3.836 \(\int \frac{(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=173 \[ \frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (6*(d + e*x)^5)/(5*e*(d^2 - e^2*x^
2)^(3/2)) + (42*(d + e*x)^3)/(5*e*Sqrt[d^2 - e^2*x^2]) + (63*d*Sqrt[d^2 - e^2*x^
2])/(2*e) + (21*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) - (63*d^2*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(2*e)

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Rubi [A]  time = 0.235747, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^8/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (6*(d + e*x)^5)/(5*e*(d^2 - e^2*x^
2)^(3/2)) + (42*(d + e*x)^3)/(5*e*Sqrt[d^2 - e^2*x^2]) + (63*d*Sqrt[d^2 - e^2*x^
2])/(2*e) + (21*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) - (63*d^2*ArcTan[(e*x)/Sqrt
[d^2 - e^2*x^2]])/(2*e)

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Rubi in Sympy [A]  time = 34.4728, size = 146, normalized size = 0.84 \[ - \frac{63 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} + \frac{63 d \sqrt{d^{2} - e^{2} x^{2}}}{2 e} + \frac{2 \left (d + e x\right )^{7}}{5 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{6 \left (d + e x\right )^{5}}{5 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{42 \left (d + e x\right )^{3}}{5 e \sqrt{d^{2} - e^{2} x^{2}}} + \frac{21 \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**8/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-63*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) + 63*d*sqrt(d**2 - e**2*x**2)/(2
*e) + 2*(d + e*x)**7/(5*e*(d**2 - e**2*x**2)**(5/2)) - 6*(d + e*x)**5/(5*e*(d**2
- e**2*x**2)**(3/2)) + 42*(d + e*x)**3/(5*e*sqrt(d**2 - e**2*x**2)) + 21*(d + e
*x)*sqrt(d**2 - e**2*x**2)/(2*e)

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Mathematica [A]  time = 0.18291, size = 99, normalized size = 0.57 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (496 d^4-1163 d^3 e x+801 d^2 e^2 x^2-65 d e^3 x^3-5 e^4 x^4\right )}{(d-e x)^3}-315 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{10 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^8/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(496*d^4 - 1163*d^3*e*x + 801*d^2*e^2*x^2 - 65*d*e^3*x^3 -
5*e^4*x^4))/(d - e*x)^3 - 315*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(10*e)

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Maple [A]  time = 0.015, size = 284, normalized size = 1.6 \[ -{\frac{63\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{6}{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{63\,{e}^{4}{d}^{2}{x}^{5}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+104\,{\frac{{d}^{3}{e}^{3}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-120\,{\frac{{d}^{5}e{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{27\,{d}^{4}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{21\,{d}^{2}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{76\,{d}^{6}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{423\,{d}^{2}x}{10}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+35\,{\frac{{d}^{4}{e}^{2}{x}^{3}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-8\,{\frac{{e}^{5}d{x}^{6}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{248\,{d}^{7}}{5\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-63/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2*e^6*x^7/(-e
^2*x^2+d^2)^(5/2)+63/10*e^4*d^2*x^5/(-e^2*x^2+d^2)^(5/2)+104*d^3*e^3*x^4/(-e^2*x
^2+d^2)^(5/2)-120*d^5*e*x^2/(-e^2*x^2+d^2)^(5/2)+27/5*d^4*x/(-e^2*x^2+d^2)^(3/2)
-21/2*e^2*d^2*x^3/(-e^2*x^2+d^2)^(3/2)-76/5*d^6*x/(-e^2*x^2+d^2)^(5/2)+423/10*d^
2*x/(-e^2*x^2+d^2)^(1/2)+35*d^4*e^2*x^3/(-e^2*x^2+d^2)^(5/2)-8*e^5*d*x^6/(-e^2*x
^2+d^2)^(5/2)+248/5*d^7/e/(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.81289, size = 485, normalized size = 2.8 \[ -\frac{e^{6} x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{21}{10} \, d^{2} e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{8 \, d e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{21}{2} \, d^{2} e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{104 \, d^{3} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{35 \, d^{4} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{120 \, d^{5} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{76 \, d^{6} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{248 \, d^{7}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{69 \, d^{4} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{39 \, d^{2} x}{10 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{63 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*e^6*x^7/(-e^2*x^2 + d^2)^(5/2) + 21/10*d^2*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)) - 8*d*e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 21/2*d^2*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)) + 104*d^3*e^3*x^4/(-e
^2*x^2 + d^2)^(5/2) + 35*d^4*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) - 120*d^5*e*x^2/(-e^
2*x^2 + d^2)^(5/2) - 76/5*d^6*x/(-e^2*x^2 + d^2)^(5/2) + 248/5*d^7/((-e^2*x^2 +
d^2)^(5/2)*e) + 69/5*d^4*x/(-e^2*x^2 + d^2)^(3/2) - 39/10*d^2*x/sqrt(-e^2*x^2 +
d^2) - 63/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2)

