3.825 \(\int \frac{(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=143 \[ \frac{2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{14 (d+e x)^3}{3 e \sqrt{d^2-e^2 x^2}}-\frac{35 \sqrt{d^2-e^2 x^2} (d+e x)}{6 e}-\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

(2*(d + e*x)^5)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (14*(d + e*x)^3)/(3*e*Sqrt[d^2 - e
^2*x^2]) - (35*d*Sqrt[d^2 - e^2*x^2])/(2*e) - (35*(d + e*x)*Sqrt[d^2 - e^2*x^2])
/(6*e) + (35*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi [A]  time = 0.187557, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{14 (d+e x)^3}{3 e \sqrt{d^2-e^2 x^2}}-\frac{35 \sqrt{d^2-e^2 x^2} (d+e x)}{6 e}-\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(d^2 - e^2*x^2)^(5/2),x]

[Out]

(2*(d + e*x)^5)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (14*(d + e*x)^3)/(3*e*Sqrt[d^2 - e
^2*x^2]) - (35*d*Sqrt[d^2 - e^2*x^2])/(2*e) - (35*(d + e*x)*Sqrt[d^2 - e^2*x^2])
/(6*e) + (35*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi in Sympy [A]  time = 28.0494, size = 121, normalized size = 0.85 \[ \frac{35 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{35 d \sqrt{d^{2} - e^{2} x^{2}}}{2 e} + \frac{2 \left (d + e x\right )^{5}}{3 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{14 \left (d + e x\right )^{3}}{3 e \sqrt{d^{2} - e^{2} x^{2}}} - \frac{35 \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{6 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(-e**2*x**2+d**2)**(5/2),x)

[Out]

35*d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e) - 35*d*sqrt(d**2 - e**2*x**2)/(2*
e) + 2*(d + e*x)**5/(3*e*(d**2 - e**2*x**2)**(3/2)) - 14*(d + e*x)**3/(3*e*sqrt(
d**2 - e**2*x**2)) - 35*(d + e*x)*sqrt(d**2 - e**2*x**2)/(6*e)

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Mathematica [A]  time = 0.147387, size = 89, normalized size = 0.62 \[ \frac{105 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{\sqrt{d^2-e^2 x^2} \left (164 d^3-229 d^2 e x+30 d e^2 x^2+3 e^3 x^3\right )}{(d-e x)^2}}{6 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(d^2 - e^2*x^2)^(5/2),x]

[Out]

(-((Sqrt[d^2 - e^2*x^2]*(164*d^3 - 229*d^2*e*x + 30*d*e^2*x^2 + 3*e^3*x^3))/(d -
e*x)^2) + 105*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e)

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Maple [A]  time = 0.06, size = 189, normalized size = 1.3 \[{\frac{16\,{d}^{4}x}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{131\,{d}^{2}x}{6}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{{e}^{4}{x}^{5}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{d}^{2}{e}^{2}{x}^{3}}{6} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-6\,{\frac{{e}^{3}d{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}+44\,{\frac{{d}^{3}e{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{82\,{d}^{5}}{3\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(-e^2*x^2+d^2)^(5/2),x)

[Out]

16/3*d^4*x/(-e^2*x^2+d^2)^(3/2)-131/6*d^2*x/(-e^2*x^2+d^2)^(1/2)-1/2*e^4*x^5/(-e
^2*x^2+d^2)^(3/2)+35/6*e^2*d^2*x^3/(-e^2*x^2+d^2)^(3/2)+35/2*d^2/(e^2)^(1/2)*arc
tan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-6*d*e^3*x^4/(-e^2*x^2+d^2)^(3/2)+44*d^3*
e*x^2/(-e^2*x^2+d^2)^(3/2)-82/3*d^5/e/(-e^2*x^2+d^2)^(3/2)

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Maxima [A]  time = 0.782216, size = 284, normalized size = 1.99 \[ \frac{35}{6} \, d^{2} e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} - \frac{e^{4} x^{5}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{6 \, d e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{44 \, d^{3} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{16 \, d^{4} x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{82 \, d^{5}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{61 \, d^{2} x}{6 \, \sqrt{-e^{2} x^{2} + d^{2}}} + \frac{35 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(-e^2*x^2 + d^2)^(5/2),x, algorithm="maxima")

