3.9.14 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^{12}} \, dx\) [814]

Optimal. Leaf size=133 \[ -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9} \]

[Out]

-1/15*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^12-1/65*(-e^2*x^2+d^2)^(9/2)/d^2/e/(e*x+d)^11-2/715*(-e^2*x^2+d^2)^(9/2
)/d^3/e/(e*x+d)^10-2/6435*(-e^2*x^2+d^2)^(9/2)/d^4/e/(e*x+d)^9

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Rubi [A]
time = 0.04, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, 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}}{65 d^2 e (d+e x)^{11}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]

[Out]

-1/15*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^12) - (d^2 - e^2*x^2)^(9/2)/(65*d^2*e*(d + e*x)^11) - (2*(d^2 - e^2
*x^2)^(9/2))/(715*d^3*e*(d + e*x)^10) - (2*(d^2 - e^2*x^2)^(9/2))/(6435*d^4*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)^{12}} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}+\frac {\int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx}{5 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{65 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{715 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}\\ \end {align*}

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Mathematica [A]
time = 0.70, size = 71, normalized size = 0.53 \begin {gather*} -\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (548 d^3+141 d^2 e x+24 d e^2 x^2+2 e^3 x^3\right )}{6435 d^4 e (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]

[Out]

-1/6435*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(548*d^3 + 141*d^2*e*x + 24*d*e^2*x^2 + 2*e^3*x^3))/(d^4*e*(d + e*x)^
8)

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Maple [A]
time = 0.58, size = 197, normalized size = 1.48

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (2 e^{3} x^{3}+24 d \,e^{2} x^{2}+141 d^{2} e x +548 d^{3}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6435 \left (e x +d \right )^{11} d^{4} e}\)	\(66\)
trager	\(-\frac {\left (2 e^{7} x^{7}+16 d \,e^{6} x^{6}+57 d^{2} e^{5} x^{5}+120 d^{3} e^{4} x^{4}-1440 d^{4} e^{3} x^{3}+2748 d^{5} e^{2} x^{2}-2051 d^{6} e x +548 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6435 d^{4} \left (e x +d \right )^{8} e}\)	\(104\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{15 d e \left (x +\frac {d}{e}\right )^{12}}+\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 d e \left (x +\frac {d}{e}\right )^{11}}+\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 d}\right )}{5 d}}{e^{12}}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^12,x,method=_RETURNVERBOSE)

[Out]

1/e^12*(-1/15/d/e/(x+d/e)^12*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+1/5*e/d*(-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. 866 vs. \(2 (113) = 226\).
time = 0.30, size = 866, normalized size = 6.51 \begin {gather*} -\frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{4 \, {\left (x^{11} e^{12} + 11 \, d x^{10} e^{11} + 55 \, d^{2} x^{9} e^{10} + 165 \, d^{3} x^{8} e^{9} + 330 \, d^{4} x^{7} e^{8} + 462 \, d^{5} x^{6} e^{7} + 462 \, d^{6} x^{5} e^{6} + 330 \, d^{7} x^{4} e^{5} + 165 \, d^{8} x^{3} e^{4} + 55 \, d^{9} x^{2} e^{3} + 11 \, d^{10} x e^{2} + d^{11} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{20 \, {\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 {3}{2}} d^{2}}{24 \, {\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 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{60 \, {\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^{2}}{1560 \, {\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}{2860 \, {\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 {7 \, \sqrt {-x^{2} e^{2} + d^{2}}}{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 {\sqrt {-x^{2} e^{2} + d^{2}}}{1287 \, {\left (d x^{4} e^{5} + 4 \, d^{2} x^{3} e^{4} + 6 \, d^{3} x^{2} e^{3} + 4 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{2145 \, {\left (d^{2} x^{3} e^{4} + 3 \, d^{3} x^{2} e^{3} + 3 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{3} x^{2} e^{3} + 2 \, d^{4} x e^{2} + d^{5} e\right )}} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{4} x e^{2} + d^{5} 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)^12,x, algorithm="maxima")

[Out]

-1/4*(-x^2*e^2 + d^2)^(7/2)/(x^11*e^12 + 11*d*x^10*e^11 + 55*d^2*x^9*e^10 + 165*d^3*x^8*e^9 + 330*d^4*x^7*e^8
+ 462*d^5*x^6*e^7 + 462*d^6*x^5*e^6 + 330*d^7*x^4*e^5 + 165*d^8*x^3*e^4 + 55*d^9*x^2*e^3 + 11*d^10*x*e^2 + d^1
1*e) + 7/20*(-x^2*e^2 + d^2)^(5/2)*d/(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) - 7/24*
(-x^2*e^2 + d^2)^(3/2)*d^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) + 7/60*sqrt(-x^2*e^2 + d^2)*d^3/(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) - 7/1560*sqrt(-x^2*e^2 + d^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) - 7/2860*sqrt(-x^2*e^2 + d^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) - 7/5148*sqrt(-x^2*e^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) - 1/1287*sqrt(-x^2*e^2 + d^2)/(d*x^4*e^5 +
4*d^2*x^3*e^4 + 6*d^3*x^2*e^3 + 4*d^4*x*e^2 + d^5*e) - 1/2145*sqrt(-x^2*e^2 + d^2)/(d^2*x^3*e^4 + 3*d^3*x^2*e
^3 + 3*d^4*x*e^2 + d^5*e) - 2/6435*sqrt(-x^2*e^2 + d^2)/(d^3*x^2*e^3 + 2*d^4*x*e^2 + d^5*e) - 2/6435*sqrt(-x^2
*e^2 + d^2)/(d^4*x*e^2 + d^5*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (113) = 226\).
time = 3.81, size = 250, normalized size = 1.88 \begin {gather*} -\frac {548 \, x^{8} e^{8} + 4384 \, d x^{7} e^{7} + 15344 \, d^{2} x^{6} e^{6} + 30688 \, d^{3} x^{5} e^{5} + 38360 \, d^{4} x^{4} e^{4} + 30688 \, d^{5} x^{3} e^{3} + 15344 \, d^{6} x^{2} e^{2} + 4384 \, d^{7} x e + 548 \, d^{8} + {\left (2 \, x^{7} e^{7} + 16 \, d x^{6} e^{6} + 57 \, d^{2} x^{5} e^{5} + 120 \, d^{3} x^{4} e^{4} - 1440 \, d^{4} x^{3} e^{3} + 2748 \, d^{5} x^{2} e^{2} - 2051 \, d^{6} x e + 548 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6435 \, {\left (d^{4} x^{8} e^{9} + 8 \, d^{5} x^{7} e^{8} + 28 \, d^{6} x^{6} e^{7} + 56 \, d^{7} x^{5} e^{6} + 70 \, d^{8} x^{4} e^{5} + 56 \, d^{9} x^{3} e^{4} + 28 \, d^{10} x^{2} e^{3} + 8 \, d^{11} x e^{2} + d^{12} 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)^12,x, algorithm="fricas")

