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

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9} \]

[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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=-\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (548 d^3+141 d^2 e x+24 d e^2 x^2+2 e^3 x^3\right )}{6435 d^4 e (d+e x)^8} \]

[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)

Maple [A] (verified)

Time = 5.96 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50

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

[In]

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

[Out]

-1/6435*(-e*x+d)*(2*e^3*x^3+24*d*e^2*x^2+141*d^2*e*x+548*d^3)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^11/d^4/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (117) = 234\).

Time = 0.52 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.02 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=-\frac {548 \, e^{8} x^{8} + 4384 \, d e^{7} x^{7} + 15344 \, d^{2} e^{6} x^{6} + 30688 \, d^{3} e^{5} x^{5} + 38360 \, d^{4} e^{4} x^{4} + 30688 \, d^{5} e^{3} x^{3} + 15344 \, d^{6} e^{2} x^{2} + 4384 \, d^{7} e x + 548 \, d^{8} + {\left (2 \, e^{7} x^{7} + 16 \, d e^{6} x^{6} + 57 \, d^{2} e^{5} x^{5} + 120 \, d^{3} e^{4} x^{4} - 1440 \, d^{4} e^{3} x^{3} + 2748 \, d^{5} e^{2} x^{2} - 2051 \, d^{6} e x + 548 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{4} e^{9} x^{8} + 8 \, d^{5} e^{8} x^{7} + 28 \, d^{6} e^{7} x^{6} + 56 \, d^{7} e^{6} x^{5} + 70 \, d^{8} e^{5} x^{4} + 56 \, d^{9} e^{4} x^{3} + 28 \, d^{10} e^{3} x^{2} + 8 \, d^{11} e^{2} x + d^{12} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^12,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\text {Timed out} \]

[In]

integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**12,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (117) = 234\).

Time = 0.20 (sec) , antiderivative size = 932, normalized size of antiderivative = 7.01 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{4 \, {\left (e^{12} x^{11} + 11 \, d e^{11} x^{10} + 55 \, d^{2} e^{10} x^{9} + 165 \, d^{3} e^{9} x^{8} + 330 \, d^{4} e^{8} x^{7} + 462 \, d^{5} e^{7} x^{6} + 462 \, d^{6} e^{6} x^{5} + 330 \, d^{7} e^{5} x^{4} + 165 \, d^{8} e^{4} x^{3} + 55 \, d^{9} e^{3} x^{2} + 11 \, d^{10} e^{2} x + d^{11} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{20 \, {\left (e^{11} x^{10} + 10 \, d e^{10} x^{9} + 45 \, d^{2} e^{9} x^{8} + 120 \, d^{3} e^{8} x^{7} + 210 \, d^{4} e^{7} x^{6} + 252 \, d^{5} e^{6} x^{5} + 210 \, d^{6} e^{5} x^{4} + 120 \, d^{7} e^{4} x^{3} + 45 \, d^{8} e^{3} x^{2} + 10 \, d^{9} e^{2} x + d^{10} e\right )}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{24 \, {\left (e^{10} x^{9} + 9 \, d e^{9} x^{8} + 36 \, d^{2} e^{8} x^{7} + 84 \, d^{3} e^{7} x^{6} + 126 \, d^{4} e^{6} x^{5} + 126 \, d^{5} e^{5} x^{4} + 84 \, d^{6} e^{4} x^{3} + 36 \, d^{7} e^{3} x^{2} + 9 \, d^{8} e^{2} x + d^{9} e\right )}} + \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{60 \, {\left (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\right )}} - \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{1560 \, {\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 {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{2860 \, {\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 {7 \, \sqrt {-e^{2} x^{2} + d^{2}}}{5148 \, {\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 {\sqrt {-e^{2} x^{2} + d^{2}}}{1287 \, {\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{2145 \, {\left (d^{2} e^{4} x^{3} + 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{3} e^{3} x^{2} + 2 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{4} e^{2} x + d^{5} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^12,x, algorithm="maxima")

[Out]

