\(\int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx\) [567]

Optimal result

Integrand size = 19, antiderivative size = 145 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=-\frac {e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {3 c d e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {c \left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 145, 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 = {759, 821, 739, 212} \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=-\frac {c \left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}}-\frac {3 c d e \sqrt {a+c x^2}}{2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac {e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )} \]

[In]

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

[Out]

-1/2*(e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)^2) - (3*c*d*e*Sqrt[a + c*x^2])/(2*(c*d^2 + a*e^2)^2*(d + e
*x)) - (c*(2*c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(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 759

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/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

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

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=-\frac {e \sqrt {a+c x^2} \left (a e^2+c d (4 d+3 e x)\right )}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c \left (2 c d^2-a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{5/2}} \]

[In]

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

[Out]

-1/2*(e*Sqrt[a + c*x^2]*(a*e^2 + c*d*(4*d + 3*e*x)))/((c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*(2*c*d^2 - a*e^2)*Ar
cTan[(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. \(443\) vs. \(2(129)=258\).

Time = 2.02 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.06

method	result	size
default	\(\frac {-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{e^{3}}\)	\(444\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (130) = 260\).

Time = 0.43 (sec) , antiderivative size = 713, normalized size of antiderivative = 4.92 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\left [-\frac {{\left (2 \, c^{2} d^{4} - a c d^{2} e^{2} + {\left (2 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 2 \, {\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (4 \, c^{2} d^{4} e + 5 \, a c d^{2} e^{3} + a^{2} e^{5} + 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{8} + 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} + a^{3} d^{2} e^{6} + {\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x\right )}}, -\frac {{\left (2 \, c^{2} d^{4} - a c d^{2} e^{2} + {\left (2 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 2 \, {\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (4 \, c^{2} d^{4} e + 5 \, a c d^{2} e^{3} + a^{2} e^{5} + 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{8} + 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} + a^{3} d^{2} e^{6} + {\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (130) = 260\).

Time = 0.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx=-c {\left (\frac {{\left (2 \, c d^{2} - a e^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a e^{3} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {3}{2}} d^{3} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a \sqrt {c} d e^{2} - 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} e^{3} + 3 \, a^{2} \sqrt {c} d e^{2}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{2}}\right )} \]

[In]

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

[Out]

-c*((2*c*d^2 - a*e^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^2*d^4 + 2
*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c*d^2 - a*e^2)) + (2*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c*d^2*e - (sqrt(c)*x - sqrt
(c*x^2 + a))^3*a*e^3 + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^2*c^(3/2)*d^3 - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*sqr
t(c)*d*e^2 - 10*(sqrt(c)*x - sqrt(c*x^2 + a))*a*c*d^2*e - (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*e^3 + 3*a^2*sqrt(c
)*d*e^2)/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 +
a))*sqrt(c)*d - a*e)^2))

Mupad [F(-1)]

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

[In]

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

[Out]

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