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

Optimal. Leaf size=138 \[ \frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

(2*(d + e*x)^6)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (14*(d + e*x)^4)/(15*e*(d^2 - e^2*
x^2)^(3/2)) + (14*(d + e*x)^2)/(3*e*Sqrt[d^2 - e^2*x^2]) + (7*Sqrt[d^2 - e^2*x^2
])/e - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Rubi [A]  time = 0.17889, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^6)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (14*(d + e*x)^4)/(15*e*(d^2 - e^2*
x^2)^(3/2)) + (14*(d + e*x)^2)/(3*e*Sqrt[d^2 - e^2*x^2]) + (7*Sqrt[d^2 - e^2*x^2
])/e - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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Rubi in Sympy [A]  time = 27.7638, size = 116, normalized size = 0.84 \[ - \frac{7 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} + \frac{2 \left (d + e x\right )^{6}}{5 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{14 \left (d + e x\right )^{4}}{15 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{14 \left (d + e x\right )^{2}}{3 e \sqrt{d^{2} - e^{2} x^{2}}} + \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*d*atan(e*x/sqrt(d**2 - e**2*x**2))/e + 2*(d + e*x)**6/(5*e*(d**2 - e**2*x**2)
**(5/2)) - 14*(d + e*x)**4/(15*e*(d**2 - e**2*x**2)**(3/2)) + 14*(d + e*x)**2/(3
*e*sqrt(d**2 - e**2*x**2)) + 7*sqrt(d**2 - e**2*x**2)/e

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Mathematica [A]  time = 0.143767, size = 86, normalized size = 0.62 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (167 d^3-381 d^2 e x+277 d e^2 x^2-15 e^3 x^3\right )}{(d-e x)^3}-105 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(167*d^3 - 381*d^2*e*x + 277*d*e^2*x^2 - 15*e^3*x^3))/(d -
e*x)^3 - 105*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e)

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Maple [B]  time = 0.01, size = 253, normalized size = 1.8 \[ -{\frac{61\,{d}^{5}x}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{71\,{d}^{3}x}{30} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{176\,dx}{15}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{{e}^{5}{x}^{6} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+27\,{\frac{{e}^{3}{d}^{2}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{73\,e{d}^{4}{x}^{2}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{167\,{d}^{6}}{15\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,d{e}^{4}{x}^{5}}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{7\,d{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-7\,{\frac{d}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }+{\frac{35\,{d}^{3}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-61/10*d^5*x/(-e^2*x^2+d^2)^(5/2)+71/30*d^3*x/(-e^2*x^2+d^2)^(3/2)+176/15*d*x/(-
e^2*x^2+d^2)^(1/2)-e^5*x^6/(-e^2*x^2+d^2)^(5/2)+27*e^3*d^2*x^4/(-e^2*x^2+d^2)^(5
/2)-73/3*e*d^4*x^2/(-e^2*x^2+d^2)^(5/2)+167/15*d^6/e/(-e^2*x^2+d^2)^(5/2)+7/5*d*
e^4*x^5/(-e^2*x^2+d^2)^(5/2)-7/3*d*e^2*x^3/(-e^2*x^2+d^2)^(3/2)-7*d/(e^2)^(1/2)*
arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+35/2*d^3*e^2*x^3/(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.813671, size = 443, normalized size = 3.21 \[ \frac{7}{15} \, d e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{7}{3} \, d e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{27 \, d^{2} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{35 \, d^{3} e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{73 \, d^{4} e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{61 \, d^{5} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{167 \, d^{6}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{127 \, d^{3} x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{22 \, d x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{7 \, d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/15*d*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)) - e^5*x^6/(-e^2*x^2 + d^2)^(5/
2) - 7/3*d*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)) + 27*d^2*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 35/2*d^3*e^2*x^3/(-e^2*x^2
+ d^2)^(5/2) - 73/3*d^4*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 61/10*d^5*x/(-e^2*x^2 + d
^2)^(5/2) + 167/15*d^6/((-e^2*x^2 + d^2)^(5/2)*e) + 127/30*d^3*x/(-e^2*x^2 + d^2
)^(3/2) + 22/15*d*x/sqrt(-e^2*x^2 + d^2) - 7*d*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(
e^2)

