3.9.9 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^7} \, dx\) [809]

Optimal. Leaf size=138 \[ -\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

[Out]

-14/3*(-e^2*x^2+d^2)^(3/2)/e/(e*x+d)^2+14/15*(-e^2*x^2+d^2)^(5/2)/e/(e*x+d)^4-2/5*(-e^2*x^2+d^2)^(7/2)/e/(e*x+
d)^6-7*d*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e-7*(-e^2*x^2+d^2)^(1/2)/e

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Rubi [A]
time = 0.04, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {677, 679, 223, 209} \begin {gather*} -\frac {7 d \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[d^2 - e^2*x^2])/e - (14*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^2) + (14*(d^2 - e^2*x^2)^(5/2))/(15*e*(
d + e*x)^4) - (2*(d^2 - e^2*x^2)^(7/2))/(5*e*(d + e*x)^6) - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 677

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

Rule 679

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

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7}{5} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx\\ &=\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {7}{3} \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\\ &=-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-7 \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx\\ &=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-(7 d) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-(7 d) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 103, normalized size = 0.75 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-167 d^3-381 d^2 e x-277 d e^2 x^2-15 e^3 x^3\right )}{15 e (d+e x)^3}+\frac {7 d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-167*d^3 - 381*d^2*e*x - 277*d*e^2*x^2 - 15*e^3*x^3))/(15*e*(d + e*x)^3) + (7*d*Log[-(Sq
rt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/Sqrt[-e^2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs. \(2(122)=244\).
time = 0.52, size = 559, normalized size = 4.05

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e}-\frac {7 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {128 d^{2} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{3} \left (x +\frac {d}{e}\right )^{2}}-\frac {232 d \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{2} \left (x +\frac {d}{e}\right )}-\frac {16 d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}\)	\(184\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{7}}-\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{6}}-\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{5}}-\frac {4 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 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}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 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}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}\right )}{d}\right )}{d}\right )}{5 d}}{e^{7}}\)	\(559\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^7*(-1/5/d/e/(x+d/e)^7*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-2/5*e/d*(-1/3/d/e/(x+d/e)^6*(-e^2*(x+d/e)^2+2*d
*e*(x+d/e))^(9/2)-e/d*(-1/d/e/(x+d/e)^5*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-4*e/d*(1/d/e/(x+d/e)^4*(-e^2*(x+d
/e)^2+2*d*e*(x+d/e))^(9/2)+5*e/d*(1/3/d/e/(x+d/e)^3*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+2*e/d*(1/5/d/e/(x+d/e
)^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+7/5*e/d*(1/7*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(7/2)+d*e*(-1/12*(-2*e^2*
(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+5/6*d^2*(-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)
^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/
(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2))))))))))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (116) = 232\).
time = 0.50, size = 377, normalized size = 2.73 \begin {gather*} -7 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{x^{6} e^{7} + 6 \, d x^{5} e^{6} + 15 \, d^{2} x^{4} e^{5} + 20 \, d^{3} x^{3} e^{4} + 15 \, d^{4} x^{2} e^{3} + 6 \, d^{5} x e^{2} + d^{6} e} - \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, {\left (x^{5} e^{6} + 5 \, d x^{4} e^{5} + 10 \, d^{2} x^{3} e^{4} + 10 \, d^{3} x^{2} e^{3} + 5 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{x^{4} e^{5} + 4 \, d x^{3} e^{4} + 6 \, d^{2} x^{2} e^{3} + 4 \, d^{3} x e^{2} + d^{4} e} + \frac {42 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{5 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d}{3 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {49 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}}{15 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} - \frac {266 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{15 \, {\left (x e^{2} + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*d*arcsin(x*e/d)*e^(-1) + (-x^2*e^2 + d^2)^(7/2)/(x^6*e^7 + 6*d*x^5*e^6 + 15*d^2*x^4*e^5 + 20*d^3*x^3*e^4 +
15*d^4*x^2*e^3 + 6*d^5*x*e^2 + d^6*e) - 7/5*(-x^2*e^2 + d^2)^(5/2)*d/(x^5*e^6 + 5*d*x^4*e^5 + 10*d^2*x^3*e^4 +
10*d^3*x^2*e^3 + 5*d^4*x*e^2 + d^5*e) - 7*(-x^2*e^2 + d^2)^(3/2)*d^2/(x^4*e^5 + 4*d*x^3*e^4 + 6*d^2*x^2*e^3 +
4*d^3*x*e^2 + d^4*e) + 42/5*sqrt(-x^2*e^2 + d^2)*d^3/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e) + 7/3*(-x^
2*e^2 + d^2)^(3/2)*d/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e) + 49/15*sqrt(-x^2*e^2 + d^2)*d^2/(x^2*e^3 +
2*d*x*e^2 + d^2*e) - 266/15*sqrt(-x^2*e^2 + d^2)*d/(x*e^2 + d*e)

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Fricas [A]
time = 2.41, size = 164, normalized size = 1.19 \begin {gather*} -\frac {167 \, d x^{3} e^{3} + 501 \, d^{2} x^{2} e^{2} + 501 \, d^{3} x e + 167 \, d^{4} - 210 \, {\left (d x^{3} e^{3} + 3 \, d^{2} x^{2} e^{2} + 3 \, d^{3} x e + d^{4}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (15 \, x^{3} e^{3} + 277 \, d x^{2} e^{2} + 381 \, d^{2} x e + 167 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(167*d*x^3*e^3 + 501*d^2*x^2*e^2 + 501*d^3*x*e + 167*d^4 - 210*(d*x^3*e^3 + 3*d^2*x^2*e^2 + 3*d^3*x*e +
d^4)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (15*x^3*e^3 + 277*d*x^2*e^2 + 381*d^2*x*e + 167*d^3)*sqrt(
-x^2*e^2 + d^2))/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.95, size = 194, normalized size = 1.41 \begin {gather*} -7 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) - \sqrt {-x^{2} e^{2} + d^{2}} e^{\left (-1\right )} + \frac {16 \, {\left (\frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{\left (-2\right )}}{x} + \frac {130 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-4\right )}}{x^{2}} + \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{\left (-6\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d e^{\left (-8\right )}}{x^{4}} + 19 \, d\right )} e^{\left (-1\right )}}{15 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*d*arcsin(x*e/d)*e^(-1)*sgn(d) - sqrt(-x^2*e^2 + d^2)*e^(-1) + 16/15*(80*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d*e^
(-2)/x + 130*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d*e^(-4)/x^2 + 60*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d*e^(-6)/x^3
+ 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d*e^(-8)/x^4 + 19*d)*e^(-1)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1
)^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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