3.1.26 \(\int \frac {d+e x}{(d^2-e^2 x^2)^{7/2}} \, dx\) [26]

Optimal. Leaf size=80 \[ \frac {d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}} \]

[Out]

1/5*(e*x+d)/d/e/(-e^2*x^2+d^2)^(5/2)+4/15*x/d^3/(-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 = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {653, 198, 197} \begin {gather*} \frac {d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\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[(d + e*x)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d + e*x)/(5*d*e*(d^2 - e^2*x^2)^(5/2)) + (4*x)/(15*d^3*(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 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps

\begin {align*} \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 x}{15 d^3 \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 {d+e x}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 x}{15 d^3 \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 = 0.01, size = 82, normalized size = 1.02 \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)^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(d^2 - e^2*x^2)^(7/2),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)^3*(d
+ e*x)^2)

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Maple [A]
time = 0.05, size = 90, normalized size = 1.12

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (8 e^{4} x^{4}-8 d \,e^{3} x^{3}-12 d^{2} x^{2} e^{2}+12 d^{3} e x +3 d^{4}\right )}{15 d^{5} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(77\)
trager	\(\frac {\left (8 e^{4} x^{4}-8 d \,e^{3} x^{3}-12 d^{2} x^{2} e^{2}+12 d^{3} e x +3 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{5} \left (-e x +d \right )^{3} \left (e x +d \right )^{2} e}\)	\(79\)
default	\(\frac {1}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 75, normalized size = 0.94 \begin {gather*} \frac {e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \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((e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

1/5*e^(-1)/(-x^2*e^2 + d^2)^(5/2) + 1/5*x/((-x^2*e^2 + d^2)^(5/2)*d) + 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. 161 vs. \(2 (65) = 130\).
time = 2.95, size = 161, normalized size = 2.01 \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((e*x+d)/(-e^2*x^2+d^2)^(7/2),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 [C] Result contains complex when optimal does not.
time = 8.11, size = 604, normalized size = 7.55 \begin {gather*} d \left (\begin {cases} - \frac {15 i d^{4} x}{15 d^{11} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {20 i d^{2} e^{2} x^{3}}{15 d^{11} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {8 i e^{4} x^{5}}{15 d^{11} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {15 d^{4} x}{15 d^{11} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {20 d^{2} e^{2} x^{3}}{15 d^{11} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {8 e^{4} x^{5}}{15 d^{11} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 30 d^{9} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 15 d^{7} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {1}{5 d^{4} e^{2} \sqrt {d^{2} - e^{2} x^{2}} - 10 d^{2} e^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 5 e^{6} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 \left (d^{2}\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((-15*I*d**4*x/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) +
15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) + 20*I*d**2*e**2*x**3/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d*
*9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)) - 8*I*e**4*x**5/(15*d**1
1*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x
**2/d**2)), Abs(e**2*x**2/d**2) > 1), (15*d**4*x/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1
- e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) - 20*d**2*e**2*x**3/(15*d**11*sqrt(1 - e**2*x
**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) + 8*e**4*
x**5/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(
1 - e**2*x**2/d**2)), True)) + e*Piecewise((1/(5*d**4*e**2*sqrt(d**2 - e**2*x**2) - 10*d**2*e**4*x**2*sqrt(d**
2 - e**2*x**2) + 5*e**6*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**2/(2*(d**2)**(7/2)), True))

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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((e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate((x*e + d)/(-x^2*e^2 + d^2)^(7/2), x)

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Mupad [B]
time = 2.58, size = 78, normalized size = 0.98 \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 )}^2\,{\left (d-e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(d^2 - e^2*x^2)^(7/2),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)^2*
(d - e*x)^3)

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