3.9.66 \(\int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx\) [866]

Optimal. Leaf size=99 \[ \frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {679, 675, 214} \begin {gather*} \frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c*d^2 - c*e^2*x^2])/(e*Sqrt[d + e*x]) - (2*Sqrt[2]*Sqrt[c]*Sqrt[d]*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sq
rt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/e

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 675

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2
), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0]

Rule 679

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx &=\frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+(2 c d) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+(4 c d e) \text {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {2 \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {c} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 98, normalized size = 0.99 \begin {gather*} \frac {2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (\frac {1}{\sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.52, size = 89, normalized size = 0.90

method	result	size
default	\(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (c d \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right )-\sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\right )}{\sqrt {e x +d}\, \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\)	\(89\)
risch	\(\frac {2 \left (-e x +d \right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c}{e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}}-\frac {2 d \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c e x +c d}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c}{e \sqrt {c d}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.36, size = 238, normalized size = 2.40 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {c d} {\left (x e + d\right )} \log \left (-\frac {c x^{2} e^{2} - 2 \, c d x e - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {c d} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{x e^{2} + d e}, -\frac {2 \, {\left (\sqrt {2} \sqrt {-c d} {\left (x e + d\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {-c d} \sqrt {x e + d}}{c x^{2} e^{2} - c d^{2}}\right ) - \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}\right )}}{x e^{2} + d e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(2)*sqrt(c*d)*(x*e + d)*log(-(c*x^2*e^2 - 2*c*d*x*e - 3*c*d^2 + 2*sqrt(2)*sqrt(-c*x^2*e^2 + c*d^2)*sqrt(
c*d)*sqrt(x*e + d))/(x^2*e^2 + 2*d*x*e + d^2)) + 2*sqrt(-c*x^2*e^2 + c*d^2)*sqrt(x*e + d))/(x*e^2 + d*e), -2*(
sqrt(2)*sqrt(-c*d)*(x*e + d)*arctan(sqrt(2)*sqrt(-c*x^2*e^2 + c*d^2)*sqrt(-c*d)*sqrt(x*e + d)/(c*x^2*e^2 - c*d
^2)) - sqrt(-c*x^2*e^2 + c*d^2)*sqrt(x*e + d))/(x*e^2 + d*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.39, size = 111, normalized size = 1.12 \begin {gather*} 2 \, {\left (\frac {\sqrt {2} d \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c}\right )} c e^{\left (-1\right )} - \frac {2 \, {\left (\sqrt {2} c d \arctan \left (\frac {\sqrt {c d}}{\sqrt {-c d}}\right ) + \sqrt {2} \sqrt {c d} \sqrt {-c d}\right )} e^{\left (-1\right )}}{\sqrt {-c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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