3.9.10 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^8} \, dx\) [810]

Optimal. Leaf size=143 \[ \frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {677, 223, 209} \begin {gather*} \frac {\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}}{7 e (d+e x)^7}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]

[Out]

(2*Sqrt[d^2 - e^2*x^2])/(e*(d + e*x)) - (2*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^3) + (2*(d^2 - e^2*x^2)^(5/2)
)/(5*e*(d + e*x)^5) - (2*(d^2 - e^2*x^2)^(7/2))/(7*e*(d + e*x)^7) + 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]

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^8} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx\\ &=\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx\\ &=-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^2} \, dx\\ &=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 102, normalized size = 0.71 \begin {gather*} \frac {8 \sqrt {d^2-e^2 x^2} \left (19 d^3+76 d^2 e x+71 d e^2 x^2+44 e^3 x^3\right )}{105 e (d+e x)^4}-\frac {\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)^8,x]

[Out]

(8*Sqrt[d^2 - e^2*x^2]*(19*d^3 + 76*d^2*e*x + 71*d*e^2*x^2 + 44*e^3*x^3))/(105*e*(d + e*x)^4) - Log[-(Sqrt[-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. \(610\) vs. \(2(127)=254\).
time = 0.48, size = 611, normalized size = 4.27

method	result	size
default	\(\text {Too long to display}\)	\(611\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

1/e^8*(-1/7/d/e/(x+d/e)^8*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-1/7*e/d*(-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. 583 vs. \(2 (122) = 244\).
time = 0.51, size = 583, normalized size = 4.08 \begin {gather*} -\frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, {\left (x^{7} e^{8} + 7 \, d x^{6} e^{7} + 21 \, d^{2} x^{5} e^{6} + 35 \, d^{3} x^{4} e^{5} + 35 \, d^{4} x^{3} e^{4} + 21 \, d^{5} x^{2} e^{3} + 7 \, d^{6} x e^{2} + d^{7} e\right )}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{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 {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{2 \, {\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 {15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{7 \, {\left (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\right )}} + \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}{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 {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d}{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 {69 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}}{70 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {34 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{105 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} + \frac {281 \, \sqrt {-x^{2} e^{2} + d^{2}}}{105 \, {\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)^8,x, algorithm="maxima")

[Out]

-1/7*(-x^2*e^2 + d^2)^(7/2)/(x^7*e^8 + 7*d*x^6*e^7 + 21*d^2*x^5*e^6 + 35*d^3*x^4*e^5 + 35*d^4*x^3*e^4 + 21*d^5
*x^2*e^3 + 7*d^6*x*e^2 + d^7*e) - (-x^2*e^2 + d^2)^(5/2)*d/(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) + 5/2*(-x^2*e^2 + d^2)^(3/2)*d^2/(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) - 15/7*sqrt(-x^2*e^2 + d^2)*d^3/(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) + arcsin(x*e/d)*e^(-1) + 1/5*(-x^2*e^2 + d^2)^(5/2)/(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) + (-x^2*e^2 + d^2)^(3/2)*d/(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) - 69/70*sqrt(-x^2*e^2 + d^2)*d^2/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*
e) - 1/3*(-x^2*e^2 + d^2)^(3/2)/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e) - 34/105*sqrt(-x^2*e^2 + d^2)*d/
(x^2*e^3 + 2*d*x*e^2 + d^2*e) + 281/105*sqrt(-x^2*e^2 + d^2)/(x*e^2 + d*e)

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Fricas [A]
time = 1.93, size = 189, normalized size = 1.32 \begin {gather*} \frac {2 \, {\left (76 \, x^{4} e^{4} + 304 \, d x^{3} e^{3} + 456 \, d^{2} x^{2} e^{2} + 304 \, d^{3} x e + 76 \, d^{4} - 105 \, {\left (x^{4} e^{4} + 4 \, d x^{3} e^{3} + 6 \, d^{2} x^{2} e^{2} + 4 \, 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 ) + 4 \, {\left (44 \, x^{3} e^{3} + 71 \, d x^{2} e^{2} + 76 \, d^{2} x e + 19 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )}}{105 \, {\left (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\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)^8,x, algorithm="fricas")

[Out]

2/105*(76*x^4*e^4 + 304*d*x^3*e^3 + 456*d^2*x^2*e^2 + 304*d^3*x*e + 76*d^4 - 105*(x^4*e^4 + 4*d*x^3*e^3 + 6*d^
2*x^2*e^2 + 4*d^3*x*e + d^4)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + 4*(44*x^3*e^3 + 71*d*x^2*e^2 + 76*
d^2*x*e + 19*d^3)*sqrt(-x^2*e^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)

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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 )^{8}}\, 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)**8,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**(7/2)/(d + e*x)**8, x)

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Giac [A]
time = 1.01, size = 199, normalized size = 1.39 \begin {gather*} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) - \frac {16 \, {\left (\frac {133 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {294 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {490 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {175 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + 19\right )} e^{\left (-1\right )}}{105 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{7}} \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)^8,x, algorithm="giac")

[Out]

arcsin(x*e/d)*e^(-1)*sgn(d) - 16/105*(133*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 294*(d*e + sqrt(-x^2*e^2 +
d^2)*e)^2*e^(-4)/x^2 + 490*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 + 175*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4
*e^(-8)/x^4 + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^(-10)/x^5 + 19)*e^(-1)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^
(-2)/x + 1)^7

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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 )}^8} \,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)^8,x)

[Out]

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^8, x)

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