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

Optimal. Leaf size=151 \[ \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}}+\frac {e \left (c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {3 c d e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{5/2}} \]

[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)

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Rubi [A]
time = 0.06, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {755, 821, 739, 212} \begin {gather*} \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 )}-\frac {3 c d e^2 \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 )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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*}

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Mathematica [A]
time = 0.70, size = 149, normalized size = 0.99 \begin {gather*} \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 \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 )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(514\) vs. \(2(141)=282\).
time = 0.52, size = 515, normalized size = 3.41

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [A]
time = 0.32, size = 280, normalized size = 1.85 \begin {gather*} \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 \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-3\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {5}{2}}} + \frac {3 \, c d}{\sqrt {c x^{2} + a} c^{2} d^{4} e^{\left (-1\right )} + 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^{3} e^{\left (-1\right )} + \sqrt {c x^{2} + a} c d^{2} x + \sqrt {c x^{2} + a} a x e^{2} + \sqrt {c x^{2} + a} a d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*ar
csinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-3)/(c*d^2*e^(-2) + a)^(5/2) + 3*c*d/(
sqrt(c*x^2 + a)*c^2*d^4*e^(-1) + 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^3*e^(-1) + sqrt(c*x^2 + a)*c*d^2*x + sqrt(c*x^2
+ a)*a*x*e^2 + sqrt(c*x^2 + a)*a*d*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (141) = 282\).
time = 2.05, size = 888, normalized size = 5.88 \begin {gather*} \left [\frac {3 \, \sqrt {c d^{2} + a e^{2}} {\left ({\left (a c^{2} d x^{3} + a^{2} c d x\right )} e^{3} + {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \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^{3} d^{5} x + 2 \, a c^{2} d^{3} x e^{2} + a^{2} c d x e^{4} - {\left (2 \, a^{2} c x^{2} + a^{3}\right )} e^{5} - {\left (a c^{2} d^{2} x^{2} - a^{2} c d^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{2} + 2 \, a c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{4} d^{7} x^{2} + a^{2} c^{3} d^{7} + {\left (a^{4} c x^{3} + a^{5} x\right )} e^{7} + {\left (a^{4} c d x^{2} + a^{5} d\right )} e^{6} + 3 \, {\left (a^{3} c^{2} d^{2} x^{3} + a^{4} c d^{2} x\right )} e^{5} + 3 \, {\left (a^{3} c^{2} d^{3} x^{2} + a^{4} c d^{3}\right )} e^{4} + 3 \, {\left (a^{2} c^{3} d^{4} x^{3} + a^{3} c^{2} d^{4} x\right )} e^{3} + 3 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{2} + {\left (a c^{4} d^{6} x^{3} + a^{2} c^{3} d^{6} x\right )} e\right )}}, \frac {3 \, \sqrt {-c d^{2} - a e^{2}} {\left ({\left (a c^{2} d x^{3} + a^{2} c d x\right )} e^{3} + {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \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^{3} d^{5} x + 2 \, a c^{2} d^{3} x e^{2} + a^{2} c d x e^{4} - {\left (2 \, a^{2} c x^{2} + a^{3}\right )} e^{5} - {\left (a c^{2} d^{2} x^{2} - a^{2} c d^{2}\right )} e^{3} + {\left (c^{3} d^{4} x^{2} + 2 \, a c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + a}}{a c^{4} d^{7} x^{2} + a^{2} c^{3} d^{7} + {\left (a^{4} c x^{3} + a^{5} x\right )} e^{7} + {\left (a^{4} c d x^{2} + a^{5} d\right )} e^{6} + 3 \, {\left (a^{3} c^{2} d^{2} x^{3} + a^{4} c d^{2} x\right )} e^{5} + 3 \, {\left (a^{3} c^{2} d^{3} x^{2} + a^{4} c d^{3}\right )} e^{4} + 3 \, {\left (a^{2} c^{3} d^{4} x^{3} + a^{3} c^{2} d^{4} x\right )} e^{3} + 3 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{2} + {\left (a c^{4} d^{6} x^{3} + a^{2} c^{3} d^{6} x\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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