3.21.15 \(\int \frac {\sqrt {d+e x}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2015]

Optimal. Leaf size=128 \[ -\frac {3 e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) \sqrt {d+e x}}+\frac {3 \sqrt {c} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

[Out]

3*e*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))*c^(1/2)*d^(1/2)/(-a*e^2+c*d^2)^(5/2)-3*e/(-a*e
^2+c*d^2)^2/(e*x+d)^(1/2)-1/(-a*e^2+c*d^2)/(c*d*x+a*e)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 44, 53, 65, 214} \begin {gather*} -\frac {3 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )^2}-\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) (a e+c d x)}+\frac {3 \sqrt {c} \sqrt {d} e \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 )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*e)/((c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*Sqrt[d + e*x]) + (3*Sqrt[c]*Sqrt[d
]*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(5/2)

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 65

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 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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) \sqrt {d+e x}}-\frac {(3 e) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=-\frac {3 e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) \sqrt {d+e x}}-\frac {(3 c d e) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {3 e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) \sqrt {d+e x}}-\frac {(3 c d) \text {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 )}{\left (c d^2-a e^2\right )^2}\\ &=-\frac {3 e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) \sqrt {d+e x}}+\frac {3 \sqrt {c} \sqrt {d} e \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 )^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.34, size = 118, normalized size = 0.92 \begin {gather*} -\frac {2 a e^2+c d (d+3 e x)}{\left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}-\frac {3 \sqrt {c} \sqrt {d} e \tan ^{-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 )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*a*e^2 + c*d*(d + 3*e*x))/((c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[d + e*x])) - (3*Sqrt[c]*Sqrt[d]*e*ArcTan[(
Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(-(c*d^2) + a*e^2)^(5/2)

________________________________________________________________________________________

Maple [A]
time = 0.72, size = 125, normalized size = 0.98

method	result	size
derivativedivides	\(2 e \left (-\frac {1}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {c d \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {3 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\right )\)	\(125\)
default	\(2 e \left (-\frac {1}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {c d \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {3 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\right )\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e*(-1/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)-1/(a*e^2-c*d^2)^2*c*d*(1/2*(e*x+d)^(1/2)/(c*d*(e*x+d)+e^2*a-c*d^2)+3/2/(
(a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*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((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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(c*d^2-%e^2*a>0)', see `assume?
` for more d