[Out]

35/6*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)) - 1/2*e^4*x^5/(-e^2*x^2 + d^2)^(3/2) - 6*d*e^3*x^4/(-e^2*x^2 + d^2)^(3/
2) + 44*d^3*e*x^2/(-e^2*x^2 + d^2)^(3/2) + 16/3*d^4*x/(-e^2*x^2 + d^2)^(3/2) - 8
2/3*d^5/((-e^2*x^2 + d^2)^(3/2)*e) - 61/6*d^2*x/sqrt(-e^2*x^2 + d^2) + 35/2*d^2*
arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2)

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Fricas [A]  time = 0.241349, size = 517, normalized size = 3.62 \[ -\frac{3 \, e^{7} x^{7} + 21 \, d e^{6} x^{6} - 179 \, d^{2} e^{5} x^{5} + 951 \, d^{3} e^{4} x^{4} - 320 \, d^{4} e^{3} x^{3} - 1332 \, d^{5} e^{2} x^{2} + 792 \, d^{6} e x + 210 \,{\left (d^{2} e^{5} x^{5} + 2 \, d^{3} e^{4} x^{4} - 11 \, d^{4} e^{3} x^{3} + 4 \, d^{5} e^{2} x^{2} + 12 \, d^{6} e x - 8 \, d^{7} -{\left (d^{2} e^{4} x^{4} - 5 \, d^{3} e^{3} x^{3} + 12 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (3 \, e^{6} x^{6} + 42 \, d e^{5} x^{5} - 285 \, d^{2} e^{4} x^{4} - 76 \, d^{3} e^{3} x^{3} + 1332 \, d^{4} e^{2} x^{2} - 792 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{6} x^{5} + 2 \, d e^{5} x^{4} - 11 \, d^{2} e^{4} x^{3} + 4 \, d^{3} e^{3} x^{2} + 12 \, d^{4} e^{2} x - 8 \, d^{5} e -{\left (e^{5} x^{4} - 5 \, d e^{4} x^{3} + 12 \, d^{3} e^{2} x - 8 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(-e^2*x^2 + d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(3*e^7*x^7 + 21*d*e^6*x^6 - 179*d^2*e^5*x^5 + 951*d^3*e^4*x^4 - 320*d^4*e^3
*x^3 - 1332*d^5*e^2*x^2 + 792*d^6*e*x + 210*(d^2*e^5*x^5 + 2*d^3*e^4*x^4 - 11*d^
4*e^3*x^3 + 4*d^5*e^2*x^2 + 12*d^6*e*x - 8*d^7 - (d^2*e^4*x^4 - 5*d^3*e^3*x^3 +
12*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)) + (3*e^6*x^6 + 42*d*e^5*x^5 - 285*d^2*e^4*x^4 - 76*d^3*e^3*x^3 + 1332*d^4*e^
2*x^2 - 792*d^5*e*x)*sqrt(-e^2*x^2 + d^2))/(e^6*x^5 + 2*d*e^5*x^4 - 11*d^2*e^4*x
^3 + 4*d^3*e^3*x^2 + 12*d^4*e^2*x - 8*d^5*e - (e^5*x^4 - 5*d*e^4*x^3 + 12*d^3*e^
2*x - 8*d^4*e)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral((d + e*x)**6/(-(-d + e*x)*(d + e*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.231348, size = 131, normalized size = 0.92 \[ \frac{35}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{{\left (164 \, d^{5} e^{\left (-1\right )} +{\left (99 \, d^{4} -{\left (264 \, d^{3} e +{\left (166 \, d^{2} e^{2} - 3 \,{\left (x e^{4} + 12 \, d e^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{6 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(-e^2*x^2 + d^2)^(5/2),x, algorithm="giac")

[Out]

35/2*d^2*arcsin(x*e/d)*e^(-1)*sign(d) - 1/6*(164*d^5*e^(-1) + (99*d^4 - (264*d^3
*e + (166*d^2*e^2 - 3*(x*e^4 + 12*d*e^3)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e
^2 - d^2)^2