\(\int \frac {1}{(d+e x)^2 (a+c x^2)^{3/2}} \, dx\) [574]

Optimal result

Integrand size = 19, antiderivative size = 151 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}}+\frac {e \left (c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {3 c d e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {755, 821, 739, 212} \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=-\frac {3 c d e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}}+\frac {e \sqrt {a+c x^2} \left (c d^2-2 a e^2\right )}{a (d+e x) \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )} \]

[In]

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

[Out]

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

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 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 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}}-\frac {\int \frac {-2 a e^2-c d e x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}}+\frac {e \left (c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}+\frac {\left (3 c d e^2\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{\left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}}+\frac {e \left (c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {\left (3 c d 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 )}{\left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}}+\frac {e \left (c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {3 c d e^2 \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 )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {-a^2 e^3+c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-2 e^2 x^2\right )}{a \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}-\frac {6 c d e^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{5/2}} \]

[In]

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

[Out]

(-(a^2*e^3) + c^2*d^2*x*(d + e*x) + a*c*e*(2*d^2 + d*e*x - 2*e^2*x^2))/(a*(c*d^2 + a*e^2)^2*(d + e*x)*Sqrt[a +
c*x^2]) - (6*c*d*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)^(5/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(514\) vs. \(2(141)=282\).

Time = 2.23 (sec) , antiderivative size = 515, normalized size of antiderivative = 3.41

method	result	size
default	\(\frac {-\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {3 c d e \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}-\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{e^{2}}\)	\(515\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (142) = 284\).

Time = 0.36 (sec) , antiderivative size = 900, normalized size of antiderivative = 5.96 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a c^{2} d e^{3} x^{3} + a c^{2} d^{2} e^{2} x^{2} + a^{2} c d e^{3} x + a^{2} c d^{2} e^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (2 \, a c^{2} d^{4} e + a^{2} c d^{2} e^{3} - a^{3} e^{5} + {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3} - 2 \, a^{2} c e^{5}\right )} x^{2} + {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{7} + 3 \, a^{3} c^{2} d^{5} e^{2} + 3 \, a^{4} c d^{3} e^{4} + a^{5} d e^{6} + {\left (a c^{4} d^{6} e + 3 \, a^{2} c^{3} d^{4} e^{3} + 3 \, a^{3} c^{2} d^{2} e^{5} + a^{4} c e^{7}\right )} x^{3} + {\left (a c^{4} d^{7} + 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} + a^{4} c d e^{6}\right )} x^{2} + {\left (a^{2} c^{3} d^{6} e + 3 \, a^{3} c^{2} d^{4} e^{3} + 3 \, a^{4} c d^{2} e^{5} + a^{5} e^{7}\right )} x\right )}}, -\frac {3 \, {\left (a c^{2} d e^{3} x^{3} + a c^{2} d^{2} e^{2} x^{2} + a^{2} c d e^{3} x + a^{2} c d^{2} e^{2}\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}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (2 \, a c^{2} d^{4} e + a^{2} c d^{2} e^{3} - a^{3} e^{5} + {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3} - 2 \, a^{2} c e^{5}\right )} x^{2} + {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{a^{2} c^{3} d^{7} + 3 \, a^{3} c^{2} d^{5} e^{2} + 3 \, a^{4} c d^{3} e^{4} + a^{5} d e^{6} + {\left (a c^{4} d^{6} e + 3 \, a^{2} c^{3} d^{4} e^{3} + 3 \, a^{3} c^{2} d^{2} e^{5} + a^{4} c e^{7}\right )} x^{3} + {\left (a c^{4} d^{7} + 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} + a^{4} c d e^{6}\right )} x^{2} + {\left (a^{2} c^{3} d^{6} e + 3 \, a^{3} c^{2} d^{4} e^{3} + 3 \, a^{4} c d^{2} e^{5} + a^{5} e^{7}\right )} x}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (142) = 284\).

Time = 0.22 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\frac {3 \, c^{2} d^{2} x}{\sqrt {c x^{2} + a} a c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{3} e^{4}} + \frac {3 \, c d}{\frac {\sqrt {c x^{2} + a} c^{2} d^{4}}{e} + 2 \, \sqrt {c x^{2} + a} a c d^{2} e + \sqrt {c x^{2} + a} a^{2} e^{3}} - \frac {2 \, c x}{\sqrt {c x^{2} + a} a c d^{2} + \sqrt {c x^{2} + a} a^{2} e^{2}} - \frac {1}{\sqrt {c x^{2} + a} c d^{2} x + \sqrt {c x^{2} + a} a e^{2} x + \frac {\sqrt {c x^{2} + a} c d^{3}}{e} + \sqrt {c x^{2} + a} a d e} + \frac {3 \, c d \operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{2}} e^{3}} \]

[In]

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

[Out]

3*c^2*d^2*x/(sqrt(c*x^2 + a)*a*c^2*d^4 + 2*sqrt(c*x^2 + a)*a^2*c*d^2*e^2 + sqrt(c*x^2 + a)*a^3*e^4) + 3*c*d/(s
qrt(c*x^2 + a)*c^2*d^4/e + 2*sqrt(c*x^2 + a)*a*c*d^2*e + sqrt(c*x^2 + a)*a^2*e^3) - 2*c*x/(sqrt(c*x^2 + a)*a*c
*d^2 + sqrt(c*x^2 + a)*a^2*e^2) - 1/(sqrt(c*x^2 + a)*c*d^2*x + sqrt(c*x^2 + a)*a*e^2*x + sqrt(c*x^2 + a)*c*d^3
/e + sqrt(c*x^2 + a)*a*d*e) + 3*c*d*arcsinh(c*d*x/(e*sqrt(a*c/e^2)*abs(e*x + d)) - a/(sqrt(a*c/e^2)*abs(e*x +
d)))/((a + c*d^2/e^2)^(5/2)*e^3)

Giac [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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