3.9.42 \(\int \frac {1}{(d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\) [842]

Optimal. Leaf size=82 \[ \frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}} \]

[Out]

4/15*x/d^3/(-e^2*x^2+d^2)^(3/2)-1/5/d/e/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+8/15*x/d^5/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, 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}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) - 1/(5*d*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(15*d^5*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) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{5 d 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}{5 d}\\ &=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d 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}{15 d^3}\\ &=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 82, normalized size = 1.00 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-3 d^4+12 d^3 e x+12 d^2 e^2 x^2-8 d e^3 x^3-8 e^4 x^4\right )}{15 d^5 e (d-e x)^2 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-3*d^4 + 12*d^3*e*x + 12*d^2*e^2*x^2 - 8*d*e^3*x^3 - 8*e^4*x^4))/(15*d^5*e*(d - e*x)^2*(
d + e*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs. \(2(70)=140\).
time = 0.49, size = 164, normalized size = 2.00

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+8 d \,e^{3} x^{3}-12 d^{2} e^{2} x^{2}-12 d^{3} e x +3 d^{4}\right )}{15 d^{5} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(70\)
trager	\(-\frac {\left (8 e^{4} x^{4}+8 d \,e^{3} x^{3}-12 d^{2} e^{2} x^{2}-12 d^{3} e x +3 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{5} \left (e x +d \right )^{3} \left (-e x +d \right )^{2} e}\)	\(79\)
default	\(\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(-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 = 0.31, size = 81, normalized size = 0.99 \begin {gather*} -\frac {1}{5 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e\right )}} + \frac {4 \, x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} + \frac {8 \, x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/((-x^2*e^2 + d^2)^(3/2)*d*x*e^2 + (-x^2*e^2 + d^2)^(3/2)*d^2*e) + 4/15*x/((-x^2*e^2 + d^2)^(3/2)*d^3) + 8
/15*x/(sqrt(-x^2*e^2 + d^2)*d^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (67) = 134\).
time = 2.27, size = 158, normalized size = 1.93 \begin {gather*} -\frac {3 \, x^{5} e^{5} + 3 \, d x^{4} e^{4} - 6 \, d^{2} x^{3} e^{3} - 6 \, d^{3} x^{2} e^{2} + 3 \, d^{4} x e + 3 \, d^{5} + {\left (8 \, x^{4} e^{4} + 8 \, d x^{3} e^{3} - 12 \, d^{2} x^{2} e^{2} - 12 \, d^{3} x e + 3 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{5} x^{5} e^{6} + d^{6} x^{4} e^{5} - 2 \, d^{7} x^{3} e^{4} - 2 \, d^{8} x^{2} e^{3} + d^{9} x e^{2} + d^{10} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(3*x^5*e^5 + 3*d*x^4*e^4 - 6*d^2*x^3*e^3 - 6*d^3*x^2*e^2 + 3*d^4*x*e + 3*d^5 + (8*x^4*e^4 + 8*d*x^3*e^3
- 12*d^2*x^2*e^2 - 12*d^3*x*e + 3*d^4)*sqrt(-x^2*e^2 + d^2))/(d^5*x^5*e^6 + d^6*x^4*e^5 - 2*d^7*x^3*e^4 - 2*d^
8*x^2*e^3 + d^9*x*e^2 + d^10*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 )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.58, size = 78, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}\,\left (3\,d^4-12\,d^3\,e\,x-12\,d^2\,e^2\,x^2+8\,d\,e^3\,x^3+8\,e^4\,x^4\right )}{15\,d^5\,e\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \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)),x)

[Out]

-((d^2 - e^2*x^2)^(1/2)*(3*d^4 + 8*e^4*x^4 + 8*d*e^3*x^3 - 12*d^2*e^2*x^2 - 12*d^3*e*x))/(15*d^5*e*(d + e*x)^3
*(d - e*x)^2)

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