\(\int \frac {x}{(d+e x) (a+c x^2)^{3/2}} \, dx\) [337]

Optimal result

Integrand size = 20, antiderivative size = 88 \[ \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {d-e x}{\left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {d e \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {837, 12, 739, 212} \[ \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {d e \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {d-e x}{\sqrt {a+c x^2} \left (a e^2+c d^2\right )} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 837

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

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12 \[ \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {-d+e x}{\left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {2 d e \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(80)=160\).

Time = 0.40 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.81

method	result	size
default	\(\frac {x}{e a \sqrt {c \,x^{2}+a}}-\frac {d \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\)	\(335\)

[In]

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

[Out]

1/e*x/a/(c*x^2+a)^(1/2)-d/e^2*(1/(a*e^2+c*d^2)*e^2/((x+d/e)^2*c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+2*e*c
*d/(a*e^2+c*d^2)*(2*c*(x+d/e)-2/e*c*d)/(4*c*(a*e^2+c*d^2)/e^2-4/e^2*c^2*d^2)/((x+d/e)^2*c-2/e*c*d*(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/e*c*d*(x+d/e)+2*(
(a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2/e*c*d*(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. 199 vs. \(2 (81) = 162\).

Time = 0.33 (sec) , antiderivative size = 425, normalized size of antiderivative = 4.83 \[ \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (c d e x^{2} + a d e\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (c d^{3} + a d e^{2} - {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} + {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{2}\right )}}, \frac {{\left (c d e x^{2} + a d e\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (c d^{3} + a d e^{2} - {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} + {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{2}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.77 \[ \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {c d^{2} x}{\sqrt {c x^{2} + a} a c d^{2} e + \sqrt {c x^{2} + a} a^{2} e^{3}} - \frac {d}{\sqrt {c x^{2} + a} c d^{2} + \sqrt {c x^{2} + a} a e^{2}} + \frac {x}{\sqrt {c x^{2} + a} a e} - \frac {d \operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{2}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (81) = 162\).

Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.91 \[ \int \frac {x}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {2 \, d e \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {\frac {{\left (c d^{2} e + a e^{3}\right )} x}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {c d^{3} + a d e^{2}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}}{\sqrt {c x^{2} + a}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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