\(\int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx\) [682]

Optimal result

Integrand size = 21, antiderivative size = 186 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=-\frac {2 e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \sqrt {-a} \sqrt {c} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}} \]

[Out]

-2*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(1/2)-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)*c^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/(a*e^2+c*d^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 = 0.06 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {759, 21, 733, 435} \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {2 e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )} \]

[In]

Int[1/((d + e*x)^(3/2)*Sqrt[a + c*x^2]),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {(2 c) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{c d^2+a e^2} \\ & = -\frac {2 e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {c \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{c d^2+a e^2} \\ & = -\frac {2 e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {\left (2 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} \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}} \\ & = -\frac {2 e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {2 \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 )}{\left (c d^2+a e^2\right ) \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 = 10.55 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 \left (-\frac {e^2 \left (a+c x^2\right )}{\sqrt {d+e x}}+\frac {i \sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} d+\sqrt {a} e}} \sqrt {d+e x} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-i \sqrt {a} e}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-i \sqrt {a} e}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {\frac {\sqrt {c} (d+e x)}{e \left (i \sqrt {a}+\sqrt {c} x\right )}}}\right )}{\left (c d^2 e+a e^3\right ) \sqrt {a+c x^2}} \]

[In]

Integrate[1/((d + e*x)^(3/2)*Sqrt[a + c*x^2]),x]

[Out]

(2*(-((e^2*(a + c*x^2))/Sqrt[d + e*x]) + (I*Sqrt[c]*(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*Sq
rt[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[(Sqrt[c]*(d + e
*x))/(e*(I*Sqrt[a] + Sqrt[c]*x))]))/((c*d^2*e + a*e^3)*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(628\) vs. \(2(154)=308\).

Time = 2.48 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.38

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c d \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 c e \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(629\)
default	\(-\frac {2 \left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a \,e^{2}+\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, E\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c \,d^{2}-\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) a \,e^{2}-\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e +c d}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{\sqrt {-a c}\, e -c d}}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-a c}\, e -c d}}, \sqrt {-\frac {\sqrt {-a c}\, e -c d}{\sqrt {-a c}\, e +c d}}\right ) c \,d^{2}+c \,x^{2} e^{2}+e^{2} a \right ) \sqrt {c \,x^{2}+a}\, \sqrt {e x +d}}{e \left (e^{2} a +c \,d^{2}\right ) \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right )}\)	\(653\)

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),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)/(a*e^2+c*d^2)/((x+d/e)*(c*e*x^2+a*e)
)^(1/2)+2/(a*e^2+c*d^2)*c*d*(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)*El
lipticF(((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))+2*c*e/(a*e^2
+c*d^2)*(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)
*EllipticE(((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 = 0.08 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c x^{2} + a} \sqrt {e x + d} e^{2} - 2 \, {\left (d e x + d^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (e^{2} x + d e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )\right )}}{3 \, {\left (c d^{3} e + a d e^{3} + {\left (c d^{2} e^{2} + a e^{4}\right )} x\right )}} \]

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),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) + 3*(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)))/(c*d^3*e + a*d*e^3 + (c*d^2*e^2 + a*e^4)*x)

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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