3.4.46 \(\int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [346]

Optimal. Leaf size=90 \[ \frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {a e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {821, 739, 212} \begin {gather*} \frac {d \sqrt {a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac {a e \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 )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.44, size = 100, normalized size = 1.11 \begin {gather*} \frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {2 a e \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(82)=164\).
time = 0.07, size = 344, normalized size = 3.82

method	result	size
default	\(-\frac {\ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {d \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\)	\(344\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/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))-d/e^3*(-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)))

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Maxima [A]
time = 0.33, size = 149, normalized size = 1.66 \begin {gather*} -\frac {c d^{2} \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 (-4\right )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} + \frac {\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 (-2\right )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} + \frac {\sqrt {c x^{2} + a} d}{c d^{2} x e + c d^{3} + a x e^{3} + a d e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (83) = 166\).
time = 1.85, size = 368, normalized size = 4.09 \begin {gather*} \left [\frac {\sqrt {c d^{2} + a e^{2}} {\left (a x e^{2} + a d e\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 d^{3} + a d e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\right )}}, \frac {\sqrt {-c d^{2} - a e^{2}} {\left (a x e^{2} + a d e\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 d^{3} + a d e^{2}\right )} \sqrt {c x^{2} + a}}{c^{2} d^{4} x e + c^{2} d^{5} + 2 \, a c d^{2} x e^{3} + 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(x/(sqrt(a + c*x**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(x/(e*x+d)^2/(c*x^2+a)^(1/2),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 {x}{\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(x/((a + c*x^2)^(1/2)*(d + e*x)^2),x)

[Out]

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

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