3.9.53 \(\int \frac {(d+e x)^2}{(d^2-e^2 x^2)^{7/2}} \, dx\) [853]

Optimal. Leaf size=77 \[ \frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 77, 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 = {667, 198, 197} \begin {gather*} \frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x))/(5*e*(d^2 - e^2*x^2)^(5/2)) + x/(5*d^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(5*d^4*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 667

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)*((a + c*x^2)^(p + 1)/(c*(p
+ 1))), x] - Dist[e^2*((p + 2)/(c*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.32, size = 70, normalized size = 0.91 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x)^3 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs. \(2(65)=130\).
time = 0.47, size = 193, normalized size = 2.51

method	result	size
gosper	\(\frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right )}{5 d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(65\)
trager	\(\frac {\left (2 e^{3} x^{3}-4 d \,e^{2} x^{2}+d^{2} e x +2 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{4} \left (-e x +d \right )^{3} e \left (e x +d \right )}\)	\(67\)
default	\(e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {2 d}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(193\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(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))))+2/5*d/e/(-e^2*x^2+d^2)^(5/2)+d^2*(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 = 0.28, size = 73, normalized size = 0.95 \begin {gather*} \frac {2 \, d e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {2 \, x}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.18, size = 111, normalized size = 1.44 \begin {gather*} \frac {2 \, x^{4} e^{4} - 4 \, d x^{3} e^{3} + 4 \, d^{3} x e - 2 \, d^{4} - {\left (2 \, x^{3} e^{3} - 4 \, d x^{2} e^{2} + d^{2} x e + 2 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d^{4} x^{4} e^{5} - 2 \, d^{5} x^{3} e^{4} + 2 \, d^{7} x e^{2} - d^{8} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(2*x^4*e^4 - 4*d*x^3*e^3 + 4*d^3*x*e - 2*d^4 - (2*x^3*e^3 - 4*d*x^2*e^2 + d^2*x*e + 2*d^3)*sqrt(-x^2*e^2 +
d^2))/(d^4*x^4*e^5 - 2*d^5*x^3*e^4 + 2*d^7*x*e^2 - d^8*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 0.60, size = 66, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^3+d^2\,e\,x-4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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