\(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^{13}} \, dx\) [815]

Optimal result

Integrand size = 24, antiderivative size = 166 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9} \]

[Out]

-1/17*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^13-4/255*(-e^2*x^2+d^2)^(9/2)/d^2/e/(e*x+d)^12-4/1105*(-e^2*x^2+d^2)^(9
/2)/d^3/e/(e*x+d)^11-8/12155*(-e^2*x^2+d^2)^(9/2)/d^4/e/(e*x+d)^10-8/109395*(-e^2*x^2+d^2)^(9/2)/d^5/e/(e*x+d)
^9

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}} \]

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]

[Out]

-1/17*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^13) - (4*(d^2 - e^2*x^2)^(9/2))/(255*d^2*e*(d + e*x)^12) - (4*(d^2
- e^2*x^2)^(9/2))/(1105*d^3*e*(d + e*x)^11) - (8*(d^2 - e^2*x^2)^(9/2))/(12155*d^4*e*(d + e*x)^10) - (8*(d^2 -
e^2*x^2)^(9/2))/(109395*d^5*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}}{17 d e (d+e x)^{13}}+\frac {4 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx}{17 d} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}+\frac {4 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx}{85 d^2} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}+\frac {8 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{1105 d^3} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}+\frac {8 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{12155 d^4} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac {4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac {8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.49 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=-\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (8627 d^4+2756 d^3 e x+660 d^2 e^2 x^2+104 d e^3 x^3+8 e^4 x^4\right )}{109395 d^5 e (d+e x)^9} \]

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]

[Out]

-1/109395*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(8627*d^4 + 2756*d^3*e*x + 660*d^2*e^2*x^2 + 104*d*e^3*x^3 + 8*e^4*
x^4))/(d^5*e*(d + e*x)^9)

Maple [A] (verified)

Time = 7.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+104 d \,e^{3} x^{3}+660 d^{2} e^{2} x^{2}+2756 d^{3} e x +8627 d^{4}\right ) \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{109395 \left (e x +d \right )^{12} d^{5} e}\)	\(77\)
trager	\(-\frac {\left (8 e^{8} x^{8}+72 d \,e^{7} x^{7}+292 d^{2} e^{6} x^{6}+708 d^{3} e^{5} x^{5}+1155 d^{4} e^{4} x^{4}-20508 d^{5} e^{3} x^{3}+41398 d^{6} e^{2} x^{2}-31752 d^{7} e x +8627 d^{8}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{109395 d^{5} \left (e x +d \right )^{9} e}\)	\(115\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{17 d e \left (x +\frac {d}{e}\right )^{13}}+\frac {4 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{15 d e \left (x +\frac {d}{e}\right )^{12}}+\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 d e \left (x +\frac {d}{e}\right )^{11}}+\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 d}\right )}{5 d}\right )}{17 d}}{e^{13}}\)	\(249\)

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^13,x,method=_RETURNVERBOSE)

[Out]

-1/109395*(-e*x+d)*(8*e^4*x^4+104*d*e^3*x^3+660*d^2*e^2*x^2+2756*d^3*e*x+8627*d^4)*(-e^2*x^2+d^2)^(7/2)/(e*x+d
)^12/d^5/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (146) = 292\).

Time = 0.70 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.82 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=-\frac {8627 \, e^{9} x^{9} + 77643 \, d e^{8} x^{8} + 310572 \, d^{2} e^{7} x^{7} + 724668 \, d^{3} e^{6} x^{6} + 1087002 \, d^{4} e^{5} x^{5} + 1087002 \, d^{5} e^{4} x^{4} + 724668 \, d^{6} e^{3} x^{3} + 310572 \, d^{7} e^{2} x^{2} + 77643 \, d^{8} e x + 8627 \, d^{9} + {\left (8 \, e^{8} x^{8} + 72 \, d e^{7} x^{7} + 292 \, d^{2} e^{6} x^{6} + 708 \, d^{3} e^{5} x^{5} + 1155 \, d^{4} e^{4} x^{4} - 20508 \, d^{5} e^{3} x^{3} + 41398 \, d^{6} e^{2} x^{2} - 31752 \, d^{7} e x + 8627 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{109395 \, {\left (d^{5} e^{10} x^{9} + 9 \, d^{6} e^{9} x^{8} + 36 \, d^{7} e^{8} x^{7} + 84 \, d^{8} e^{7} x^{6} + 126 \, d^{9} e^{6} x^{5} + 126 \, d^{10} e^{5} x^{4} + 84 \, d^{11} e^{4} x^{3} + 36 \, d^{12} e^{3} x^{2} + 9 \, d^{13} e^{2} x + d^{14} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^13,x, algorithm="fricas")