-1/4*(-e^2*x^2 + d^2)^(7/2)/(e^12*x^11 + 11*d*e^11*x^10 + 55*d^2*e^10*x^9 + 165*d^3*e^9*x^8 + 330*d^4*e^8*x^7
+ 462*d^5*e^7*x^6 + 462*d^6*e^6*x^5 + 330*d^7*e^5*x^4 + 165*d^8*e^4*x^3 + 55*d^9*e^3*x^2 + 11*d^10*e^2*x + d^1
1*e) + 7/20*(-e^2*x^2 + d^2)^(5/2)*d/(e^11*x^10 + 10*d*e^10*x^9 + 45*d^2*e^9*x^8 + 120*d^3*e^8*x^7 + 210*d^4*e
^7*x^6 + 252*d^5*e^6*x^5 + 210*d^6*e^5*x^4 + 120*d^7*e^4*x^3 + 45*d^8*e^3*x^2 + 10*d^9*e^2*x + d^10*e) - 7/24*
(-e^2*x^2 + d^2)^(3/2)*d^2/(e^10*x^9 + 9*d*e^9*x^8 + 36*d^2*e^8*x^7 + 84*d^3*e^7*x^6 + 126*d^4*e^6*x^5 + 126*d
^5*e^5*x^4 + 84*d^6*e^4*x^3 + 36*d^7*e^3*x^2 + 9*d^8*e^2*x + d^9*e) + 7/60*sqrt(-e^2*x^2 + d^2)*d^3/(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) - 7/1560*sqrt(-e^2*x^2 + d^2)*d^2/(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) - 7/2860*sqrt(-e^2*x^2 + d^2)*d/(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) - 7/5148*sqrt(-e^2*x^2 + d^2)/(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) - 1/1287*sqrt(-e^2*x^2 + d^2)/(d*e^5*x^4 +
4*d^2*e^4*x^3 + 6*d^3*e^3*x^2 + 4*d^4*e^2*x + d^5*e) - 1/2145*sqrt(-e^2*x^2 + d^2)/(d^2*e^4*x^3 + 3*d^3*e^3*x
^2 + 3*d^4*e^2*x + d^5*e) - 2/6435*sqrt(-e^2*x^2 + d^2)/(d^3*e^3*x^2 + 2*d^4*e^2*x + d^5*e) - 2/6435*sqrt(-e^2
*x^2 + d^2)/(d^4*e^2*x + d^5*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (117) = 234\).

Time = 0.30 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.57 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\frac {2 \, {\left (\frac {1785 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {38235 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {99190 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {426270 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {735735 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + \frac {1492205 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {1621620 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{14} x^{7}} + \frac {1904760 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + \frac {1250535 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{18} x^{9}} + \frac {909909 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10}}{e^{20} x^{10}} + \frac {321750 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{11}}{e^{22} x^{11}} + \frac {150150 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{12}}{e^{24} x^{12}} + \frac {19305 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{13}}{e^{26} x^{13}} + \frac {6435 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{14}}{e^{28} x^{14}} + 548\right )}}{6435 \, d^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{15} {\left | e \right |}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^12,x, algorithm="giac")

[Out]

2/6435*(1785*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 38235*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^
2) + 99190*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) + 426270*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8
*x^4) + 735735*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^10*x^5) + 1492205*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^
6/(e^12*x^6) + 1621620*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7/(e^14*x^7) + 1904760*(d*e + sqrt(-e^2*x^2 + d^2)*
abs(e))^8/(e^16*x^8) + 1250535*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^18*x^9) + 909909*(d*e + sqrt(-e^2*x^2
+ d^2)*abs(e))^10/(e^20*x^10) + 321750*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^11/(e^22*x^11) + 150150*(d*e + sqrt
(-e^2*x^2 + d^2)*abs(e))^12/(e^24*x^12) + 19305*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^13/(e^26*x^13) + 6435*(d*e
+ sqrt(-e^2*x^2 + d^2)*abs(e))^14/(e^28*x^14) + 548)/(d^4*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^1
5*abs(e))

Mupad [B] (verification not implemented)

Time = 12.02 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.71 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx=\frac {320\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^5}-\frac {824\,d\,\sqrt {d^2-e^2\,x^2}}{715\,e\,{\left (d+e\,x\right )}^6}-\frac {\sqrt {d^2-e^2\,x^2}}{1287\,d\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{2145\,d^2\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{6435\,d^3\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{6435\,d^4\,e\,\left (d+e\,x\right )}+\frac {368\,d^2\,\sqrt {d^2-e^2\,x^2}}{195\,e\,{\left (d+e\,x\right )}^7}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{15\,e\,{\left (d+e\,x\right )}^8} \]

[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)