3.9.13 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^{11}} \, dx\) [813]

Optimal. Leaf size=100 \[ -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \]

[Out]

-1/13*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^11-2/143*(-e^2*x^2+d^2)^(9/2)/d^2/e/(e*x+d)^10-2/1287*(-e^2*x^2+d^2)^(9
/2)/d^3/e/(e*x+d)^9

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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \begin {gather*} -\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]

[Out]

-1/13*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(143*d^2*e*(d + e*x)^10) - (2*(d^2
- e^2*x^2)^(9/2))/(1287*d^3*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]

Rule 673

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

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{13 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{143 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9}\\ \end {align*}

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Mathematica [A]
time = 0.63, size = 60, normalized size = 0.60 \begin {gather*} -\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (119 d^2+22 d e x+2 e^2 x^2\right )}{1287 d^3 e (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]

[Out]

-1/1287*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(119*d^2 + 22*d*e*x + 2*e^2*x^2))/(d^3*e*(d + e*x)^7)

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Maple [A]
time = 0.48, size = 145, normalized size = 1.45

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (2 e^{2} x^{2}+22 d x e +119 d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{1287 \left (e x +d \right )^{10} d^{3} e}\)	\(55\)
trager	\(-\frac {\left (2 e^{6} x^{6}+14 d \,e^{5} x^{5}+43 d^{2} e^{4} x^{4}-352 d^{3} e^{3} x^{3}+628 d^{4} e^{2} x^{2}-454 d^{5} e x +119 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{1287 d^{3} \left (e x +d \right )^{7} e}\)	\(93\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 d e \left (x +\frac {d}{e}\right )^{11}}+\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 d}}{e^{11}}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x,method=_RETURNVERBOSE)

[Out]

1/e^11*(-1/13/d/e/(x+d/e)^11*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+2/13*e/d*(-1/11/d/e/(x+d/e)^10*(-e^2*(x+d/e)
^2+2*d*e*(x+d/e))^(9/2)-1/99/d^2/(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. 735 vs. \(2 (85) = 170\).
time = 0.32, size = 735, normalized size = 7.35 \begin {gather*} -\frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{3 \, {\left (x^{10} e^{11} + 10 \, d x^{9} e^{10} + 45 \, d^{2} x^{8} e^{9} + 120 \, d^{3} x^{7} e^{8} + 210 \, d^{4} x^{6} e^{7} + 252 \, d^{5} x^{5} e^{6} + 210 \, d^{6} x^{4} e^{5} + 120 \, d^{7} x^{3} e^{4} + 45 \, d^{8} x^{2} e^{3} + 10 \, d^{9} x e^{2} + d^{10} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{12 \, {\left (x^{9} e^{10} + 9 \, d x^{8} e^{9} + 36 \, d^{2} x^{7} e^{8} + 84 \, d^{3} x^{6} e^{7} + 126 \, d^{4} x^{5} e^{6} + 126 \, d^{5} x^{4} e^{5} + 84 \, d^{6} x^{3} e^{4} + 36 \, d^{7} x^{2} e^{3} + 9 \, d^{8} x e^{2} + d^{9} e\right )}} - \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{12 \, {\left (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\right )}} + \frac {7 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{26 \, {\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 {7 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}}{572 \, {\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}{5148 \, {\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}}}{1287 \, {\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}}}{429 \, {\left (d x^{3} e^{4} + 3 \, d^{2} x^{2} e^{3} + 3 \, d^{3} x e^{2} + d^{4} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{1287 \, {\left (d^{2} x^{2} e^{3} + 2 \, d^{3} x e^{2} + d^{4} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{1287 \, {\left (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)^11,x, algorithm="maxima")

[Out]

