3.17.89 \(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)} \, dx\)
Optimal. Leaf size=91 \[ \frac {2}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used =
{626, 51, 63, 208} \begin {gather*} \frac {2}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
[Out]
2/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - (2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*
e^2]])/(c*d^2 - a*e^2)^(3/2)
Rule 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
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 626
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx\\ &=\frac {2}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {(c d) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{c d^2-a e^2}\\ &=\frac {2}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}+\frac {(2 c d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e \left (c d^2-a e^2\right )}\\ &=\frac {2}{\left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {2 \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 55, normalized size = 0.60 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\sqrt {d+e x} \left (a e^2-c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
[Out]
(-2*Hypergeometric2F1[-1/2, 1, 1/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((-(c*d^2) + a*e^2)*Sqrt[d + e*x])
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.13, size = 105, normalized size = 1.15 \begin {gather*} \frac {2}{\sqrt {d+e x} \left (c d^2-a e^2\right )}+\frac {2 \sqrt {c} \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e^2-c d^2}}{c d^2-a e^2}\right )}{\left (a e^2-c d^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
[Out]
2/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + (2*Sqrt[c]*Sqrt[d]*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[-(c*d^2) + a*e^2]*Sqrt[d +
e*x])/(c*d^2 - a*e^2)])/(-(c*d^2) + a*e^2)^(3/2)
________________________________________________________________________________________
fricas [A] time = 0.42, size = 261, normalized size = 2.87 \begin {gather*} \left [-\frac {{\left (e x + d\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} + 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) - 2 \, \sqrt {e x + d}}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x}, -\frac {2 \, {\left ({\left (e x + d\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - \sqrt {e x + d}\right )}}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")
[Out]
[-((e*x + d)*sqrt(c*d/(c*d^2 - a*e^2))*log((c*d*e*x + 2*c*d^2 - a*e^2 + 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c
*d/(c*d^2 - a*e^2)))/(c*d*x + a*e)) - 2*sqrt(e*x + d))/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x), -2*((e*x + d)*
sqrt(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*e*x + c*d^2))
- sqrt(e*x + d))/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((4*a^4*c*d*exp(2)^4+2*a^4*sqrt(-c^2*d
^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^4-32*a^3*c^
2*d^3*exp(1)^2*exp(2)^2+16*a^3*c^2*d^3*exp(2)^3-4*a^3*c^2*d^2*exp(2)^3-16*a^3*c*d^2*sqrt(-c^2*d^3-c*d*sqrt(c^2
*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^2+8*a^3*c*d^2*sqr
t(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^3-4
*a^3*c*d*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2
)*exp(2)^3+2*a^3*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2)
)*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)^3+64*a^2*c^3*d^5*exp(1)^4-64*a
^2*c^3*d^5*exp(1)^2*exp(2)+24*a^2*c^3*d^5*exp(2)^2+16*a^2*c^3*d^4*exp(1)^2*exp(2)-4*a^2*c^3*d^4*exp(2)^2+32*a^
2*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(
2)*exp(1)^4-32*a^2*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*
c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)+12*a^2*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2
)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2+16*a^2*c^2*d^3*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*e
xp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)-4*a^2*c^2*d^3*sqrt(-c^2*d^3-c*d*s
qrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2+2*a^2*c^2*d^2*sqr
t(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2-8
*a^2*c*d^2*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt
(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)+2*a^2*c*d^2*sqrt(-c^2*d^3-c
*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2
*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)^2-4*a^2*c*d*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+
a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*ex
p(2))*exp(2)^2-4*a^2*c*d*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)^2-32*a*c^4*d^7*exp(
1)^2+16*a*c^4*d^7*exp(2)-16*a*c^4*d^6*exp(1)^2+4*a*c^4*d^6*exp(2)-16*a*c^3*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-
4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2+8*a*c^3*d^6*sqrt(-c^2*d^3-c*d
*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)-16*a*c^3*d^5*sqrt
