3.4.48 \(\int \frac {1}{x (d+e x)^2 \sqrt {a+c x^2}} \, dx\) [348]

Optimal. Leaf size=179 \[ \frac {e^2 \sqrt {a+c x^2}}{d \left (c d^2+a e^2\right ) (d+e x)}+\frac {c e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}}+\frac {e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \sqrt {c d^2+a e^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^2} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {975, 272, 65, 214, 745, 739, 212} \begin {gather*} \frac {e^2 \sqrt {a+c x^2}}{d (d+e x) \left (a e^2+c d^2\right )}+\frac {e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \sqrt {a e^2+c d^2}}+\frac {c e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 745

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*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 975

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rubi steps

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

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Mathematica [A]
time = 0.70, size = 153, normalized size = 0.85 \begin {gather*} \frac {e \left (\frac {d e \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {2 \left (2 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}\right )+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*((d*e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)) - (2*(2*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqr
t[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e^2)^(3/2)) + (2*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sq
rt[a]])/Sqrt[a])/d^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(159)=318\).
time = 0.07, size = 376, normalized size = 2.10

method	result	size
default	\(\frac {\ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}}{e d}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{2} \sqrt {a}}\)	\(376\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.19, size = 1213, normalized size = 6.78 \begin {gather*} \left [\frac {{\left (2 \, a c d^{2} x e^{2} + 2 \, a c d^{3} e + a^{2} x e^{4} + a^{2} d e^{3}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + {\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (a c d^{3} e^{2} + a^{2} d e^{4}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{6} x e + a c^{2} d^{7} + 2 \, a^{2} c d^{4} x e^{3} + 2 \, a^{2} c d^{5} e^{2} + a^{3} d^{2} x e^{5} + a^{3} d^{3} e^{4}\right )}}, -\frac {2 \, {\left (2 \, a c d^{2} x e^{2} + 2 \, a c d^{3} e + a^{2} x e^{4} + a^{2} d e^{3}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) - {\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a c d^{3} e^{2} + a^{2} d e^{4}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{6} x e + a c^{2} d^{7} + 2 \, a^{2} c d^{4} x e^{3} + 2 \, a^{2} c d^{5} e^{2} + a^{3} d^{2} x e^{5} + a^{3} d^{3} e^{4}\right )}}, \frac {2 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, a c d^{2} x e^{2} + 2 \, a c d^{3} e + a^{2} x e^{4} + a^{2} d e^{3}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (a c d^{3} e^{2} + a^{2} d e^{4}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{6} x e + a c^{2} d^{7} + 2 \, a^{2} c d^{4} x e^{3} + 2 \, a^{2} c d^{5} e^{2} + a^{3} d^{2} x e^{5} + a^{3} d^{3} e^{4}\right )}}, -\frac {{\left (2 \, a c d^{2} x e^{2} + 2 \, a c d^{3} e + a^{2} x e^{4} + a^{2} d e^{3}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) - {\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (a c d^{3} e^{2} + a^{2} d e^{4}\right )} \sqrt {c x^{2} + a}}{a c^{2} d^{6} x e + a c^{2} d^{7} + 2 \, a^{2} c d^{4} x e^{3} + 2 \, a^{2} c d^{5} e^{2} + a^{3} d^{2} x e^{5} + a^{3} d^{3} e^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((2*a*c*d^2*x*e^2 + 2*a*c*d^3*e + a^2*x*e^4 + a^2*d*e^3)*sqrt(c*d^2 + a*e^2)*log(-(2*c^2*d^2*x^2 - 2*a*c*
d*x*e + a*c*d^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2)/(x^2*e^2 + 2*d*
x*e + d^2)) + (c^2*d^4*x*e + c^2*d^5 + 2*a*c*d^2*x*e^3 + 2*a*c*d^3*e^2 + a^2*x*e^5 + a^2*d*e^4)*sqrt(a)*log(-(
c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(a*c*d^3*e^2 + a^2*d*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^6*x*e +
a*c^2*d^7 + 2*a^2*c*d^4*x*e^3 + 2*a^2*c*d^5*e^2 + a^3*d^2*x*e^5 + a^3*d^3*e^4), -1/2*(2*(2*a*c*d^2*x*e^2 + 2*a
*c*d^3*e + a^2*x*e^4 + a^2*d*e^3)*sqrt(-c*d^2 - a*e^2)*arctan(-sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 +