[Out]

-1/109395*(8627*e^9*x^9 + 77643*d*e^8*x^8 + 310572*d^2*e^7*x^7 + 724668*d^3*e^6*x^6 + 1087002*d^4*e^5*x^5 + 10
87002*d^5*e^4*x^4 + 724668*d^6*e^3*x^3 + 310572*d^7*e^2*x^2 + 77643*d^8*e*x + 8627*d^9 + (8*e^8*x^8 + 72*d*e^7
*x^7 + 292*d^2*e^6*x^6 + 708*d^3*e^5*x^5 + 1155*d^4*e^4*x^4 - 20508*d^5*e^3*x^3 + 41398*d^6*e^2*x^2 - 31752*d^
7*e*x + 8627*d^8)*sqrt(-e^2*x^2 + d^2))/(d^5*e^10*x^9 + 9*d^6*e^9*x^8 + 36*d^7*e^8*x^7 + 84*d^8*e^7*x^6 + 126*
d^9*e^6*x^5 + 126*d^10*e^5*x^4 + 84*d^11*e^4*x^3 + 36*d^12*e^3*x^2 + 9*d^13*e^2*x + d^14*e)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=\text {Timed out} \]

[In]

integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**13,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (146) = 292\).

Time = 0.21 (sec) , antiderivative size = 1085, normalized size of antiderivative = 6.54 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{5 \, {\left (e^{13} x^{12} + 12 \, d e^{12} x^{11} + 66 \, d^{2} e^{11} x^{10} + 220 \, d^{3} e^{10} x^{9} + 495 \, d^{4} e^{9} x^{8} + 792 \, d^{5} e^{8} x^{7} + 924 \, d^{6} e^{7} x^{6} + 792 \, d^{7} e^{6} x^{5} + 495 \, d^{8} e^{5} x^{4} + 220 \, d^{9} e^{4} x^{3} + 66 \, d^{10} e^{3} x^{2} + 12 \, d^{11} e^{2} x + d^{12} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{30 \, {\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 {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\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 {\sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{17 \, {\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 {\sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{510 \, {\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}{6630 \, {\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}}}{12155 \, {\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}}}{21879 \, {\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{21879 \, {\left (d^{2} e^{5} x^{4} + 4 \, d^{3} e^{4} x^{3} + 6 \, d^{4} e^{3} x^{2} + 4 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{36465 \, {\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{109395 \, {\left (d^{4} e^{3} x^{2} + 2 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{109395 \, {\left (d^{5} e^{2} x + d^{6} e\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^13,x, algorithm="maxima")

[Out]

-1/5*(-e^2*x^2 + d^2)^(7/2)/(e^13*x^12 + 12*d*e^12*x^11 + 66*d^2*e^11*x^10 + 220*d^3*e^10*x^9 + 495*d^4*e^9*x^
8 + 792*d^5*e^8*x^7 + 924*d^6*e^7*x^6 + 792*d^7*e^6*x^5 + 495*d^8*e^5*x^4 + 220*d^9*e^4*x^3 + 66*d^10*e^3*x^2
+ 12*d^11*e^2*x + d^12*e) + 7/30*(-e^2*x^2 + d^2)^(5/2)*d/(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) - 1/6*(-e^2*x^2 + d^2)^(3/2)*d^2/(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) + 1/17*sqrt(-e^2*x^2 + d^2)*d^3/(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) - 1/510*sqrt(
-e^2*x^2 + d^2)*d^2/(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) - 7/6630*sqrt(-e^2*x^2 + d^2)*d/(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) - 7/12155*sqrt(-e^2*x^2 + d^2)/(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) - 7/21879*sqrt(
-e^2*x^2 + d^2)/(d*e^6*x^5 + 5*d^2*e^5*x^4 + 10*d^3*e^4*x^3 + 10*d^4*e^3*x^2 + 5*d^5*e^2*x + d^6*e) - 4/21879*
sqrt(-e^2*x^2 + d^2)/(d^2*e^5*x^4 + 4*d^3*e^4*x^3 + 6*d^4*e^3*x^2 + 4*d^5*e^2*x + d^6*e) - 4/36465*sqrt(-e^2*x
^2 + d^2)/(d^3*e^4*x^3 + 3*d^4*e^3*x^2 + 3*d^5*e^2*x + d^6*e) - 8/109395*sqrt(-e^2*x^2 + d^2)/(d^4*e^3*x^2 + 2
*d^5*e^2*x + d^6*e) - 8/109395*sqrt(-e^2*x^2 + d^2)/(d^5*e^2*x + d^6*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (146) = 292\).