[Out]

-1/6435*(548*x^8*e^8 + 4384*d*x^7*e^7 + 15344*d^2*x^6*e^6 + 30688*d^3*x^5*e^5 + 38360*d^4*x^4*e^4 + 30688*d^5*
x^3*e^3 + 15344*d^6*x^2*e^2 + 4384*d^7*x*e + 548*d^8 + (2*x^7*e^7 + 16*d*x^6*e^6 + 57*d^2*x^5*e^5 + 120*d^3*x^
4*e^4 - 1440*d^4*x^3*e^3 + 2748*d^5*x^2*e^2 - 2051*d^6*x*e + 548*d^7)*sqrt(-x^2*e^2 + d^2))/(d^4*x^8*e^9 + 8*d
^5*x^7*e^8 + 28*d^6*x^6*e^7 + 56*d^7*x^5*e^6 + 70*d^8*x^4*e^5 + 56*d^9*x^3*e^4 + 28*d^10*x^2*e^3 + 8*d^11*x*e^
2 + d^12*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)**12,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (113) = 226\).
time = 0.95, size = 458, normalized size = 3.44 \begin {gather*} \frac {2 \, {\left (\frac {1785 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {38235 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {99190 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {426270 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + \frac {735735 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-10\right )}}{x^{5}} + \frac {1492205 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-12\right )}}{x^{6}} + \frac {1621620 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-14\right )}}{x^{7}} + \frac {1904760 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-16\right )}}{x^{8}} + \frac {1250535 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-18\right )}}{x^{9}} + \frac {909909 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-20\right )}}{x^{10}} + \frac {321750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} e^{\left (-22\right )}}{x^{11}} + \frac {150150 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{12} e^{\left (-24\right )}}{x^{12}} + \frac {19305 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{13} e^{\left (-26\right )}}{x^{13}} + \frac {6435 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{14} e^{\left (-28\right )}}{x^{14}} + 548\right )} e^{\left (-1\right )}}{6435 \, d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{15}} \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)^12,x, algorithm="giac")

[Out]

2/6435*(1785*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 38235*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 991
90*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 + 426270*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 + 735735*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^(-10)/x^5 + 1492205*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^(-12)/x^6 + 1621620*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e^(-14)/x^7 + 1904760*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^(-16)/x^8 + 1250535*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^9*e^(-18)/x^9 + 909909*(d*e + sqrt(-x^2*e^2 + d^2)*e)^10*e^(-20)/x^10 + 321750*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^11*e^(-22)/x^11 + 150150*(d*e + sqrt(-x^2*e^2 + d^2)*e)^12*e^(-24)/x^12 + 19305*
(d*e + sqrt(-x^2*e^2 + d^2)*e)^13*e^(-26)/x^13 + 6435*(d*e + sqrt(-x^2*e^2 + d^2)*e)^14*e^(-28)/x^14 + 548)*e^
(-1)/(d^4*((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x + 1)^15)

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Mupad [B]
time = 3.02, size = 228, normalized size = 1.71 \begin {gather*} \frac {320\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^5}-\frac {824\,d\,\sqrt {d^2-e^2\,x^2}}{715\,e\,{\left (d+e\,x\right )}^6}-\frac {\sqrt {d^2-e^2\,x^2}}{1287\,d\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{2145\,d^2\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{6435\,d^3\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{6435\,d^4\,e\,\left (d+e\,x\right )}+\frac {368\,d^2\,\sqrt {d^2-e^2\,x^2}}{195\,e\,{\left (d+e\,x\right )}^7}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{15\,e\,{\left (d+e\,x\right )}^8} \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)^12,x)

[Out]

(320*(d^2 - e^2*x^2)^(1/2))/(1287*e*(d + e*x)^5) - (824*d*(d^2 - e^2*x^2)^(1/2))/(715*e*(d + e*x)^6) - (d^2 -
e^2*x^2)^(1/2)/(1287*d*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(1/2)/(2145*d^2*e*(d + e*x)^3) - (2*(d^2 - e^2*x^2)^(1
/2))/(6435*d^3*e*(d + e*x)^2) - (2*(d^2 - e^2*x^2)^(1/2))/(6435*d^4*e*(d + e*x)) + (368*d^2*(d^2 - e^2*x^2)^(1
/2))/(195*e*(d + e*x)^7) - (16*d^3*(d^2 - e^2*x^2)^(1/2))/(15*e*(d + e*x)^8)

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