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Fricas [A]  time = 0.246151, size = 724, normalized size = 4.18 \[ -\frac{5 \, e^{9} x^{9} + 90 \, d e^{8} x^{8} - 1012 \, d^{2} e^{7} x^{7} + 10 \, d^{3} e^{6} x^{6} + 8979 \, d^{4} e^{5} x^{5} - 10540 \, d^{5} e^{4} x^{4} - 4820 \, d^{6} e^{3} x^{3} + 12360 \, d^{7} e^{2} x^{2} - 5200 \, d^{8} e x - 630 \,{\left (d^{2} e^{7} x^{7} - 7 \, d^{3} e^{6} x^{6} + 3 \, d^{4} e^{5} x^{5} + 31 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} - 12 \, d^{7} e^{2} x^{2} + 40 \, d^{8} e x - 16 \, d^{9} +{\left (d^{2} e^{6} x^{6} + 2 \, d^{3} e^{5} x^{5} - 19 \, d^{4} e^{4} x^{4} + 20 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 40 \, d^{7} e x + 16 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (5 \, e^{8} x^{8} + 45 \, d e^{7} x^{7} - 625 \, d^{2} e^{6} x^{6} + 4619 \, d^{3} e^{5} x^{5} - 4360 \, d^{4} e^{4} x^{4} - 7420 \, d^{5} e^{3} x^{3} + 12360 \, d^{6} e^{2} x^{2} - 5200 \, d^{7} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (e^{8} x^{7} - 7 \, d e^{7} x^{6} + 3 \, d^{2} e^{6} x^{5} + 31 \, d^{3} e^{5} x^{4} - 40 \, d^{4} e^{4} x^{3} - 12 \, d^{5} e^{3} x^{2} + 40 \, d^{6} e^{2} x - 16 \, d^{7} e +{\left (e^{7} x^{6} + 2 \, d e^{6} x^{5} - 19 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 20 \, d^{4} e^{3} x^{2} - 40 \, d^{5} e^{2} x + 16 \, d^{6} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/10*(5*e^9*x^9 + 90*d*e^8*x^8 - 1012*d^2*e^7*x^7 + 10*d^3*e^6*x^6 + 8979*d^4*e
^5*x^5 - 10540*d^5*e^4*x^4 - 4820*d^6*e^3*x^3 + 12360*d^7*e^2*x^2 - 5200*d^8*e*x
- 630*(d^2*e^7*x^7 - 7*d^3*e^6*x^6 + 3*d^4*e^5*x^5 + 31*d^5*e^4*x^4 - 40*d^6*e^
3*x^3 - 12*d^7*e^2*x^2 + 40*d^8*e*x - 16*d^9 + (d^2*e^6*x^6 + 2*d^3*e^5*x^5 - 19
*d^4*e^4*x^4 + 20*d^5*e^3*x^3 + 20*d^6*e^2*x^2 - 40*d^7*e*x + 16*d^8)*sqrt(-e^2*
x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (5*e^8*x^8 + 45*d*e^7*x^
7 - 625*d^2*e^6*x^6 + 4619*d^3*e^5*x^5 - 4360*d^4*e^4*x^4 - 7420*d^5*e^3*x^3 + 1
2360*d^6*e^2*x^2 - 5200*d^7*e*x)*sqrt(-e^2*x^2 + d^2))/(e^8*x^7 - 7*d*e^7*x^6 +
3*d^2*e^6*x^5 + 31*d^3*e^5*x^4 - 40*d^4*e^4*x^3 - 12*d^5*e^3*x^2 + 40*d^6*e^2*x
- 16*d^7*e + (e^7*x^6 + 2*d*e^6*x^5 - 19*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 20*d^4*e
^3*x^2 - 40*d^5*e^2*x + 16*d^6*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{8}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**8/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**8/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.233936, size = 159, normalized size = 0.92 \[ -\frac{63}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{{\left (496 \, d^{7} e^{\left (-1\right )} +{\left (325 \, d^{6} -{\left (1200 \, d^{5} e +{\left (655 \, d^{4} e^{2} -{\left (1040 \, d^{3} e^{3} +{\left (591 \, d^{2} e^{4} - 5 \,{\left (x e^{6} + 16 \, d e^{5}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{10 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-63/2*d^2*arcsin(x*e/d)*e^(-1)*sign(d) - 1/10*(496*d^7*e^(-1) + (325*d^6 - (1200
*d^5*e + (655*d^4*e^2 - (1040*d^3*e^3 + (591*d^2*e^4 - 5*(x*e^6 + 16*d*e^5)*x)*x
)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3