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

Optimal. Leaf size=33 \[ -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

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Mathematica [A]
time = 0.50, size = 41, normalized size = 1.24 \begin {gather*} -\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [A]
time = 0.47, size = 46, normalized size = 1.39

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9/e^10/d/(x+d/e)^9*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (28) = 56\).
time = 0.33, size = 503, normalized size = 15.24 \begin {gather*} -\frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{x^{8} e^{9} + 8 \, d x^{7} e^{8} + 28 \, d^{2} x^{6} e^{7} + 56 \, d^{3} x^{5} e^{6} + 70 \, d^{4} x^{4} e^{5} + 56 \, d^{5} x^{3} e^{4} + 28 \, d^{6} x^{2} e^{3} + 8 \, d^{7} x e^{2} + d^{8} e} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\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 {35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (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\right )}} + \frac {35 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{9 \, {\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 {5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}}{18 \, {\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 )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} d}{6 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{9 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{9 \, {\left (d x e^{2} + d^{2} 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)^9,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (28) = 56\).
time = 2.43, size = 152, normalized size = 4.61 \begin {gather*} -\frac {x^{5} e^{5} + 5 \, d x^{4} e^{4} + 10 \, d^{2} x^{3} e^{3} + 10 \, d^{3} x^{2} e^{2} + 5 \, d^{4} x e + d^{5} + {\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 )} \sqrt {-x^{2} e^{2} + d^{2}}}{9 \, {\left (d x^{5} e^{6} + 5 \, d^{2} x^{4} e^{5} + 10 \, d^{3} x^{3} e^{4} + 10 \, d^{4} x^{2} e^{3} + 5 \, d^{5} x e^{2} + d^{6} 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)^9,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**9,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (28) = 56\).
time = 0.97, size = 160, normalized size = 4.85 \begin {gather*} \frac {2 \, {\left (\frac {36 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {126 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {84 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + 1\right )} e^{\left (-1\right )}}{9 \, d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{9}} \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)^9,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.62, size = 141, normalized size = 4.27 \begin {gather*} \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} \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)^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)

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