3.7.76 \(\int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3} \]

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Rubi [A]  time = 0.07, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 651} \begin {gather*} -\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[d^2 - e^2*x^2]/(9*d*e*(d + e*x)^5) - (4*Sqrt[d^2 - e^2*x^2])/(63*d^2*e*(d + e*x)^4) - (4*Sqrt[d^2 - e^2*
x^2])/(105*d^3*e*(d + e*x)^3) - (8*Sqrt[d^2 - e^2*x^2])/(315*d^4*e*(d + e*x)^2) - (8*Sqrt[d^2 - e^2*x^2])/(315
*d^5*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)^5 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}+\frac {4 \int \frac {1}{(d+e x)^4 \sqrt {d^2-e^2 x^2}} \, dx}{9 d}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}+\frac {4 \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{21 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}+\frac {8 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{105 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}+\frac {8 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{315 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 74, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \left (83 d^4+100 d^3 e x+84 d^2 e^2 x^2+40 d e^3 x^3+8 e^4 x^4\right )}{315 d^5 e (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*(Sqrt[d^2 - e^2*x^2]*(83*d^4 + 100*d^3*e*x + 84*d^2*e^2*x^2 + 40*d*e^3*x^3 + 8*e^4*x^4))/(d^5*e*(d + e*
x)^5)

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IntegrateAlgebraic [A]  time = 0.57, size = 74, normalized size = 0.45 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-83 d^4-100 d^3 e x-84 d^2 e^2 x^2-40 d e^3 x^3-8 e^4 x^4\right )}{315 d^5 e (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-83*d^4 - 100*d^3*e*x - 84*d^2*e^2*x^2 - 40*d*e^3*x^3 - 8*e^4*x^4))/(315*d^5*e*(d + e*x)
^5)

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fricas [A]  time = 0.44, size = 170, normalized size = 1.02 \begin {gather*} -\frac {83 \, e^{5} x^{5} + 415 \, d e^{4} x^{4} + 830 \, d^{2} e^{3} x^{3} + 830 \, d^{3} e^{2} x^{2} + 415 \, d^{4} e x + 83 \, d^{5} + {\left (8 \, e^{4} x^{4} + 40 \, d e^{3} x^{3} + 84 \, d^{2} e^{2} x^{2} + 100 \, d^{3} e x + 83 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(83*e^5*x^5 + 415*d*e^4*x^4 + 830*d^2*e^3*x^3 + 830*d^3*e^2*x^2 + 415*d^4*e*x + 83*d^5 + (8*e^4*x^4 + 4
0*d*e^3*x^3 + 84*d^2*e^2*x^2 + 100*d^3*e*x + 83*d^4)*sqrt(-e^2*x^2 + d^2))/(d^5*e^6*x^5 + 5*d^6*e^5*x^4 + 10*d
^7*e^4*x^3 + 10*d^8*e^3*x^2 + 5*d^9*e^2*x + d^10*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.06, size = 77, normalized size = 0.46 \begin {gather*} -\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+40 e^{3} x^{3} d +84 e^{2} x^{2} d^{2}+100 x \,d^{3} e +83 d^{4}\right )}{315 \left (e x +d \right )^{4} \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(-e*x+d)*(8*e^4*x^4+40*d*e^3*x^3+84*d^2*e^2*x^2+100*d^3*e*x+83*d^4)/(e*x+d)^4/d^5/e/(-e^2*x^2+d^2)^(1/2
)

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maxima [A]  time = 3.08, size = 269, normalized size = 1.62 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{63 \, {\left (d^{2} e^{5} x^{4} + 4 \, d^{3} e^{4} x^{3} + 6 \, d^{4} e^{3} x^{2} + 4 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{4} e^{3} x^{2} + 2 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{5} e^{2} x + d^{6} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

-1/9*sqrt(-e^2*x^2 + d^2)/(d*e^6*x^5 + 5*d^2*e^5*x^4 + 10*d^3*e^4*x^3 + 10*d^4*e^3*x^2 + 5*d^5*e^2*x + d^6*e)
- 4/63*sqrt(-e^2*x^2 + d^2)/(d^2*e^5*x^4 + 4*d^3*e^4*x^3 + 6*d^4*e^3*x^2 + 4*d^5*e^2*x + d^6*e) - 4/105*sqrt(-
e^2*x^2 + d^2)/(d^3*e^4*x^3 + 3*d^4*e^3*x^2 + 3*d^5*e^2*x + d^6*e) - 8/315*sqrt(-e^2*x^2 + d^2)/(d^4*e^3*x^2 +
2*d^5*e^2*x + d^6*e) - 8/315*sqrt(-e^2*x^2 + d^2)/(d^5*e^2*x + d^6*e)

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mupad [B]  time = 0.47, size = 146, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}}{9\,d\,e\,{\left (d+e\,x\right )}^5}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{63\,d^2\,e\,{\left (d+e\,x\right )}^4}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{105\,d^3\,e\,{\left (d+e\,x\right )}^3}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{315\,d^4\,e\,{\left (d+e\,x\right )}^2}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e\,\left (d+e\,x\right )} \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)^5),x)

[Out]

- (d^2 - e^2*x^2)^(1/2)/(9*d*e*(d + e*x)^5) - (4*(d^2 - e^2*x^2)^(1/2))/(63*d^2*e*(d + e*x)^4) - (4*(d^2 - e^2
*x^2)^(1/2))/(105*d^3*e*(d + e*x)^3) - (8*(d^2 - e^2*x^2)^(1/2))/(315*d^4*e*(d + e*x)^2) - (8*(d^2 - e^2*x^2)^
(1/2))/(315*d^5*e*(d + e*x))

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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 )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**5), x)

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