3.834 \(\int \frac {1}{(d+e x)^4 \sqrt {d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=133 \[ -\frac {3 \sqrt {d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^3 e (d+e x)^2} \]

[Out]

-1/7*(-e^2*x^2+d^2)^(1/2)/d/e/(e*x+d)^4-3/35*(-e^2*x^2+d^2)^(1/2)/d^2/e/(e*x+d)^3-2/35*(-e^2*x^2+d^2)^(1/2)/d^
3/e/(e*x+d)^2-2/35*(-e^2*x^2+d^2)^(1/2)/d^4/e/(e*x+d)

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Rubi [A]  time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac {3 \sqrt {d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-Sqrt[d^2 - e^2*x^2]/(7*d*e*(d + e*x)^4) - (3*Sqrt[d^2 - e^2*x^2])/(35*d^2*e*(d + e*x)^3) - (2*Sqrt[d^2 - e^2*
x^2])/(35*d^3*e*(d + e*x)^2) - (2*Sqrt[d^2 - e^2*x^2])/(35*d^4*e*(d + e*x))

Rule 651

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 659

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)^4 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}+\frac {3 \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{7 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac {3 \sqrt {d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}+\frac {6 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{35 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac {3 \sqrt {d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}+\frac {2 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{35 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac {3 \sqrt {d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac {2 \sqrt {d^2-e^2 x^2}}{35 d^4 e (d+e x)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 63, normalized size = 0.47 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (12 d^3+13 d^2 e x+8 d e^2 x^2+2 e^3 x^3\right )}{35 d^4 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-1/35*(Sqrt[d^2 - e^2*x^2]*(12*d^3 + 13*d^2*e*x + 8*d*e^2*x^2 + 2*e^3*x^3))/(d^4*e*(d + e*x)^4)

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fricas [A]  time = 1.09, size = 137, normalized size = 1.03 \[ -\frac {12 \, e^{4} x^{4} + 48 \, d e^{3} x^{3} + 72 \, d^{2} e^{2} x^{2} + 48 \, d^{3} e x + 12 \, d^{4} + {\left (2 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} + 13 \, d^{2} e x + 12 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{35 \, {\left (d^{4} e^{5} x^{4} + 4 \, d^{5} e^{4} x^{3} + 6 \, d^{6} e^{3} x^{2} + 4 \, d^{7} e^{2} x + d^{8} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/35*(12*e^4*x^4 + 48*d*e^3*x^3 + 72*d^2*e^2*x^2 + 48*d^3*e*x + 12*d^4 + (2*e^3*x^3 + 8*d*e^2*x^2 + 13*d^2*e*
x + 12*d^3)*sqrt(-e^2*x^2 + d^2))/(d^4*e^5*x^4 + 4*d^5*e^4*x^3 + 6*d^6*e^3*x^2 + 4*d^7*e^2*x + d^8*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (36*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^2*exp(2)^9+8*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^16*e
xp(2)+12*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^14*exp(2)^2+6*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^12*exp(2)^3+12*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^2*exp(1)^14*exp(2)^2-8*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^12
*exp(2)^3-36*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^10*exp(2)^4-18*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^8*exp(2)^5-18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^2*exp(1)^10*exp(2)^4+42*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(
1)^8*exp(2)^5+81*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^6*exp(2)^6+27*(-1/2*(-2*
d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^4*exp(2)^7+2*exp(1)^10*exp(2)^4+120*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^6*exp(2)^6+108*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))
*exp(1))/x/exp(2))^3*exp(1)^4*exp(2)^7+18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)
^9-5*exp(1)^6*exp(2)^6+18*exp(2)^9-81/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^7/x/exp(2)
+6*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^8*exp(2)^5/x/exp(2)-3*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))*exp(1)^12*exp(2)^3/x/exp(2))/((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)-(
-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))^3/(3*d^4*exp(1)^13-9*d^4*exp(1)^9*exp(2)^2+9*d^4*exp(1)^5
*exp(2)^4-3*d^4*exp(1)*exp(2)^6)+1/2*(6*exp(1)^4*exp(2)^5+4*exp(2)^7)*atan((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/sqrt(-exp(1)^4+exp(2)^2)/(d^4*exp(1)^13-3*d^4*exp(1)^9*exp(
2)^2+3*d^4*exp(1)^5*exp(2)^4-d^4*exp(1)*exp(2)^6)

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maple [A]  time = 0.05, size = 66, normalized size = 0.50 \[ -\frac {\left (-e x +d \right ) \left (2 e^{3} x^{3}+8 e^{2} x^{2} d +13 x \,d^{2} e +12 d^{3}\right )}{35 \left (e x +d \right )^{3} \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/35*(-e*x+d)*(2*e^3*x^3+8*d*e^2*x^2+13*d^2*e*x+12*d^3)/(e*x+d)^3/d^4/e/(-e^2*x^2+d^2)^(1/2)

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maxima [A]  time = 3.06, size = 193, normalized size = 1.45 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{7 \, {\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}}}{35 \, {\left (d^{2} e^{4} x^{3} + 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{35 \, {\left (d^{3} e^{3} x^{2} + 2 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{35 \, {\left (d^{4} e^{2} x + d^{5} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

-1/7*sqrt(-e^2*x^2 + d^2)/(d*e^5*x^4 + 4*d^2*e^4*x^3 + 6*d^3*e^3*x^2 + 4*d^4*e^2*x + d^5*e) - 3/35*sqrt(-e^2*x
^2 + d^2)/(d^2*e^4*x^3 + 3*d^3*e^3*x^2 + 3*d^4*e^2*x + d^5*e) - 2/35*sqrt(-e^2*x^2 + d^2)/(d^3*e^3*x^2 + 2*d^4
*e^2*x + d^5*e) - 2/35*sqrt(-e^2*x^2 + d^2)/(d^4*e^2*x + d^5*e)

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mupad [B]  time = 0.48, size = 117, normalized size = 0.88 \[ -\frac {\sqrt {d^2-e^2\,x^2}}{7\,d\,e\,{\left (d+e\,x\right )}^4}-\frac {3\,\sqrt {d^2-e^2\,x^2}}{35\,d^2\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{35\,d^3\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{35\,d^4\,e\,\left (d+e\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d^2 - e^2*x^2)^(1/2)*(d + e*x)^4),x)

[Out]

- (d^2 - e^2*x^2)^(1/2)/(7*d*e*(d + e*x)^4) - (3*(d^2 - e^2*x^2)^(1/2))/(35*d^2*e*(d + e*x)^3) - (2*(d^2 - e^2
*x^2)^(1/2))/(35*d^3*e*(d + e*x)^2) - (2*(d^2 - e^2*x^2)^(1/2))/(35*d^4*e*(d + e*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**4), x)

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