3.835 \(\int \frac{(d+e x)^9}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=206 \[ \frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(2*(d + e*x)^8)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (22*(d + e*x)^6)/(15*e*(d^2 - e^2*
x^2)^(3/2)) + (66*(d + e*x)^4)/(5*e*Sqrt[d^2 - e^2*x^2]) + (231*d^2*Sqrt[d^2 - e
^2*x^2])/(2*e) + (77*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) + (77*(d + e*x)^2*Sq
rt[d^2 - e^2*x^2])/(5*e) - (231*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.305335, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^8)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (22*(d + e*x)^6)/(15*e*(d^2 - e^2*
x^2)^(3/2)) + (66*(d + e*x)^4)/(5*e*Sqrt[d^2 - e^2*x^2]) + (231*d^2*Sqrt[d^2 - e
^2*x^2])/(2*e) + (77*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) + (77*(d + e*x)^2*Sq
rt[d^2 - e^2*x^2])/(5*e) - (231*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.5715, size = 175, normalized size = 0.85 \[ - \frac{231 d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} + \frac{231 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{2 e} + \frac{77 d \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{2 e} + \frac{2 \left (d + e x\right )^{8}}{5 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{22 \left (d + e x\right )^{6}}{15 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{66 \left (d + e x\right )^{4}}{5 e \sqrt{d^{2} - e^{2} x^{2}}} + \frac{77 \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**9/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-231*d**3*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) + 231*d**2*sqrt(d**2 - e**2*x**
2)/(2*e) + 77*d*(d + e*x)*sqrt(d**2 - e**2*x**2)/(2*e) + 2*(d + e*x)**8/(5*e*(d*
*2 - e**2*x**2)**(5/2)) - 22*(d + e*x)**6/(15*e*(d**2 - e**2*x**2)**(3/2)) + 66*
(d + e*x)**4/(5*e*sqrt(d**2 - e**2*x**2)) + 77*(d + e*x)**2*sqrt(d**2 - e**2*x**
2)/(5*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.14292, size = 111, normalized size = 0.54 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-5446 d^5+12843 d^4 e x-8711 d^3 e^2 x^2+815 d^2 e^3 x^3+105 d e^4 x^4+10 e^5 x^5\right )}{(e x-d)^3}-3465 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{30 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-5446*d^5 + 12843*d^4*e*x - 8711*d^3*e^2*x^2 + 815*d^2*e^
3*x^3 + 105*d*e^4*x^4 + 10*e^5*x^5))/(-d + e*x)^3 - 3465*d^3*ArcTan[(e*x)/Sqrt[d
^2 - e^2*x^2]])/(30*e)

_______________________________________________________________________________________

Maple [A]  time = 0.267, size = 309, normalized size = 1.5 \[{\frac{4093\,{d}^{3}x}{30}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{116\,{e}^{5}{d}^{2}{x}^{6}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+358\,{\frac{{e}^{3}{d}^{4}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{1348\,e{d}^{6}{x}^{2}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{9\,d{e}^{6}{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{231\,{d}^{3}{e}^{4}{x}^{5}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{77\,{d}^{3}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{{e}^{7}{x}^{8}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+63\,{\frac{{d}^{5}{e}^{2}{x}^{3}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{2723\,{d}^{8}}{15\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{152\,{d}^{7}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{157\,{d}^{5}x}{15} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{231\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x)

