3.4.49 \(\int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx\) [349]

Optimal. Leaf size=212 \[ -\frac {\sqrt {a+c x^2}}{a d^2 x}-\frac {e^3 \sqrt {a+c x^2}}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {c e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \left (c d^2+a e^2\right )^{3/2}}-\frac {2 e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}+\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {975, 270, 272, 65, 214, 745, 739, 212} \begin {gather*} \frac {2 e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3}-\frac {c e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}-\frac {e^3 \sqrt {a+c x^2}}{d^2 (d+e x) \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2}}{a d^2 x}-\frac {2 e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \sqrt {a e^2+c d^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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

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Mathematica [A]
time = 1.01, size = 183, normalized size = 0.86 \begin {gather*} \frac {-\frac {d \sqrt {a+c x^2} \left (c d^2 (d+e x)+a e^2 (d+2 e x)\right )}{a \left (c d^2+a e^2\right ) x (d+e x)}+\frac {2 e^2 \left (3 c d^2+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}}-\frac {4 e \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(190)=380\).
time = 0.08, size = 395, normalized size = 1.86

method	result	size
default	\(-\frac {2 e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {\sqrt {c \,x^{2}+a}}{a \,d^{2} x}+\frac {-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}}{d^{2}}+\frac {2 e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{3} \sqrt {a}}\)	\(395\)
risch	\(-\frac {\sqrt {c \,x^{2}+a}}{a \,d^{2} x}-\frac {2 e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{d^{2} \left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {2 e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{3} \sqrt {a}}\)	\(395\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.18, size = 1479, normalized size = 6.98 \begin {gather*} \left [\frac {{\left (3 \, a c d^{2} x^{2} e^{3} + 3 \, a c d^{3} x e^{2} + 2 \, a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4}\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 (c^{2} d^{4} x^{2} e^{2} + c^{2} d^{5} x e + 2 \, a c d^{2} x^{2} e^{4} + 2 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + a^{2} d x e^{5}\right )} \sqrt {a} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{7} x^{2} e + a c^{2} d^{8} x + 2 \, a^{2} c d^{5} x^{2} e^{3} + 2 \, a^{2} c d^{6} x e^{2} + a^{3} d^{3} x^{2} e^{5} + a^{3} d^{4} x e^{4}\right )}}, \frac {{\left (3 \, a c d^{2} x^{2} e^{3} + 3 \, a c d^{3} x e^{2} + 2 \, a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4}\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^{2} e^{2} + c^{2} d^{5} x e + 2 \, a c d^{2} x^{2} e^{4} + 2 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + a^{2} d x e^{5}\right )} \sqrt {a} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {c x^{2} + a}}{a c^{2} d^{7} x^{2} e + a c^{2} d^{8} x + 2 \, a^{2} c d^{5} x^{2} e^{3} + 2 \, a^{2} c d^{6} x e^{2} + a^{3} d^{3} x^{2} e^{5} + a^{3} d^{4} x e^{4}}, -\frac {4 \, {\left (c^{2} d^{4} x^{2} e^{2} + c^{2} d^{5} x e + 2 \, a c d^{2} x^{2} e^{4} + 2 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + a^{2} d x e^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, a