3.9.43 \(\int \frac {1}{(d+e x)^2 (d^2-e^2 x^2)^{5/2}} \, dx\) [843]

Optimal. Leaf size=115 \[ \frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}} \]

[Out]

4/21*x/d^4/(-e^2*x^2+d^2)^(3/2)-1/7/d/e/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2)-1/7/d^2/e/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+
8/21*x/d^6/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \begin {gather*} -\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(21*d^4*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d^2*e*(d + e*x)*(d^2
- e^2*x^2)^(3/2)) + (8*x)/(21*d^6*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

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)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d}\\ &=-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d^2}\\ &=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{21 d^4}\\ &=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 93, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^5 + 9*d^4*e*x + 24*d^3*e^2*x^2 + 4*d^2*e^3*x^3 - 16*d*e^4*x^4 - 8*e^5*x^5))/(21*d^6
*e*(d - e*x)^2*(d + e*x)^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(99)=198\).
time = 0.48, size = 216, normalized size = 1.88

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (8 e^{5} x^{5}+16 d \,e^{4} x^{4}-4 d^{2} e^{3} x^{3}-24 d^{3} e^{2} x^{2}-9 d^{4} e x +6 d^{5}\right )}{21 \left (e x +d \right ) d^{6} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(88\)
trager	\(-\frac {\left (8 e^{5} x^{5}+16 d \,e^{4} x^{4}-4 d^{2} e^{3} x^{3}-24 d^{3} e^{2} x^{2}-9 d^{4} e x +6 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{21 d^{6} \left (e x +d \right )^{4} \left (-e x +d \right )^{2} e}\)	\(90\)
default	\(\frac {-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}}{e^{2}}\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 148, normalized size = 1.29 \begin {gather*} -\frac {1}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} - \frac {1}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} + \frac {4 \, x}{21 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {8 \, x}{21 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/((-x^2*e^2 + d^2)^(3/2)*d*x^2*e^3 + 2*(-x^2*e^2 + d^2)^(3/2)*d^2*x*e^2 + (-x^2*e^2 + d^2)^(3/2)*d^3*e) -
1/7/((-x^2*e^2 + d^2)^(3/2)*d^2*x*e^2 + (-x^2*e^2 + d^2)^(3/2)*d^3*e) + 4/21*x/((-x^2*e^2 + d^2)^(3/2)*d^4) +
8/21*x/(sqrt(-x^2*e^2 + d^2)*d^6)

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Fricas [A]
time = 1.82, size = 190, normalized size = 1.65 \begin {gather*} -\frac {6 \, x^{6} e^{6} + 12 \, d x^{5} e^{5} - 6 \, d^{2} x^{4} e^{4} - 24 \, d^{3} x^{3} e^{3} - 6 \, d^{4} x^{2} e^{2} + 12 \, d^{5} x e + 6 \, d^{6} + {\left (8 \, x^{5} e^{5} + 16 \, d x^{4} e^{4} - 4 \, d^{2} x^{3} e^{3} - 24 \, d^{3} x^{2} e^{2} - 9 \, d^{4} x e + 6 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{21 \, {\left (d^{6} x^{6} e^{7} + 2 \, d^{7} x^{5} e^{6} - d^{8} x^{4} e^{5} - 4 \, d^{9} x^{3} e^{4} - d^{10} x^{2} e^{3} + 2 \, d^{11} x e^{2} + d^{12} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(6*x^6*e^6 + 12*d*x^5*e^5 - 6*d^2*x^4*e^4 - 24*d^3*x^3*e^3 - 6*d^4*x^2*e^2 + 12*d^5*x*e + 6*d^6 + (8*x^5
*e^5 + 16*d*x^4*e^4 - 4*d^2*x^3*e^3 - 24*d^3*x^2*e^2 - 9*d^4*x*e + 6*d^5)*sqrt(-x^2*e^2 + d^2))/(d^6*x^6*e^7 +
2*d^7*x^5*e^6 - d^8*x^4*e^5 - 4*d^9*x^3*e^4 - d^10*x^2*e^3 + 2*d^11*x*e^2 + d^12*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 1.55, size = 220, normalized size = 1.91 \begin {gather*} \frac {1}{672} \, {\left ({\left (\frac {14 \, {\left (\frac {15 \, d}{x e + d} - 7\right )} e^{\left (-5\right )}}{d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (3 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 21 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 70 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 210 \, d^{36} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6}\right )} e^{\left (-35\right )}}{d^{42} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{7}}\right )} e^{5} + \frac {256 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{6}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/672*((14*(15*d/(x*e + d) - 7)*e^(-5)/(d^6*(2*d/(x*e + d) - 1)^(3/2)*sgn(1/(x*e + d))) - (3*d^36*(2*d/(x*e +
d) - 1)^(7/2)*e^30*sgn(1/(x*e + d))^6 + 21*d^36*(2*d/(x*e + d) - 1)^(5/2)*e^30*sgn(1/(x*e + d))^6 + 70*d^36*(2
*d/(x*e + d) - 1)^(3/2)*e^30*sgn(1/(x*e + d))^6 + 210*d^36*sqrt(2*d/(x*e + d) - 1)*e^30*sgn(1/(x*e + d))^6)*e^
(-35)/(d^42*sgn(1/(x*e + d))^7))*e^5 + 256*I*sgn(1/(x*e + d))/d^6)*e^(-1)

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Mupad [B]
time = 0.64, size = 139, normalized size = 1.21 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {11\,x}{42\,d^4}-\frac {5}{28\,d^3\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{28\,d^3\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{14\,d^4\,e\,{\left (d+e\,x\right )}^3}+\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{21\,d^6\,\left (d+e\,x\right )\,\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)^(5/2)*(d + e*x)^2),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*((11*x)/(42*d^4) - 5/(28*d^3*e)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(28
*d^3*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(1/2)/(14*d^4*e*(d + e*x)^3) + (8*x*(d^2 - e^2*x^2)^(1/2))/(21*d^6*(d +
e*x)*(d - e*x))

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