(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2+4*
a*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(
2)*exp(2)-8*a*c^3*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*e
xp(2))*sqrt(2)*exp(1)^2+4*a*c^3*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*e
xp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)+8*a*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2
+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)
^2-2*a*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*
sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+8*a*c^2*d^3*sqrt(-c^2*d^3-c*d*sq
rt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(
1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+16*a*c^2*d^3*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp
(2))*exp(1)^2-8*a*c^2*d^3*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+2*a*c^2*d^2*sqrt(-
c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-
4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+4*a*c^2*d^2*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+
2*a*c*d^2*exp(2))*exp(2)+4*c^5*d^9+4*c^5*d^8+2*c^4*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*e
xp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)+4*c^4*d^7*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^
2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)+2*c^4*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2
+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)-2*c^3*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1
)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^
2*exp(2))-4*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp
(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))-4*c^3*d^5*(c^2*d^4-4*a*c*d^2*exp(1
)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))-2*c^3*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*
a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))-4*c^3*d^4
*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2)))/(8*a^5*d*exp(1)^2*exp(2)^4-8*a^5*d*exp(2)^5-64*a^
4*c*d^3*exp(1)^4*exp(2)^2+96*a^4*c*d^3*exp(1)^2*exp(2)^3-32*a^4*c*d^3*exp(2)^4-16*a^4*c*d^2*exp(1)^2*exp(2)^3+
16*a^4*c*d^2*exp(2)^4+128*a^3*c^2*d^5*exp(1)^6-256*a^3*c^2*d^5*exp(1)^4*exp(2)+176*a^3*c^2*d^5*exp(1)^2*exp(2)
^2-48*a^3*c^2*d^5*exp(2)^3+64*a^3*c^2*d^4*exp(1)^4*exp(2)-80*a^3*c^2*d^4*exp(1)^2*exp(2)^2+16*a^3*c^2*d^4*exp(
2)^3+8*a^3*c^2*d^3*exp(1)^2*exp(2)^2-8*a^3*c^2*d^3*exp(2)^3-64*a^2*c^3*d^7*exp(1)^4+96*a^2*c^3*d^7*exp(1)^2*ex
p(2)-32*a^2*c^3*d^7*exp(2)^2-64*a^2*c^3*d^6*exp(1)^4+80*a^2*c^3*d^6*exp(1)^2*exp(2)-16*a^2*c^3*d^6*exp(2)^2-32
*a^2*c^3*d^5*exp(1)^4+48*a^2*c^3*d^5*exp(1)^2*exp(2)-16*a^2*c^3*d^5*exp(2)^2+8*a*c^4*d^9*exp(1)^2-8*a*c^4*d^9*
exp(2)+16*a*c^4*d^8*exp(1)^2-16*a*c^4*d^8*exp(2)+8*a*c^4*d^7*exp(1)^2-8*a*c^4*d^7*exp(2))/abs(c)/abs(d)*atan(s
qrt(d+x*exp(1))/sqrt(-(c*d^2-a*exp(2)+sqrt((-c*d^2+a*exp(2))*(-c*d^2+a*exp(2))-4*c*d*(d*exp(1)^2*a-d*a*exp(2))
))/2/c/d))-(4*a^4*c*d*exp(2)^4-2*a^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*
exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^4-32*a^3*c^2*d^3*exp(1)^2*exp(2)^2+16*a^3*c^2*d^3*exp(2)^3-4*a^3*c^2*d^2*
exp(2)^3+16*a^3*c*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*e
xp(2))*sqrt(2)*exp(1)^2*exp(2)^2-8*a^3*c*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*
a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^3+4*a^3*c*d*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2
*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^3+2*a^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(
1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d
^2*exp(2))*exp(2)^3+64*a^2*c^3*d^5*exp(1)^4-64*a^2*c^3*d^5*exp(1)^2*exp(2)+24*a^2*c^3*d^5*exp(2)^2+16*a^2*c^3*
d^4*exp(1)^2*exp(2)-4*a^2*c^3*d^4*exp(2)^2-32*a^2*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^
2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4+32*a^2*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*
c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)-12*a^2*c^2*d^4*sqrt(-c^2*d
^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2-16*a^2*c^
2*d^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*e
xp(1)^2*exp(2)+4*a^2*c^2*d^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+
a*c*d*exp(2))*sqrt(2)*exp(2)^2-2*a^2*c^2*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*
a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2-8*a^2*c*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a
^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp
(2))*exp(1)^2*exp(2)+2*a^2*c*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(
2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)^2-4*a^2*c*d*sq
rt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*
d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)^2-4*a^2*c*d*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2
)^2+2*a*c*d^2*exp(2))*exp(2)^2-32*a*c^4*d^7*exp(1)^2+16*a*c^4*d^7*exp(2)-16*a*c^4*d^6*exp(1)^2+4*a*c^4*d^6*exp