c d^{2} x^{2} e^{3} + 3 \, a c d^{3} x e^{2} + 2 \, a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4}\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 (c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{7} x^{2} e + a c^{2} d^{8} x + 2 \, a^{2} c d^{5} x^{2} e^{3} + 2 \, a^{2} c d^{6} x e^{2} + a^{3} d^{3} x^{2} e^{5} + a^{3} d^{4} x e^{4}\right )}}, \frac {{\left (3 \, a c d^{2} x^{2} e^{3} + 3 \, a c d^{3} x e^{2} + 2 \, a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4}\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 ) - 2 \, {\left (c^{2} d^{4} x^{2} e^{2} + c^{2} d^{5} x e + 2 \, a c d^{2} x^{2} e^{4} + 2 \, a c d^{3} x e^{3} + a^{2} x^{2} e^{6} + a^{2} d x e^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} \sqrt {c x^{2} + a}}{a c^{2} d^{7} x^{2} e + a c^{2} d^{8} x + 2 \, a^{2} c d^{5} x^{2} e^{3} + 2 \, a^{2} c d^{6} x e^{2} + a^{3} d^{3} x^{2} e^{5} + a^{3} d^{4} x e^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((3*a*c*d^2*x^2*e^3 + 3*a*c*d^3*x*e^2 + 2*a^2*x^2*e^5 + 2*a^2*d*x*e^4)*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)) + 2*(c^2*d^4*x^2*e^2 + c^2*d^5*x*e + 2*a*c*d^2*x^2*e^4 + 2*a*c*d^3*x*e^3 + a^2*x^2*e
^6 + a^2*d*x*e^5)*sqrt(a)*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(c^2*d^5*x*e + c^2*d^6 + 3*a
*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + 2*a^2*d*x*e^5 + a^2*d^2*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^7*x^2*e + a*c^2*d^8*x +
2*a^2*c*d^5*x^2*e^3 + 2*a^2*c*d^6*x*e^2 + a^3*d^3*x^2*e^5 + a^3*d^4*x*e^4), ((3*a*c*d^2*x^2*e^3 + 3*a*c*d^3*x*
e^2 + 2*a^2*x^2*e^5 + 2*a^2*d*x*e^4)*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^2*e^2 + c^2*d^5*x*e + 2*a*c*d^2*x^2*e^4 + 2
*a*c*d^3*x*e^3 + a^2*x^2*e^6 + a^2*d*x*e^5)*sqrt(a)*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - (c^2
*d^5*x*e + c^2*d^6 + 3*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2 + 2*a^2*d*x*e^5 + a^2*d^2*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^
7*x^2*e + a*c^2*d^8*x + 2*a^2*c*d^5*x^2*e^3 + 2*a^2*c*d^6*x*e^2 + a^3*d^3*x^2*e^5 + a^3*d^4*x*e^4), -1/2*(4*(c
^2*d^4*x^2*e^2 + c^2*d^5*x*e + 2*a*c*d^2*x^2*e^4 + 2*a*c*d^3*x*e^3 + a^2*x^2*e^6 + a^2*d*x*e^5)*sqrt(-a)*arcta
n(sqrt(-a)/sqrt(c*x^2 + a)) - (3*a*c*d^2*x^2*e^3 + 3*a*c*d^3*x*e^2 + 2*a^2*x^2*e^5 + 2*a^2*d*x*e^4)*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)) + 2*(c^2*d^5*x*e + c^2*d^6 + 3*a*c*d^3*x*e^3 + 2*a*c*d^4*e^2
+ 2*a^2*d*x*e^5 + a^2*d^2*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^7*x^2*e + a*c^2*d^8*x + 2*a^2*c*d^5*x^2*e^3 + 2*a^2*
c*d^6*x*e^2 + a^3*d^3*x^2*e^5 + a^3*d^4*x*e^4), ((3*a*c*d^2*x^2*e^3 + 3*a*c*d^3*x*e^2 + 2*a^2*x^2*e^5 + 2*a^2*
d*x*e^4)*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)) - 2*(c^2*d^4*x^2*e^2 + c^2*d^5*x*e + 2*a*c*d^2*x^2*e^4 + 2*a*c*d^3*x*e^3 + a^2*x^2*e
^6 + a^2*d*x*e^5)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (c^2*d^5*x*e + c^2*d^6 + 3*a*c*d^3*x*e^3 + 2*a*c
*d^4*e^2 + 2*a^2*d*x*e^5 + a^2*d^2*e^4)*sqrt(c*x^2 + a))/(a*c^2*d^7*x^2*e + a*c^2*d^8*x + 2*a^2*c*d^5*x^2*e^3
+ 2*a^2*c*d^6*x*e^2 + a^3*d^3*x^2*e^5 + a^3*d^4*x*e^4)]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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

[Out]

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

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