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (112) = 224\).
time = 1.88, size = 494, normalized size = 3.86 \begin {gather*} \left [\frac {3 \, {\left (c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d x e + 2 \, c d^{2} + 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} - a e^{2}}{c d x + a e}\right ) - 2 \, {\left (3 \, c d x e + c d^{2} + 2 \, a e^{2}\right )} \sqrt {x e + d}}{2 \, {\left (c^{3} d^{6} x - a c^{2} d^{4} x e^{2} - a^{2} c d^{2} x e^{4} + a^{3} x e^{6} + {\left (a^{2} c d x^{2} + a^{3} d\right )} e^{5} - 2 \, {\left (a c^{2} d^{3} x^{2} + a^{2} c d^{3}\right )} e^{3} + {\left (c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e\right )}}, \frac {3 \, {\left (c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d x e + c d^{2}}\right ) - {\left (3 \, c d x e + c d^{2} + 2 \, a e^{2}\right )} \sqrt {x e + d}}{c^{3} d^{6} x - a c^{2} d^{4} x e^{2} - a^{2} c d^{2} x e^{4} + a^{3} x e^{6} + {\left (a^{2} c d x^{2} + a^{3} d\right )} e^{5} - 2 \, {\left (a c^{2} d^{3} x^{2} + a^{2} c d^{3}\right )} e^{3} + {\left (c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*(c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2)*sqrt(c*d/(c*d^2 - a*e^2))*log((c*d*x*e + 2*c*d^2 + 2*(c*d^
2 - a*e^2)*sqrt(x*e + d)*sqrt(c*d/(c*d^2 - a*e^2)) - a*e^2)/(c*d*x + a*e)) - 2*(3*c*d*x*e + c*d^2 + 2*a*e^2)*s
qrt(x*e + d))/(c^3*d^6*x - a*c^2*d^4*x*e^2 - a^2*c*d^2*x*e^4 + a^3*x*e^6 + (a^2*c*d*x^2 + a^3*d)*e^5 - 2*(a*c^
2*d^3*x^2 + a^2*c*d^3)*e^3 + (c^3*d^5*x^2 + a*c^2*d^5)*e), (3*(c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2)*sqrt
(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*x*e + c*d^2)) - (
3*c*d*x*e + c*d^2 + 2*a*e^2)*sqrt(x*e + d))/(c^3*d^6*x - a*c^2*d^4*x*e^2 - a^2*c*d^2*x*e^4 + a^3*x*e^6 + (a^2*
c*d*x^2 + a^3*d)*e^5 - 2*(a*c^2*d^3*x^2 + a^2*c*d^3)*e^3 + (c^3*d^5*x^2 + a*c^2*d^5)*e)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (112) = 224\).
time = 12.89, size = 452, normalized size = 3.53 \begin {gather*} \frac {c d e \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} \log {\left (- a^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + 2 a c d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} - c^{2} d^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + \sqrt {d + e x} \right )}}{2 a e^{2} - 2 c d^{2}} - \frac {c d e \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} \log {\left (a^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} - 2 a c d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + c^{2} d^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + \sqrt {d + e x} \right )}}{2 a e^{2} - 2 c d^{2}} - \frac {2 c d e \sqrt {d + e x}}{2 a^{3} e^{6} - 4 a^{2} c d^{2} e^{4} + 2 a^{2} c d e^{5} x + 2 a c^{2} d^{4} e^{2} - 4 a c^{2} d^{3} e^{3} x + 2 c^{3} d^{5} e x} - \frac {2 e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2}}{c d} - d}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2} \sqrt {\frac {a e^{2}}{c d} - d}} - \frac {2 e}{\sqrt {d + e x} \left (a e^{2} - c d^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*d*e*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3))*log(-a**2*e**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + 2*a*c*d**2*e**2
*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) - c**2*d**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + sqrt(d + e*x))/(2*a*e**
2 - 2*c*d**2) - c*d*e*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3))*log(a**2*e**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) -
2*a*c*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + c**2*d**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + sqrt(d +
e*x))/(2*a*e**2 - 2*c*d**2) - 2*c*d*e*sqrt(d + e*x)/(2*a**3*e**6 - 4*a**2*c*d**2*e**4 + 2*a**2*c*d*e**5*x + 2
*a*c**2*d**4*e**2 - 4*a*c**2*d**3*e**3*x + 2*c**3*d**5*e*x) - 2*e*atan(sqrt(d + e*x)/sqrt(a*e**2/(c*d) - d))/(
(a*e**2 - c*d**2)**2*sqrt(a*e**2/(c*d) - d)) - 2*e/(sqrt(d + e*x)*(a*e**2 - c*d**2)**2)

________________________________________________________________________________________

Giac [A]
time = 1.11, size = 171, normalized size = 1.34 \begin {gather*} -\frac {3 \, c d \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} - \frac {3 \, {\left (x e + d\right )} c d e - 2 \, c d^{2} e + 2 \, a e^{3}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} c d - \sqrt {x e + d} c d^{2} + \sqrt {x e + d} a e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.75, size = 155, normalized size = 1.21 \begin {gather*} -\frac {\frac {2\,e}{a\,e^2-c\,d^2}+\frac {3\,c\,d\,e\,\left (d+e\,x\right )}{{\left (a\,e^2-c\,d^2\right )}^2}}{\left (a\,e^2-c\,d^2\right )\,\sqrt {d+e\,x}+c\,d\,{\left (d+e\,x\right )}^{3/2}}-\frac {3\,\sqrt {c}\,\sqrt {d}\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^{5/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*e)/(a*e^2 - c*d^2) + (3*c*d*e*(d + e*x))/(a*e^2 - c*d^2)^2)/((a*e^2 - c*d^2)*(d + e*x)^(1/2) + c*d*(d +
e*x)^(3/2)) - (3*c^(1/2)*d^(1/2)*e*atan((c^(1/2)*d^(1/2)*(d + e*x)^(1/2)*(a^2*e^4 + c^2*d^4 - 2*a*c*d^2*e^2))/
(a*e^2 - c*d^2)^(5/2)))/(a*e^2 - c*d^2)^(5/2)

________________________________________________________________________________________