3.9.12 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^{10}} \, dx\) [812]

Optimal. Leaf size=67 \[ -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9} \]

[Out]

-1/11*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^10-1/99*(-e^2*x^2+d^2)^(9/2)/d^2/e/(e*x+d)^9

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, 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 {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]

[Out]

-1/11*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*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)^{10}} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}+\frac {\int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{11 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.59, size = 48, normalized size = 0.72 \begin {gather*} -\frac {(d-e x)^4 (10 d+e x) \sqrt {d^2-e^2 x^2}}{99 d^2 e (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]

[Out]

-1/99*((d - e*x)^4*(10*d + e*x)*Sqrt[d^2 - e^2*x^2])/(d^2*e*(d + e*x)^6)

________________________________________________________________________________________

Maple [A]
time = 0.47, size = 93, normalized size = 1.39

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +10 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{99 \left (e x +d \right )^{9} d^{2} e}\)	\(43\)
trager	\(-\frac {\left (e^{5} x^{5}+6 d \,e^{4} x^{4}-34 d^{2} e^{3} x^{3}+56 d^{3} e^{2} x^{2}-39 d^{4} e x +10 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{99 d^{2} \left (e x +d \right )^{6} e}\)	\(81\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}}{e^{10}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

1/e^10*(-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))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (57) = 114\).
time = 0.35, size = 614, normalized size = 9.16 \begin {gather*} -\frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{2 \, {\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 {5}{2}} d}{6 \, {\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 {35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{24 \, {\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 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{44 \, {\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^{2}}{792 \, {\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}{198 \, {\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}}}{66 \, {\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}}}{99 \, {\left (d x^{2} e^{3} + 2 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{99 \, {\left (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)^10,x, algorithm="maxima")

[Out]

-1/2*(-x^2*e^2 + d^2)^(7/2)/(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) + 7/6*(-x^2*e^2 + d^2)^(5/2)*d/(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) - 35/24*(-x^2*e^2 + d^2)^(3/2)*d^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) + 35/44*sqrt(-x^2*e^2 + d^2)*d^3/(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) - 35/792*sqrt(-x^2*e^2 + d^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) - 5/198*sqrt(-x^2*e^2 + d^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) - 1/66*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) - 1/99*sqrt(-x^2*e^2 + d^2)/(d*x^2*e^3 + 2*d^2*x*e^2 + d^3*e) - 1/99*sqrt(-x^2*e^2 + d^2)
/(d^2*x*e^2 + d^3*e)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (57) = 114\).
time = 3.14, size = 189, normalized size = 2.82 \begin {gather*} -\frac {10 \, x^{6} e^{6} + 60 \, d x^{5} e^{5} + 150 \, d^{2} x^{4} e^{4} + 200 \, d^{3} x^{3} e^{3} + 150 \, d^{4} x^{2} e^{2} + 60 \, d^{5} x e + 10 \, d^{6} + {\left (x^{5} e^{5} + 6 \, d x^{4} e^{4} - 34 \, d^{2} x^{3} e^{3} + 56 \, d^{3} x^{2} e^{2} - 39 \, d^{4} x e + 10 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{99 \, {\left (d^{2} x^{6} e^{7} + 6 \, d^{3} x^{5} e^{6} + 15 \, d^{4} x^{4} e^{5} + 20 \, d^{5} x^{3} e^{4} + 15 \, d^{6} x^{2} e^{3} + 6 \, d^{7} x e^{2} + d^{8} 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)^10,x, algorithm="fricas")

[Out]

-1/99*(10*x^6*e^6 + 60*d*x^5*e^5 + 150*d^2*x^4*e^4 + 200*d^3*x^3*e^3 + 150*d^4*x^2*e^2 + 60*d^5*x*e + 10*d^6 +
(x^5*e^5 + 6*d*x^4*e^4 - 34*d^2*x^3*e^3 + 56*d^3*x^2*e^2 - 39*d^4*x*e + 10*d^5)*sqrt(-x^2*e^2 + d^2))/(d^2*x^
6*e^7 + 6*d^3*x^5*e^6 + 15*d^4*x^4*e^5 + 20*d^5*x^3*e^4 + 15*d^6*x^2*e^3 + 6*d^7*x*e^2 + d^8*e)

________________________________________________________________________________________

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)**10,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (57) = 114\).
time = 1.58, size = 338, normalized size = 5.04 \begin {gather*} \frac {2 \, {\left (\frac {11 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {451 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {396 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {2376 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {1386 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {3234 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {924 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-14\right )}}{x^{7}} + \frac {1254 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + \frac {99 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-18\right )}}{x^{9}} + \frac {99 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-20\right )}}{x^{10}} + 10\right )} e^{\left (-1\right )}}{99 \, d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{11}} \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)^10,x, algorithm="giac")

[Out]

2/99*(11*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 451*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 396*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 + 2376*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 + 1386*(d*e + sqrt(-
x^2*e^2 + d^2)*e)^5*e^(-10)/x^5 + 3234*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^(-12)/x^6 + 924*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^7*e^(-14)/x^7 + 1254*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^(-16)/x^8 + 99*(d*e + sqrt(-x^2*e^2 + d^2)*
e)^9*e^(-18)/x^9 + 99*(d*e + sqrt(-x^2*e^2 + d^2)*e)^10*e^(-20)/x^10 + 10)*e^(-1)/(d^2*((d*e + sqrt(-x^2*e^2 +
d^2)*e)*e^(-2)/x + 1)^11)

________________________________________________________________________________________

Mupad [B]
time = 2.12, size = 170, normalized size = 2.54 \begin {gather*} \frac {16\,\sqrt {d^2-e^2\,x^2}}{33\,e\,{\left (d+e\,x\right )}^3}-\frac {184\,d\,\sqrt {d^2-e^2\,x^2}}{99\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{99\,d\,e\,{\left (d+e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{99\,d^2\,e\,\left (d+e\,x\right )}+\frac {272\,d^2\,\sqrt {d^2-e^2\,x^2}}{99\,e\,{\left (d+e\,x\right )}^5}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{11\,e\,{\left (d+e\,x\right )}^6} \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)^10,x)

[Out]

(16*(d^2 - e^2*x^2)^(1/2))/(33*e*(d + e*x)^3) - (184*d*(d^2 - e^2*x^2)^(1/2))/(99*e*(d + e*x)^4) - (d^2 - e^2*
x^2)^(1/2)/(99*d*e*(d + e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(99*d^2*e*(d + e*x)) + (272*d^2*(d^2 - e^2*x^2)^(1/2))
/(99*e*(d + e*x)^5) - (16*d^3*(d^2 - e^2*x^2)^(1/2))/(11*e*(d + e*x)^6)

________________________________________________________________________________________