\(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^9} \, dx\) [811]

Optimal result

Integrand size = 24, antiderivative size = 33 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]

[Out]

-1/9*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^9

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]

[Out]

-1/9*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^9)

Rule 665

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*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5} \]

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x]

[Out]

-1/9*((d - e*x)^4*Sqrt[d^2 - e^2*x^2])/(d*e*(d + e*x)^5)

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{9 \left (e x +d \right )^{8} d e}\)	\(36\)
default	\(-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{9 e^{10} d \left (x +\frac {d}{e}\right )^{9}}\)	\(46\)
trager	\(-\frac {\left (e^{4} x^{4}-4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}-4 d^{3} e x +d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{9 d \left (e x +d \right )^{5} e}\)	\(68\)

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

-1/9/(e*x+d)^8*(-e*x+d)/d/e*(-e^2*x^2+d^2)^(7/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (29) = 58\).

Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 4.91 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5} + {\left (e^{4} x^{4} - 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/9*(e^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x + d^5 + (e^4*x^4 - 4*d*e^3*x^3 + 6*d
^2*e^2*x^2 - 4*d^3*e*x + d^4)*sqrt(-e^2*x^2 + d^2))/(d*e^6*x^5 + 5*d^2*e^5*x^4 + 10*d^3*e^4*x^3 + 10*d^4*e^3*x
^2 + 5*d^5*e^2*x + d^6*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=\text {Timed out} \]

[In]

integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**9,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (29) = 58\).

Time = 0.20 (sec) , antiderivative size = 539, normalized size of antiderivative = 16.33 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{e^{9} x^{8} + 8 \, d e^{8} x^{7} + 28 \, d^{2} e^{7} x^{6} + 56 \, d^{3} e^{6} x^{5} + 70 \, d^{4} e^{5} x^{4} + 56 \, d^{5} e^{4} x^{3} + 28 \, d^{6} e^{3} x^{2} + 8 \, d^{7} e^{2} x + d^{8} e} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} + \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{9 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{18 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{6 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{2} x + d^{2} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^9,x, algorithm="maxima")

[Out]

-(-e^2*x^2 + d^2)^(7/2)/(e^9*x^8 + 8*d*e^8*x^7 + 28*d^2*e^7*x^6 + 56*d^3*e^6*x^5 + 70*d^4*e^5*x^4 + 56*d^5*e^4
*x^3 + 28*d^6*e^3*x^2 + 8*d^7*e^2*x + d^8*e) + 7/2*(-e^2*x^2 + d^2)^(5/2)*d/(e^8*x^7 + 7*d*e^7*x^6 + 21*d^2*e^
6*x^5 + 35*d^3*e^5*x^4 + 35*d^4*e^4*x^3 + 21*d^5*e^3*x^2 + 7*d^6*e^2*x + d^7*e) - 35/6*(-e^2*x^2 + d^2)^(3/2)*
d^2/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x + d^6*e) + 35/9*sq
rt(-e^2*x^2 + d^2)*d^3/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) - 5/18*
sqrt(-e^2*x^2 + d^2)*d^2/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e) - 1/6*sqrt(-e^2*x^2 + d
^2)*d/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/9*sqrt(-e^2*x^2 + d^2)/(e^3*x^2 + 2*d*e^2*x + d^2*e) -
1/9*sqrt(-e^2*x^2 + d^2)/(d*e^2*x + d^2*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (29) = 58\).

Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.06 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=\frac {2 \, {\left (\frac {36 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {126 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {84 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + 1\right )}}{9 \, d {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{9} {\left | e \right |}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

2/9*(36*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 126*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8*x^4)
+ 84*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^12*x^6) + 9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^16*x^8) + 1
)/(d*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^9*abs(e))

Mupad [B] (verification not implemented)

Time = 10.62 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.27 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=\frac {8\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d+e\,x\right )}^2}-\frac {8\,d\,\sqrt {d^2-e^2\,x^2}}{3\,e\,{\left (d+e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{9\,d\,e\,\left (d+e\,x\right )}+\frac {32\,d^2\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d+e\,x\right )}^4}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d+e\,x\right )}^5} \]

[In]

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^9,x)

[Out]

(8*(d^2 - e^2*x^2)^(1/2))/(9*e*(d + e*x)^2) - (8*d*(d^2 - e^2*x^2)^(1/2))/(3*e*(d + e*x)^3) - (d^2 - e^2*x^2)^
(1/2)/(9*d*e*(d + e*x)) + (32*d^2*(d^2 - e^2*x^2)^(1/2))/(9*e*(d + e*x)^4) - (16*d^3*(d^2 - e^2*x^2)^(1/2))/(9
*e*(d + e*x)^5)