\(\int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx\) [660]

Optimal result

Integrand size = 21, antiderivative size = 305 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {a+c x^2}}{e \sqrt {d+e x}}-\frac {4 \sqrt {-a} \sqrt {c} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{e^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \sqrt {c} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{e^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

-2*(c*x^2+a)^(1/2)/e/(e*x+d)^(1/2)-4*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)*c^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^2/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2
)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+4*d*EllipticF(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)*c^(1/2)*(1+c*x^2/a)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)
/e^2/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {747, 858, 733, 435, 430} \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {4 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{e^2 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{e^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {2 \sqrt {a+c x^2}}{e \sqrt {d+e x}} \]

[In]

Int[Sqrt[a + c*x^2]/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[a + c*x^2])/(e*Sqrt[d + e*x]) - (4*Sqrt[-a]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSi
n[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(e^2*Sqrt[(Sqrt[c]*(d + e*x))
/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*Sqrt[c]*d*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*a*e)/(Sqrt[-a]*Sqrt
[c]*d - a*e)])/(e^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

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]

Rule 747

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 1))), x] - Dist[2*c*(p/(e*(m + 1))), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+c x^2}}{e \sqrt {d+e x}}+\frac {(2 c) \int \frac {x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{e} \\ & = -\frac {2 \sqrt {a+c x^2}}{e \sqrt {d+e x}}+\frac {(2 c) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{e^2}-\frac {(2 c d) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{e^2} \\ & = -\frac {2 \sqrt {a+c x^2}}{e \sqrt {d+e x}}+\frac {\left (4 a \sqrt {c} \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} e^2 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (4 a \sqrt {c} d \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} e^2 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {2 \sqrt {a+c x^2}}{e \sqrt {d+e x}}-\frac {4 \sqrt {-a} \sqrt {c} \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 )}{e^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \sqrt {c} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\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 )}{e^2 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.45 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \left (e^2 \left (a+c x^2\right )+\frac {2 \sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}-\frac {2 \sqrt {a} \sqrt {c} e \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{e^3 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[In]

Integrate[Sqrt[a + c*x^2]/(d + e*x)^(3/2),x]

[Out]

(2*(e^2*(a + c*x^2) + (2*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sq
rt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqr
t[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]] -
(2*Sqrt[a]*Sqrt[c]*e*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d +
e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqr
t[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(e^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.26

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right )}{e^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {4 c \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(383\)
default	\(\frac {2 \sqrt {c \,x^{2}+a}\, \sqrt {e x +d}\, \left (2 \sqrt {-a c}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) d e -2 \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a \,e^{2}-2 \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c \,d^{2}+2 \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a \,e^{2}-c \,x^{2} e^{2}-e^{2} a \right )}{\left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) e^{3}}\)	\(648\)

[In]

int((c*x^2+a)^(1/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2*(c*e*x^2+a*e)/e^2/((x+d/e)*(c*e*x^2+a*e))^(1/2)+4*
c/e*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)
*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-(-a*c)^(1/2)/c)*Ell
ipticE(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/
c*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c x^{2} + a} \sqrt {e x + d} e^{2} + 2 \, {\left (d e x + d^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (e^{2} x + d e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )\right )}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(3*sqrt(c*x^2 + a)*sqrt(e*x + d)*e^2 + 2*(d*e*x + d^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) + 6*(e^2*x + d*e)*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)))/(e^4*x + d*e^3)

Sympy [F]

\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((c*x**2+a)**(1/2)/(e*x+d)**(3/2),x)

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**(3/2), x)

Maxima [F]

\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)/(e*x + d)^(3/2), x)

Giac [F]

\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + a)/(e*x + d)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

int((a + c*x^2)^(1/2)/(d + e*x)^(3/2),x)

[Out]

int((a + c*x^2)^(1/2)/(d + e*x)^(3/2), x)