3.9.55 \(\int \frac {1}{(d+e x) (d^2-e^2 x^2)^{7/2}} \, dx\) [855]

Optimal. Leaf size=106 \[ \frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}} \]

[Out]

6/35*x/d^3/(-e^2*x^2+d^2)^(5/2)-1/7/d/e/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+8/35*x/d^5/(-e^2*x^2+d^2)^(3/2)+16/35*x/d
^7/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 106, 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 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(6*x)/(35*d^3*(d^2 - e^2*x^2)^(5/2)) - 1/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (8*x)/(35*d^5*(d^2 - e^2*x^
2)^(3/2)) + (16*x)/(35*d^7*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 )^{7/2}} \, dx &=-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d^3}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{35 d^5}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 104, normalized size = 0.98 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5 d^6+30 d^5 e x+30 d^4 e^2 x^2-40 d^3 e^3 x^3-40 d^2 e^4 x^4+16 d e^5 x^5+16 e^6 x^6\right )}{35 d^7 e (d-e x)^3 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5*d^6 + 30*d^5*e*x + 30*d^4*e^2*x^2 - 40*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 16*d*e^5*x^5 +
16*e^6*x^6))/(35*d^7*e*(d - e*x)^3*(d + e*x)^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(90)=180\).
time = 0.48, size = 223, normalized size = 2.10

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-16 e^{6} x^{6}-16 d \,e^{5} x^{5}+40 e^{4} x^{4} d^{2}+40 d^{3} e^{3} x^{3}-30 d^{4} e^{2} x^{2}-30 d^{5} e x +5 d^{6}\right )}{35 d^{7} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(92\)
trager	\(-\frac {\left (-16 e^{6} x^{6}-16 d \,e^{5} x^{5}+40 e^{4} x^{4} d^{2}+40 d^{3} e^{3} x^{3}-30 d^{4} e^{2} x^{2}-30 d^{5} e x +5 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{35 d^{7} \left (e x +d \right )^{4} \left (-e x +d \right )^{3} e}\)	\(101\)
default	\(\frac {-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}}{e}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-1/7/d/e/(x+d/e)/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+6/7*e/d*(-1/10*(-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))^(5/2)+4/5/d^2*(-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 = 100, normalized size = 0.94 \begin {gather*} -\frac {1}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e\right )}} + \frac {6 \, x}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, x}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {16 \, x}{35 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/7/((-x^2*e^2 + d^2)^(5/2)*d*x*e^2 + (-x^2*e^2 + d^2)^(5/2)*d^2*e) + 6/35*x/((-x^2*e^2 + d^2)^(5/2)*d^3) + 8
/35*x/((-x^2*e^2 + d^2)^(3/2)*d^5) + 16/35*x/(sqrt(-x^2*e^2 + d^2)*d^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (86) = 172\).
time = 2.37, size = 220, normalized size = 2.08 \begin {gather*} -\frac {5 \, x^{7} e^{7} + 5 \, d x^{6} e^{6} - 15 \, d^{2} x^{5} e^{5} - 15 \, d^{3} x^{4} e^{4} + 15 \, d^{4} x^{3} e^{3} + 15 \, d^{5} x^{2} e^{2} - 5 \, d^{6} x e - 5 \, d^{7} + {\left (16 \, x^{6} e^{6} + 16 \, d x^{5} e^{5} - 40 \, d^{2} x^{4} e^{4} - 40 \, d^{3} x^{3} e^{3} + 30 \, d^{4} x^{2} e^{2} + 30 \, d^{5} x e - 5 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{35 \, {\left (d^{7} x^{7} e^{8} + d^{8} x^{6} e^{7} - 3 \, d^{9} x^{5} e^{6} - 3 \, d^{10} x^{4} e^{5} + 3 \, d^{11} x^{3} e^{4} + 3 \, d^{12} x^{2} e^{3} - d^{13} x e^{2} - d^{14} 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)^(7/2),x, algorithm="fricas")

[Out]

-1/35*(5*x^7*e^7 + 5*d*x^6*e^6 - 15*d^2*x^5*e^5 - 15*d^3*x^4*e^4 + 15*d^4*x^3*e^3 + 15*d^5*x^2*e^2 - 5*d^6*x*e
- 5*d^7 + (16*x^6*e^6 + 16*d*x^5*e^5 - 40*d^2*x^4*e^4 - 40*d^3*x^3*e^3 + 30*d^4*x^2*e^2 + 30*d^5*x*e - 5*d^6)
*sqrt(-x^2*e^2 + d^2))/(d^7*x^7*e^8 + d^8*x^6*e^7 - 3*d^9*x^5*e^6 - 3*d^10*x^4*e^5 + 3*d^11*x^3*e^4 + 3*d^12*x
^2*e^3 - d^13*x*e^2 - d^14*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 {7}{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)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(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)^(7/2),x, algorithm="giac")

[Out]

integrate(1/((-x^2*e^2 + d^2)^(7/2)*(x*e + d)), x)

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Mupad [B]
time = 0.63, size = 155, normalized size = 1.46 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {17\,x}{70\,d^3}-\frac {1}{7\,d^2\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {8\,x}{35\,d^5}+\frac {1}{56\,d^4\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{56\,d^4\,e\,{\left (d+e\,x\right )}^4}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{35\,d^7\,\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)^(7/2)*(d + e*x)),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*((17*x)/(70*d^3) - 1/(7*d^2*e)))/((d + e*x)^3*(d - e*x)^3) + ((d^2 - e^2*x^2)^(1/2)*((8
*x)/(35*d^5) + 1/(56*d^4*e)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(56*d^4*e*(d + e*x)^4) + (16*x
*(d^2 - e^2*x^2)^(1/2))/(35*d^7*(d + e*x)*(d - e*x))

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