\(\int \frac {1}{(d+e x)^{3/2} (a-c x^2)} \, dx\) [616]

Optimal result

Integrand size = 20, antiderivative size = 160 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\frac {2 e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]

[Out]

-c^(1/4)*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))/a^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(3/2)+c^(1
/4)*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))/a^(1/2)/(e*a^(1/2)+d*c^(1/2))^(3/2)+2*e/(-a*e^2
+c*d^2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {724, 841, 1180, 214} \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=-\frac {\sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}+\frac {2 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

(2*e)/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - (c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])
/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) + (c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e
]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^(3/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 724

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

Rule 841

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

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {2 e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {c \int \frac {d-e x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{c d^2-a e^2} \\ & = \frac {2 e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {2 d e-e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{c d^2-a e^2} \\ & = \frac {2 e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {c \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )}+\frac {c \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )} \\ & = \frac {2 e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\frac {2 e}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^2}+\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(-2 e \left (\frac {1}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}-\frac {c^{2} \left (-\frac {\left (c d -\sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-c d -\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{e^{2} a -c \,d^{2}}\right )\)	\(188\)
default	\(2 e \left (-\frac {1}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {e x +d}}+\frac {c^{2} \left (-\frac {\left (c d -\sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-c d -\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{e^{2} a -c \,d^{2}}\right )\)	\(188\)
pseudoelliptic	\(-\frac {2 \left (\frac {c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \left (c d +\sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\frac {c \sqrt {e x +d}\, \left (c d -\sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\right )\right ) e}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {e x +d}\, \left (e^{2} a -c \,d^{2}\right )}\)	\(226\)

[In]

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

[Out]

-2*e*(1/(a*e^2-c*d^2)/(e*x+d)^(1/2)-c^2/(a*e^2-c*d^2)*(-1/2*(c*d-(a*c*e^2)^(1/2))/(a*c*e^2)^(1/2)/c/((c*d+(a*c
*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+1/2*(-c*d-(a*c*e^2)^(1/2))/(a*c
*e^2)^(1/2)/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2861 vs. \(2 (118) = 236\).

Time = 0.33 (sec) , antiderivative size = 2861, normalized size of antiderivative = 17.88 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*
a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 +
15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*
a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 + 2
*a^2*c*d*e^4 - (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e
^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^
8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*
e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^
4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^
4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (
a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/
(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*
e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sq
rt(e*x + d) - (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 - (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*s
qrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a
^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6
- 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^1
2 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7
*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) + (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*
x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e
^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 +
15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^
6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 + (a*c^4*d^8 - 2*a^2*c^3*d^6*
e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^
10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c
^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c
^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^
2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (c*
d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^
2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*
c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*
d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2 + 2*a^2*c*d
*e^4 + (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2
*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^
6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^
4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^
4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 +
3*a^3*c*d^2*e^4 - a^4*e^6))) + 4*sqrt(e*x + d)*e)/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (118) = 236\).

Time = 0.32 (sec) , antiderivative size = 656, normalized size of antiderivative = 4.10 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\frac {2 \, e}{{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d}} - \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} a e {\left | c \right |} - 2 \, {\left (\sqrt {a c} c d^{3} e - \sqrt {a c} a d e^{3}\right )} {\left | -c d^{2} e + a e^{3} \right |} {\left | c \right |} + {\left (c^{3} d^{6} e - 2 \, a c^{2} d^{4} e^{3} + a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} + \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} + \sqrt {a c} c^{2} d^{5} - 2 \, \sqrt {a c} a c d^{3} e^{2} + \sqrt {a c} a^{2} d e^{4}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | -c d^{2} e + a e^{3} \right |}} - \frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} a e {\left | c \right |} + 2 \, {\left (\sqrt {a c} c d^{3} e - \sqrt {a c} a d e^{3}\right )} {\left | -c d^{2} e + a e^{3} \right |} {\left | c \right |} + {\left (c^{3} d^{6} e - 2 \, a c^{2} d^{4} e^{3} + a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} - a c d e^{2} - \sqrt {{\left (c^{2} d^{3} - a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - a c e^{2}\right )}}}{c^{2} d^{2} - a c e^{2}}}}\right )}{{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} - \sqrt {a c} c^{2} d^{5} + 2 \, \sqrt {a c} a c d^{3} e^{2} - \sqrt {a c} a^{2} d e^{4}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | -c d^{2} e + a e^{3} \right |}} \]