Time = 0.30 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.23 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=\frac {2 \, {\left (\frac {37264 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {735692 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {2511580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {11197220 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {24999520 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + \frac {55979300 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {80466100 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{14} x^{7}} + \frac {108787250 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + \frac {100935120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{18} x^{9}} + \frac {87311796 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10}}{e^{20} x^{10}} + \frac {50788452 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{11}}{e^{22} x^{11}} + \frac {28384356 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{12}}{e^{24} x^{12}} + \frac {9335040 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{13}}{e^{26} x^{13}} + \frac {3354780 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{14}}{e^{28} x^{14}} + \frac {437580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{15}}{e^{30} x^{15}} + \frac {109395 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{16}}{e^{32} x^{16}} + 8627\right )}}{109395 \, d^{5} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{17} {\left | e \right |}} \]

[In]

integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^13,x, algorithm="giac")

[Out]

2/109395*(37264*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 735692*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^
4*x^2) + 2511580*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) + 11197220*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)
)^4/(e^8*x^4) + 24999520*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^10*x^5) + 55979300*(d*e + sqrt(-e^2*x^2 + d^
2)*abs(e))^6/(e^12*x^6) + 80466100*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7/(e^14*x^7) + 108787250*(d*e + sqrt(-e
^2*x^2 + d^2)*abs(e))^8/(e^16*x^8) + 100935120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^18*x^9) + 87311796*(d*
e + sqrt(-e^2*x^2 + d^2)*abs(e))^10/(e^20*x^10) + 50788452*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^11/(e^22*x^11)
+ 28384356*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^12/(e^24*x^12) + 9335040*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^13
/(e^26*x^13) + 3354780*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^14/(e^28*x^14) + 437580*(d*e + sqrt(-e^2*x^2 + d^2)
*abs(e))^15/(e^30*x^15) + 109395*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^16/(e^32*x^16) + 8627)/(d^5*((d*e + sqrt(
-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^17*abs(e))

Mupad [B] (verification not implemented)

Time = 12.65 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.55 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx=\frac {2424\,\sqrt {d^2-e^2\,x^2}}{12155\,e\,{\left (d+e\,x\right )}^6}-\frac {3208\,d\,\sqrt {d^2-e^2\,x^2}}{3315\,e\,{\left (d+e\,x\right )}^7}-\frac {7\,\sqrt {d^2-e^2\,x^2}}{21879\,d\,e\,{\left (d+e\,x\right )}^5}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{21879\,d^2\,e\,{\left (d+e\,x\right )}^4}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{36465\,d^3\,e\,{\left (d+e\,x\right )}^3}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{109395\,d^4\,e\,{\left (d+e\,x\right )}^2}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{109395\,d^5\,e\,\left (d+e\,x\right )}+\frac {416\,d^2\,\sqrt {d^2-e^2\,x^2}}{255\,e\,{\left (d+e\,x\right )}^8}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{17\,e\,{\left (d+e\,x\right )}^9} \]

[In]

int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x)

[Out]

(2424*(d^2 - e^2*x^2)^(1/2))/(12155*e*(d + e*x)^6) - (3208*d*(d^2 - e^2*x^2)^(1/2))/(3315*e*(d + e*x)^7) - (7*
(d^2 - e^2*x^2)^(1/2))/(21879*d*e*(d + e*x)^5) - (4*(d^2 - e^2*x^2)^(1/2))/(21879*d^2*e*(d + e*x)^4) - (4*(d^2
- e^2*x^2)^(1/2))/(36465*d^3*e*(d + e*x)^3) - (8*(d^2 - e^2*x^2)^(1/2))/(109395*d^4*e*(d + e*x)^2) - (8*(d^2
- e^2*x^2)^(1/2))/(109395*d^5*e*(d + e*x)) + (416*d^2*(d^2 - e^2*x^2)^(1/2))/(255*e*(d + e*x)^8) - (16*d^3*(d^
2 - e^2*x^2)^(1/2))/(17*e*(d + e*x)^9)