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

Optimal. Leaf size=160 \[ \frac {2 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {\sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {710, 827, 1166, 208} \begin {gather*} \frac {2 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\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 {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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 208

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

Rule 710

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 827

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 1166

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*} \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 {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) \operatorname {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 \operatorname {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 \operatorname {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 [C]  time = 0.09, size = 136, normalized size = 0.85 \begin {gather*} \frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {a} e}\right )+\left (\sqrt {a} e-\sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {d+e x} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqr
t[c]*d) + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*(c
*d^2 - a*e^2)*Sqrt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.47, size = 238, normalized size = 1.49 \begin {gather*} \frac {2 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]*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)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) - (Sqrt[c]*Arc
Tan[(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
)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

________________________________________________________________________________________

fricas [B]  time = 0.47, size = 2861, normalized size = 17.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [B]  time = 0.72, size = 686, normalized size = 4.29 \begin {gather*} -\frac {{\left ({\left (c d^{2} e - a e^{3}\right )}^{2} \sqrt {a c} a {\left | c \right |} e - 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} + {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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^{3} d^{5} - \sqrt {a c} a c^{2} d^{4} e - 2 \, a^{2} c^{2} d^{3} e^{2} + 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} + a^{3} c d e^{4} - \sqrt {a c} a^{3} e^{5}\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} \sqrt {a c} a {\left | c \right |} e + 2 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} {\left | c d^{2} e - a e^{3} \right |} {\left | c \right |} + {\left (\sqrt {a c} c^{3} d^{6} e - 2 \, \sqrt {a c} a c^{2} d^{4} e^{3} + \sqrt {a c} a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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^{3} d^{5} + \sqrt {a c} a c^{2} d^{4} e - 2 \, a^{2} c^{2} d^{3} e^{2} - 2 \, \sqrt {a c} a^{2} c d^{2} e^{3} + a^{3} c d e^{4} + \sqrt {a c} a^{3} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c d^{2} e - a e^{3} \right |}} + \frac {2 \, e}{{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.07, size = 291, normalized size = 1.82 \begin {gather*} -\frac {c^{2} d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {c e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {c e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 e}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{{\left (c x^{2} - a\right )} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 1.33, size = 4412, normalized size = 27.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________