(2)+16*a*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2)
)*sqrt(2)*exp(1)^2-8*a*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2)
)+a*c*d*exp(2))*sqrt(2)*exp(2)+16*a*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a
*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2-4*a*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^
2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)+8*a*c^3*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2
*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2-4*a*c^3*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2
*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)+8*a*c^2*d^4*sqrt(-c^2*d^3+
c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^
2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-2*a*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)
^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2
*exp(2))*exp(2)+8*a*c^2*d^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a
*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+16*a*c^2*d^3*(c^2*d
^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-8*a*c^2*d^3*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(
2)^2+2*a*c*d^2*exp(2))*exp(2)+2*a*c^2*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c
*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+4*a*c
^2*d^2*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(2)+4*c^5*d^9+4*c^5*d^8-2*c^4*d^8*sqrt(-c
^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)-4*c^4*d^7*sqrt
(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)-2*c^4*d^6*s
qrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)-2*c^3*d^
6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(
c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))-4*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*e
xp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*
c*d^2*exp(2))-4*c^3*d^5*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))-2*c^3*d^4*sqrt(-c^2*d^3+c*d
*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*e
xp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))-4*c^3*d^4*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2)))/(
8*a^5*d*exp(1)^2*exp(2)^4-8*a^5*d*exp(2)^5-64*a^4*c*d^3*exp(1)^4*exp(2)^2+96*a^4*c*d^3*exp(1)^2*exp(2)^3-32*a^
4*c*d^3*exp(2)^4-16*a^4*c*d^2*exp(1)^2*exp(2)^3+16*a^4*c*d^2*exp(2)^4+128*a^3*c^2*d^5*exp(1)^6-256*a^3*c^2*d^5
*exp(1)^4*exp(2)+176*a^3*c^2*d^5*exp(1)^2*exp(2)^2-48*a^3*c^2*d^5*exp(2)^3+64*a^3*c^2*d^4*exp(1)^4*exp(2)-80*a
^3*c^2*d^4*exp(1)^2*exp(2)^2+16*a^3*c^2*d^4*exp(2)^3+8*a^3*c^2*d^3*exp(1)^2*exp(2)^2-8*a^3*c^2*d^3*exp(2)^3-64
*a^2*c^3*d^7*exp(1)^4+96*a^2*c^3*d^7*exp(1)^2*exp(2)-32*a^2*c^3*d^7*exp(2)^2-64*a^2*c^3*d^6*exp(1)^4+80*a^2*c^
3*d^6*exp(1)^2*exp(2)-16*a^2*c^3*d^6*exp(2)^2-32*a^2*c^3*d^5*exp(1)^4+48*a^2*c^3*d^5*exp(1)^2*exp(2)-16*a^2*c^
3*d^5*exp(2)^2+8*a*c^4*d^9*exp(1)^2-8*a*c^4*d^9*exp(2)+16*a*c^4*d^8*exp(1)^2-16*a*c^4*d^8*exp(2)+8*a*c^4*d^7*e
xp(1)^2-8*a*c^4*d^7*exp(2))/abs(c)/abs(d)*atan(sqrt(d+x*exp(1))/sqrt(-(c*d^2-a*exp(2)-sqrt((-c*d^2+a*exp(2))*(
-c*d^2+a*exp(2))-4*c*d*(d*exp(1)^2*a-d*a*exp(2))))/2/c/d)))
________________________________________________________________________________________
maple [A] time = 0.05, size = 88, normalized size = 0.97 \begin {gather*} -\frac {2 c d \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}-\frac {2}{\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)^(1/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x),x)
[Out]
-2*c*d/(a*e^2-c*d^2)/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)*c*d)-2/(a*e^2-c*
d^2)/(e*x+d)^(1/2)
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 positive or negative?
________________________________________________________________________________________
mupad [B] time = 0.63, size = 75, normalized size = 0.82 \begin {gather*} -\frac {2}{\left (a\,e^2-c\,d^2\right )\,\sqrt {d+e\,x}}-\frac {2\,\sqrt {c}\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}}{\sqrt {a\,e^2-c\,d^2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)),x)
[Out]
- 2/((a*e^2 - c*d^2)*(d + e*x)^(1/2)) - (2*c^(1/2)*d^(1/2)*atan((c^(1/2)*d^(1/2)*(d + e*x)^(1/2))/(a*e^2 - c*d
^2)^(1/2)))/(a*e^2 - c*d^2)^(3/2)
________________________________________________________________________________________
sympy [A] time = 15.81, size = 82, normalized size = 0.90 \begin {gather*} \frac {2 c d \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {c d}{a e^{2} - c d^{2}}} \sqrt {d + e x}} \right )}}{\sqrt {\frac {c d}{a e^{2} - c d^{2}}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac {2}{\sqrt {d + e x} \left (a e^{2} - c d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
2*c*d*atan(1/(sqrt(c*d/(a*e**2 - c*d**2))*sqrt(d + e*x)))/(sqrt(c*d/(a*e**2 - c*d**2))*(a*e**2 - c*d**2)**2) -
2/(sqrt(d + e*x)*(a*e**2 - c*d**2))
________________________________________________________________________________________