-1/3*(-x^2*e^2 + d^2)^(7/2)/(x^10*e^11 + 10*d*x^9*e^10 + 45*d^2*x^8*e^9 + 120*d^3*x^7*e^8 + 210*d^4*x^6*e^7 +
252*d^5*x^5*e^6 + 210*d^6*x^4*e^5 + 120*d^7*x^3*e^4 + 45*d^8*x^2*e^3 + 10*d^9*x*e^2 + d^10*e) + 7/12*(-x^2*e^2
+ d^2)^(5/2)*d/(x^9*e^10 + 9*d*x^8*e^9 + 36*d^2*x^7*e^8 + 84*d^3*x^6*e^7 + 126*d^4*x^5*e^6 + 126*d^5*x^4*e^5
+ 84*d^6*x^3*e^4 + 36*d^7*x^2*e^3 + 9*d^8*x*e^2 + d^9*e) - 7/12*(-x^2*e^2 + d^2)^(3/2)*d^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
) + 7/26*sqrt(-x^2*e^2 + d^2)*d^3/(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) - 7/572*sqrt(-x^2*e^2 + d^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) - 35/5148*sqrt(-x^2*e^2 + d^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) - 5/1287*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) - 1/429*sqrt(-x^2*e^2 + d^2)/(d*x^3*e^4 + 3*d^2*x^2*e^3 + 3*d^3*x*e
^2 + d^4*e) - 2/1287*sqrt(-x^2*e^2 + d^2)/(d^2*x^2*e^3 + 2*d^3*x*e^2 + d^4*e) - 2/1287*sqrt(-x^2*e^2 + d^2)/(d
^3*x*e^2 + d^4*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (85) = 170\).
time = 3.16, size = 220, normalized size = 2.20 \begin {gather*} -\frac {119 \, x^{7} e^{7} + 833 \, d x^{6} e^{6} + 2499 \, d^{2} x^{5} e^{5} + 4165 \, d^{3} x^{4} e^{4} + 4165 \, d^{4} x^{3} e^{3} + 2499 \, d^{5} x^{2} e^{2} + 833 \, d^{6} x e + 119 \, d^{7} + {\left (2 \, x^{6} e^{6} + 14 \, d x^{5} e^{5} + 43 \, d^{2} x^{4} e^{4} - 352 \, d^{3} x^{3} e^{3} + 628 \, d^{4} x^{2} e^{2} - 454 \, d^{5} x e + 119 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{1287 \, {\left (d^{3} x^{7} e^{8} + 7 \, d^{4} x^{6} e^{7} + 21 \, d^{5} x^{5} e^{6} + 35 \, d^{6} x^{4} e^{5} + 35 \, d^{7} x^{3} e^{4} + 21 \, d^{8} x^{2} e^{3} + 7 \, d^{9} x e^{2} + d^{10} 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)^11,x, algorithm="fricas")

[Out]

-1/1287*(119*x^7*e^7 + 833*d*x^6*e^6 + 2499*d^2*x^5*e^5 + 4165*d^3*x^4*e^4 + 4165*d^4*x^3*e^3 + 2499*d^5*x^2*e
^2 + 833*d^6*x*e + 119*d^7 + (2*x^6*e^6 + 14*d*x^5*e^5 + 43*d^2*x^4*e^4 - 352*d^3*x^3*e^3 + 628*d^4*x^2*e^2 -
454*d^5*x*e + 119*d^6)*sqrt(-x^2*e^2 + d^2))/(d^3*x^7*e^8 + 7*d^4*x^6*e^7 + 21*d^5*x^5*e^6 + 35*d^6*x^4*e^5 +
35*d^7*x^3*e^4 + 21*d^8*x^2*e^3 + 7*d^9*x*e^2 + d^10*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)**11,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (85) = 170\).
time = 0.94, size = 398, normalized size = 3.98 \begin {gather*} \frac {2 \, {\left (\frac {260 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {6708 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {11726 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {52481 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {61776 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {120120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {84084 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-14\right )}}{x^{7}} + \frac {91377 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + \frac {32604 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-18\right )}}{x^{9}} + \frac {22308 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-20\right )}}{x^{10}} + \frac {2574 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} e^{\left (-22\right )}}{x^{11}} + \frac {1287 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{12} e^{\left (-24\right )}}{x^{12}} + 119\right )} e^{\left (-1\right )}}{1287 \, d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{13}} \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)^11,x, algorithm="giac")

[Out]

2/1287*(260*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 6708*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 11726
*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 + 52481*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 + 61776*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)^5*e^(-10)/x^5 + 120120*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^(-12)/x^6 + 84084*(d*e + s
qrt(-x^2*e^2 + d^2)*e)^7*e^(-14)/x^7 + 91377*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^(-16)/x^8 + 32604*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^9*e^(-18)/x^9 + 22308*(d*e + sqrt(-x^2*e^2 + d^2)*e)^10*e^(-20)/x^10 + 2574*(d*e + sqrt(-x^
2*e^2 + d^2)*e)^11*e^(-22)/x^11 + 1287*(d*e + sqrt(-x^2*e^2 + d^2)*e)^12*e^(-24)/x^12 + 119)*e^(-1)/(d^3*((d*e
+ sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1)^13)

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Mupad [B]
time = 2.53, size = 199, normalized size = 1.99 \begin {gather*} \frac {424\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^4}-\frac {1832\,d\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}}{429\,d\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{1287\,d^2\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{1287\,d^3\,e\,\left (d+e\,x\right )}+\frac {320\,d^2\,\sqrt {d^2-e^2\,x^2}}{143\,e\,{\left (d+e\,x\right )}^6}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{13\,e\,{\left (d+e\,x\right )}^7} \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)^11,x)

[Out]

(424*(d^2 - e^2*x^2)^(1/2))/(1287*e*(d + e*x)^4) - (1832*d*(d^2 - e^2*x^2)^(1/2))/(1287*e*(d + e*x)^5) - (d^2
- e^2*x^2)^(1/2)/(429*d*e*(d + e*x)^3) - (2*(d^2 - e^2*x^2)^(1/2))/(1287*d^2*e*(d + e*x)^2) - (2*(d^2 - e^2*x^
2)^(1/2))/(1287*d^3*e*(d + e*x)) + (320*d^2*(d^2 - e^2*x^2)^(1/2))/(143*e*(d + e*x)^6) - (16*d^3*(d^2 - e^2*x^
2)^(1/2))/(13*e*(d + e*x)^7)

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