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Fricas [A]  time = 0.248594, size = 598, normalized size = 4.33 \[ \frac{15 \, e^{7} x^{7} - 155 \, d e^{6} x^{6} + 1259 \, d^{2} e^{5} x^{5} - 1205 \, d^{3} e^{4} x^{4} - 1030 \, d^{4} e^{3} x^{3} + 1980 \, d^{5} e^{2} x^{2} - 960 \, d^{6} e x + 210 \,{\left (d e^{6} x^{6} + d^{2} e^{5} x^{5} - 13 \, d^{3} e^{4} x^{4} + 15 \, d^{4} e^{3} x^{3} + 8 \, d^{5} e^{2} x^{2} - 20 \, d^{6} e x + 8 \, d^{7} -{\left (d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} + 12 \, d^{4} e^{2} x^{2} - 20 \, d^{5} e x + 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{6} x^{6} - 384 \, d e^{5} x^{5} + 215 \, d^{2} e^{4} x^{4} + 1510 \, d^{3} e^{3} x^{3} - 1980 \, d^{4} e^{2} x^{2} + 960 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{7} x^{6} + d e^{6} x^{5} - 13 \, d^{2} e^{5} x^{4} + 15 \, d^{3} e^{4} x^{3} + 8 \, d^{4} e^{3} x^{2} - 20 \, d^{5} e^{2} x + 8 \, d^{6} e -{\left (e^{6} x^{5} - 6 \, d e^{5} x^{4} + 5 \, d^{2} e^{4} x^{3} + 12 \, d^{3} e^{3} x^{2} - 20 \, d^{4} e^{2} x + 8 \, d^{5} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(15*e^7*x^7 - 155*d*e^6*x^6 + 1259*d^2*e^5*x^5 - 1205*d^3*e^4*x^4 - 1030*d^
4*e^3*x^3 + 1980*d^5*e^2*x^2 - 960*d^6*e*x + 210*(d*e^6*x^6 + d^2*e^5*x^5 - 13*d
^3*e^4*x^4 + 15*d^4*e^3*x^3 + 8*d^5*e^2*x^2 - 20*d^6*e*x + 8*d^7 - (d*e^5*x^5 -
6*d^2*e^4*x^4 + 5*d^3*e^3*x^3 + 12*d^4*e^2*x^2 - 20*d^5*e*x + 8*d^6)*sqrt(-e^2*x
^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^6*x^6 - 384*d*e^5*x
^5 + 215*d^2*e^4*x^4 + 1510*d^3*e^3*x^3 - 1980*d^4*e^2*x^2 + 960*d^5*e*x)*sqrt(-
e^2*x^2 + d^2))/(e^7*x^6 + d*e^6*x^5 - 13*d^2*e^5*x^4 + 15*d^3*e^4*x^3 + 8*d^4*e
^3*x^2 - 20*d^5*e^2*x + 8*d^6*e - (e^6*x^5 - 6*d*e^5*x^4 + 5*d^2*e^4*x^3 + 12*d^
3*e^3*x^2 - 20*d^4*e^2*x + 8*d^5*e)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232678, size = 144, normalized size = 1.04 \[ -7 \, d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{{\left (167 \, d^{6} e^{\left (-1\right )} +{\left (120 \, d^{5} -{\left (365 \, d^{4} e +{\left (160 \, d^{3} e^{2} -{\left (405 \, d^{2} e^{3} -{\left (15 \, x e^{5} - 232 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*d*arcsin(x*e/d)*e^(-1)*sign(d) - 1/15*(167*d^6*e^(-1) + (120*d^5 - (365*d^4*e
+ (160*d^3*e^2 - (405*d^2*e^3 - (15*x*e^5 - 232*d*e^4)*x)*x)*x)*x)*x)*sqrt(-x^2
*e^2 + d^2)/(x^2*e^2 - d^2)^3