac [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + a}} \,d x } \]

[In]

integrate((e*x+d)^(1/2)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)/sqrt(c*x^2 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {c\,x^2+a}} \,d x \]

[In]

int((d + e*x)^(1/2)/(a + c*x^2)^(1/2),x)

[Out]

int((d + e*x)^(1/2)/(a + c*x^2)^(1/2), x)