[In]

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

[Out]

2*e/((c*d^2 - a*e^2)*sqrt(e*x + d)) - ((c*d^2*e - a*e^3)^2*a*e*abs(c) - 2*(sqrt(a*c)*c*d^3*e - sqrt(a*c)*a*d*e
^3)*abs(-c*d^2*e + a*e^3)*abs(c) + (c^3*d^6*e - 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*abs(c))*arctan(sqrt(e*x + d)/
sqrt(-(c^2*d^3 - a*c*d*e^2 + sqrt((c^2*d^3 - a*c*d*e^2)^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 - a*c
*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5 + sqrt(a*c)*c^2*d^5 - 2*sqrt(a*c)*a*c*
d^3*e^2 + sqrt(a*c)*a^2*d*e^4)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(-c*d^2*e + a*e^3)) - ((c*d^2*e - a*e^3)^2*a*e*
abs(c) + 2*(sqrt(a*c)*c*d^3*e - sqrt(a*c)*a*d*e^3)*abs(-c*d^2*e + a*e^3)*abs(c) + (c^3*d^6*e - 2*a*c^2*d^4*e^3
+ a^2*c*d^2*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d^3 - a*c*d*e^2 - sqrt((c^2*d^3 - a*c*d*e^2)^2 - (c^
2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 - a*c*e^2)))/(c^2*d^2 - a*c*e^2)))/((a*c^2*d^4*e - 2*a^2*c*d^2*e^3 +
a^3*e^5 - sqrt(a*c)*c^2*d^5 + 2*sqrt(a*c)*a*c*d^3*e^2 - sqrt(a*c)*a^2*d*e^4)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs
(-c*d^2*e + a*e^3))

Mupad [B] (verification not implemented)

Time = 10.61 (sec) , antiderivative size = 4412, normalized size of antiderivative = 27.58 \[ \int \frac {1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan((((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2
*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*
e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2
*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2
- 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10)
- 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 - 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(
a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c
*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*1i + ((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e
^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a
^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(a^3
*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d
^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c
^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 - 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*
d^5*e^7 - 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/
(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*1i)/(((d + e*x)^(1/2)*(16*a^4*c^4*e^1
0 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e
^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e
*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*
d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^
4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) - 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 +
256*a^2*c^7*d^7*e^5 - 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*
c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2) -
((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 + a
*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*
a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*
c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6
*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c
^8*d^9*e^3 + 64*a^5*c^4*d*e^11 - 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 - 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^
3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4
+ 3*a^3*c^2*d^4*e^2)))^(1/2) + 16*a^3*c^4*e^9 - 16*c^7*d^6*e^3 + 48*a*c^6*d^4*e^5 - 48*a^2*c^5*d^2*e^7))*(-(a
*c^2*d^3 + a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 + 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*
d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*2i - atan((((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*
d^6*e^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/
(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3
*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*
c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*
a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) - 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 - 384*a^3
*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1
/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*1i + ((d + e*x)^(1/2)*(16*a^4*c^
4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*
c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((
d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2
*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d
^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e
^11 - 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 - 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3
*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/
2)*1i)/(((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e^4 - 32*a^3*c^5*d^2*e^8) + (-(a*c^2
*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*
e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2
*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2
- 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10)
- 64*a*c^8*d^9*e^3 - 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 - 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(
a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c
*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2) - ((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 + 32*a*c^7*d^6*e^4
- 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*
e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3*(a^3*c)
^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*
e^2)))^(1/2)*(64*a*c^9*d^11*e^2 - 64*a^6*c^4*d*e^12 - 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 - 640*a^4*c^6*
d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 - 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5
*e^7 - 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/2))/(4*
(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2) + 16*a^3*c^4*e^9 - 16*c^7*d^6*e^3 + 48*a
*c^6*d^4*e^5 - 48*a^2*c^5*d^2*e^7))*(-(a*c^2*d^3 - a*e^3*(a^3*c)^(1/2) + 3*a^2*c*d*e^2 - 3*c*d^2*e*(a^3*c)^(1/
2))/(4*(a^5*e^6 - a^2*c^3*d^6 - 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*2i - (2*e)/((a*e^2 - c*d^2)*(d +
e*x)^(1/2))