a)/(c^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) - (c^2*d^4*x*e + c^2*d^5 + 2*a*c*d^2*x*e^3 + 2*a*c*d^3*e^2
+ a^2*x*e^5 + a^2*d*e^4)*sqrt(a)*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*c*d^3*e^2 + a^2*d*
e^4)*sqrt(c*x^2 + a))/(a*c^2*d^6*x*e + a*c^2*d^7 + 2*a^2*c*d^4*x*e^3 + 2*a^2*c*d^5*e^2 + a^3*d^2*x*e^5 + a^3*d
^3*e^4), 1/2*(2*(c^2*d^4*x*e + c^2*d^5 + 2*a*c*d^2*x*e^3 + 2*a*c*d^3*e^2 + a^2*x*e^5 + a^2*d*e^4)*sqrt(-a)*arc
tan(sqrt(-a)/sqrt(c*x^2 + a)) + (2*a*c*d^2*x*e^2 + 2*a*c*d^3*e + a^2*x*e^4 + a^2*d*e^3)*sqrt(c*d^2 + a*e^2)*lo
g(-(2*c^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2
*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(a*c*d^3*e^2 + a^2*d*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^6*x*e + a*c^2*d^
7 + 2*a^2*c*d^4*x*e^3 + 2*a^2*c*d^5*e^2 + a^3*d^2*x*e^5 + a^3*d^3*e^4), -((2*a*c*d^2*x*e^2 + 2*a*c*d^3*e + a^2
*x*e^4 + a^2*d*e^3)*sqrt(-c*d^2 - a*e^2)*arctan(-sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(c^2*d^2*x
^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) - (c^2*d^4*x*e + c^2*d^5 + 2*a*c*d^2*x*e^3 + 2*a*c*d^3*e^2 + a^2*x*e^5 +
a^2*d*e^4)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (a*c*d^3*e^2 + a^2*d*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^6*x
*e + a*c^2*d^7 + 2*a^2*c*d^4*x*e^3 + 2*a^2*c*d^5*e^2 + a^3*d^2*x*e^5 + a^3*d^3*e^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (160) = 320\).
time = 1.23, size = 600, normalized size = 3.35 \begin {gather*} {\left (\frac {\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}} d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{c d^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{2} + a d^{3} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{2}} - \frac {{\left (2 \, \sqrt {c d^{2} + a e^{2}} c d^{2} \arctan \left (\frac {{\left (\sqrt {c} d - \sqrt {c d^{2} + a e^{2}}\right )} e^{\left (-1\right )}}{\sqrt {-a}}\right ) e + 2 \, \sqrt {-a} c d^{2} e^{2} \log \left ({\left | -c d + \sqrt {c d^{2} + a e^{2}} \sqrt {c} \right |}\right ) + 2 \, \sqrt {c d^{2} + a e^{2}} a \arctan \left (\frac {{\left (\sqrt {c} d - \sqrt {c d^{2} + a e^{2}}\right )} e^{\left (-1\right )}}{\sqrt {-a}}\right ) e^{3} + \sqrt {c d^{2} + a e^{2}} \sqrt {-a} \sqrt {c} d e^{2} + \sqrt {-a} a e^{4} \log \left ({\left | -c d + \sqrt {c d^{2} + a e^{2}} \sqrt {c} \right |}\right )\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c d^{2} + a e^{2}} \sqrt {-a} c d^{4} + \sqrt {c d^{2} + a e^{2}} \sqrt {-a} a d^{2} e^{2}} + \frac {{\left (2 \, c d^{2} e^{2} + a e^{4}\right )} \log \left ({\left | -c d + \sqrt {c d^{2} + a e^{2}} {\left (\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} + a e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{{\left (c d^{4} + a d^{2} e^{2}\right )} \sqrt {c d^{2} + a e^{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {2 \, \arctan \left (\frac {{\left (d {\left (\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} + a e^{4}} e^{\left (-1\right )}}{x e + d}\right )} - \sqrt {c d^{2} + a e^{2}}\right )} e^{\left (-1\right )}}{\sqrt {-a}}\right ) e}{\sqrt {-a} d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)*d^2*e^2*sgn(1/(x*e + d))/(c*d^5*sgn(1/(x*e
+ d))^2 + a*d^3*e^2*sgn(1/(x*e + d))^2) - (2*sqrt(c*d^2 + a*e^2)*c*d^2*arctan((sqrt(c)*d - sqrt(c*d^2 + a*e^2)
)*e^(-1)/sqrt(-a))*e + 2*sqrt(-a)*c*d^2*e^2*log(abs(-c*d + sqrt(c*d^2 + a*e^2)*sqrt(c))) + 2*sqrt(c*d^2 + a*e^
2)*a*arctan((sqrt(c)*d - sqrt(c*d^2 + a*e^2))*e^(-1)/sqrt(-a))*e^3 + sqrt(c*d^2 + a*e^2)*sqrt(-a)*sqrt(c)*d*e^
2 + sqrt(-a)*a*e^4*log(abs(-c*d + sqrt(c*d^2 + a*e^2)*sqrt(c))))*sgn(1/(x*e + d))/(sqrt(c*d^2 + a*e^2)*sqrt(-a
)*c*d^4 + sqrt(c*d^2 + a*e^2)*sqrt(-a)*a*d^2*e^2) + (2*c*d^2*e^2 + a*e^4)*log(abs(-c*d + sqrt(c*d^2 + a*e^2)*(
sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 + a*e^4)*e^(-1)/(x*e + d)))
)/((c*d^4 + a*d^2*e^2)*sqrt(c*d^2 + a*e^2)*sgn(1/(x*e + d))) + 2*arctan((d*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(
x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 + a*e^4)*e^(-1)/(x*e + d)) - sqrt(c*d^2 + a*e^2))*e^(-1)/sqrt
(-a))*e/(sqrt(-a)*d^2*sgn(1/(x*e + d))))*e^(-1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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