3.9.33 \(\int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx\) [833]

Optimal. Leaf size=100 \[ -\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]

[Out]

-1/5*(-e^2*x^2+d^2)^(1/2)/d/e/(e*x+d)^3-2/15*(-e^2*x^2+d^2)^(1/2)/d^2/e/(e*x+d)^2-2/15*(-e^2*x^2+d^2)^(1/2)/d^
3/e/(e*x+d)

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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \begin {gather*} -\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)^3) - (2*Sqrt[d^2 - e^2*x^2])/(15*d^2*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^
2*x^2])/(15*d^3*e*(d + e*x))

Rule 665

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

Rule 673

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}+\frac {2 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{5 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}+\frac {2 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{15 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac {2 \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 52, normalized size = 0.52 \begin {gather*} \frac {\left (-7 d^2-6 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{15 d^3 e (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*d^2 - 6*d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(15*d^3*e*(d + e*x)^3)

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Maple [A]
time = 0.48, size = 145, normalized size = 1.45

method	result	size
trager	\(-\frac {\left (2 e^{2} x^{2}+6 d x e +7 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} \left (e x +d \right )^{3} e}\)	\(49\)
gosper	\(-\frac {\left (-e x +d \right ) \left (2 e^{2} x^{2}+6 d x e +7 d^{2}\right )}{15 \left (e x +d \right )^{2} d^{3} e \sqrt {-e^{2} x^{2}+d^{2}}}\)	\(55\)
default	\(\frac {-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}\right )}{5 d}}{e^{3}}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^3*(-1/5/d/e/(x+d/e)^3*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)+2/5*e/d*(-1/3/d/e/(x+d/e)^2*(-e^2*(x+d/e)^2+2*d
*e*(x+d/e))^(1/2)-1/3/d^2/(x+d/e)*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)))

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Maxima [A]
time = 0.50, size = 122, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d x^{3} e^{4} + 3 \, d^{2} x^{2} e^{3} + 3 \, d^{3} x e^{2} + d^{4} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x^{2} e^{3} + 2 \, d^{3} x e^{2} + d^{4} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x e^{2} + d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(-x^2*e^2 + d^2)/(d*x^3*e^4 + 3*d^2*x^2*e^3 + 3*d^3*x*e^2 + d^4*e) - 2/15*sqrt(-x^2*e^2 + d^2)/(d^2*x
^2*e^3 + 2*d^3*x*e^2 + d^4*e) - 2/15*sqrt(-x^2*e^2 + d^2)/(d^3*x*e^2 + d^4*e)

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Fricas [A]
time = 2.94, size = 100, normalized size = 1.00 \begin {gather*} -\frac {7 \, x^{3} e^{3} + 21 \, d x^{2} e^{2} + 21 \, d^{2} x e + 7 \, d^{3} + {\left (2 \, x^{2} e^{2} + 6 \, d x e + 7 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{3} e^{4} + 3 \, d^{4} x^{2} e^{3} + 3 \, d^{5} x e^{2} + d^{6} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(7*x^3*e^3 + 21*d*x^2*e^2 + 21*d^2*x*e + 7*d^3 + (2*x^2*e^2 + 6*d*x*e + 7*d^2)*sqrt(-x^2*e^2 + d^2))/(d^
3*x^3*e^4 + 3*d^4*x^2*e^3 + 3*d^5*x*e^2 + d^6*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**3), x)

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Giac [A]
time = 1.39, size = 158, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (\frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 7\right )} e^{\left (-1\right )}}{15 \, d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 40*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 30*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 + 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 + 7)*e^(-1)/(d^3*((d*e +
sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1)^5)

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Mupad [B]
time = 0.49, size = 48, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}\,\left (7\,d^2+6\,d\,e\,x+2\,e^2\,x^2\right )}{15\,d^3\,e\,{\left (d+e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d^2 - e^2*x^2)^(1/2)*(7*d^2 + 2*e^2*x^2 + 6*d*e*x))/(15*d^3*e*(d + e*x)^3)

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