[Out]

4093/30*d^3*x/(-e^2*x^2+d^2)^(1/2)-116/3*e^5*d^2*x^6/(-e^2*x^2+d^2)^(5/2)+358*e^
3*d^4*x^4/(-e^2*x^2+d^2)^(5/2)-1348/3*e*d^6*x^2/(-e^2*x^2+d^2)^(5/2)-9/2*d*e^6*x
^7/(-e^2*x^2+d^2)^(5/2)+231/10*d^3*e^4*x^5/(-e^2*x^2+d^2)^(5/2)-77/2*d^3*e^2*x^3
/(-e^2*x^2+d^2)^(3/2)-1/3*e^7*x^8/(-e^2*x^2+d^2)^(5/2)+63*d^5*e^2*x^3/(-e^2*x^2+
d^2)^(5/2)+2723/15*d^8/e/(-e^2*x^2+d^2)^(5/2)-152/5*d^7*x/(-e^2*x^2+d^2)^(5/2)+1
57/15*d^5*x/(-e^2*x^2+d^2)^(3/2)-231/2*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^
2*x^2+d^2)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.816189, size = 518, normalized size = 2.51 \[ -\frac{e^{7} x^{8}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{9 \, d e^{6} x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{77}{10} \, d^{3} e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{116 \, d^{2} e^{5} x^{6}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{77}{2} \, d^{3} e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{358 \, d^{4} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{63 \, d^{5} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{1348 \, d^{6} e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{152 \, d^{7} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2723 \, d^{8}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{619 \, d^{5} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{989 \, d^{3} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{231 \, d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^9/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/3*e^7*x^8/(-e^2*x^2 + d^2)^(5/2) - 9/2*d*e^6*x^7/(-e^2*x^2 + d^2)^(5/2) + 77/
10*d^3*e^6*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)
^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 116/3*d^2*e^5*x^6/(-e^2*x^2
+ d^2)^(5/2) - 77/2*d^3*e^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2
*x^2 + d^2)^(3/2)*e^4)) + 358*d^4*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 63*d^5*e^2*x^
3/(-e^2*x^2 + d^2)^(5/2) - 1348/3*d^6*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 152/5*d^7*x
/(-e^2*x^2 + d^2)^(5/2) + 2723/15*d^8/((-e^2*x^2 + d^2)^(5/2)*e) + 619/15*d^5*x/
(-e^2*x^2 + d^2)^(3/2) - 989/30*d^3*x/sqrt(-e^2*x^2 + d^2) - 231/2*d^3*arcsin(e^
2*x/sqrt(d^2*e^2))/sqrt(e^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.255288, size = 841, normalized size = 4.08 \[ \frac{10 \, e^{11} x^{11} + 55 \, d e^{10} x^{10} + 110 \, d^{2} e^{9} x^{9} - 9030 \, d^{3} e^{8} x^{8} + 60646 \, d^{4} e^{7} x^{7} - 38725 \, d^{5} e^{6} x^{6} - 226678 \, d^{6} e^{5} x^{5} + 301580 \, d^{7} e^{4} x^{4} + 76240 \, d^{8} e^{3} x^{3} - 275280 \, d^{9} e^{2} x^{2} + 111840 \, d^{10} e x + 6930 \,{\left (d^{3} e^{8} x^{8} + 3 \, d^{4} e^{7} x^{7} - 27 \, d^{5} e^{6} x^{6} + 21 \, d^{6} e^{5} x^{5} + 70 \, d^{7} e^{4} x^{4} - 100 \, d^{8} e^{3} x^{3} - 16 \, d^{9} e^{2} x^{2} + 80 \, d^{10} e x - 32 \, d^{11} -{\left (d^{3} e^{7} x^{7} - 8 \, d^{4} e^{6} x^{6} + d^{5} e^{5} x^{5} + 50 \, d^{6} e^{4} x^{4} - 60 \, d^{7} e^{3} x^{3} - 32 \, d^{8} e^{2} x^{2} + 80 \, d^{9} e x - 32 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (10 \, e^{10} x^{10} + 165 \, d e^{9} x^{9} + 1325 \, d^{2} e^{8} x^{8} - 10847 \, d^{3} e^{7} x^{7} - 8835 \, d^{4} e^{6} x^{6} + 146618 \, d^{5} e^{5} x^{5} - 163940 \, d^{6} e^{4} x^{4} - 132160 \, d^{7} e^{3} x^{3} + 275280 \, d^{8} e^{2} x^{2} - 111840 \, d^{9} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{9} x^{8} + 3 \, d e^{8} x^{7} - 27 \, d^{2} e^{7} x^{6} + 21 \, d^{3} e^{6} x^{5} + 70 \, d^{4} e^{5} x^{4} - 100 \, d^{5} e^{4} x^{3} - 16 \, d^{6} e^{3} x^{2} + 80 \, d^{7} e^{2} x - 32 \, d^{8} e -{\left (e^{8} x^{7} - 8 \, d e^{7} x^{6} + d^{2} e^{6} x^{5} + 50 \, d^{3} e^{5} x^{4} - 60 \, d^{4} e^{4} x^{3} - 32 \, d^{5} e^{3} x^{2} + 80 \, d^{6} e^{2} x - 32 \, d^{7} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^9/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(10*e^11*x^11 + 55*d*e^10*x^10 + 110*d^2*e^9*x^9 - 9030*d^3*e^8*x^8 + 60646
*d^4*e^7*x^7 - 38725*d^5*e^6*x^6 - 226678*d^6*e^5*x^5 + 301580*d^7*e^4*x^4 + 762
40*d^8*e^3*x^3 - 275280*d^9*e^2*x^2 + 111840*d^10*e*x + 6930*(d^3*e^8*x^8 + 3*d^
4*e^7*x^7 - 27*d^5*e^6*x^6 + 21*d^6*e^5*x^5 + 70*d^7*e^4*x^4 - 100*d^8*e^3*x^3 -
16*d^9*e^2*x^2 + 80*d^10*e*x - 32*d^11 - (d^3*e^7*x^7 - 8*d^4*e^6*x^6 + d^5*e^5
*x^5 + 50*d^6*e^4*x^4 - 60*d^7*e^3*x^3 - 32*d^8*e^2*x^2 + 80*d^9*e*x - 32*d^10)*
sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (10*e^10*x^10
+ 165*d*e^9*x^9 + 1325*d^2*e^8*x^8 - 10847*d^3*e^7*x^7 - 8835*d^4*e^6*x^6 + 1466
18*d^5*e^5*x^5 - 163940*d^6*e^4*x^4 - 132160*d^7*e^3*x^3 + 275280*d^8*e^2*x^2 -
111840*d^9*e*x)*sqrt(-e^2*x^2 + d^2))/(e^9*x^8 + 3*d*e^8*x^7 - 27*d^2*e^7*x^6 +
21*d^3*e^6*x^5 + 70*d^4*e^5*x^4 - 100*d^5*e^4*x^3 - 16*d^6*e^3*x^2 + 80*d^7*e^2*
x - 32*d^8*e - (e^8*x^7 - 8*d*e^7*x^6 + d^2*e^6*x^5 + 50*d^3*e^5*x^4 - 60*d^4*e^
4*x^3 - 32*d^5*e^3*x^2 + 80*d^6*e^2*x - 32*d^7*e)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{9}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**9/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**9/(-(-d + e*x)*(d + e*x))**(7/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232912, size = 174, normalized size = 0.84 \[ -\frac{231}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{{\left (5446 \, d^{8} e^{\left (-1\right )} +{\left (3495 \, d^{7} -{\left (13480 \, d^{6} e +{\left (7765 \, d^{5} e^{2} -{\left (10740 \, d^{4} e^{3} +{\left (5941 \, d^{3} e^{4} - 5 \,{\left (232 \, d^{2} e^{5} +{\left (2 \, x e^{7} + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{30 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^9/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-231/2*d^3*arcsin(x*e/d)*e^(-1)*sign(d) - 1/30*(5446*d^8*e^(-1) + (3495*d^7 - (1
3480*d^6*e + (7765*d^5*e^2 - (10740*d^4*e^3 + (5941*d^3*e^4 - 5*(232*d^2*e^5 + (
2*x*e^7 + 27*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3