3.800 \(\int (d+e x)^3 (d^2-e^2 x^2)^{7/2} \, dx\)

Optimal. Leaf size=212 \[ \frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{91 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e} \]

[Out]

(91*d^9*x*Sqrt[d^2 - e^2*x^2])/256 + (91*d^7*x*(d^2 - e^2*x^2)^(3/2))/384 + (91*d^5*x*(d^2 - e^2*x^2)^(5/2))/4
80 + (13*d^3*x*(d^2 - e^2*x^2)^(7/2))/80 - (13*d^2*(d^2 - e^2*x^2)^(9/2))/(90*e) - (13*d*(d + e*x)*(d^2 - e^2*
x^2)^(9/2))/(110*e) - ((d + e*x)^2*(d^2 - e^2*x^2)^(9/2))/(11*e) + (91*d^11*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])
/(256*e)

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Rubi [A]  time = 0.0900898, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \[ \frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{91 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]

[Out]

(91*d^9*x*Sqrt[d^2 - e^2*x^2])/256 + (91*d^7*x*(d^2 - e^2*x^2)^(3/2))/384 + (91*d^5*x*(d^2 - e^2*x^2)^(5/2))/4
80 + (13*d^3*x*(d^2 - e^2*x^2)^(7/2))/80 - (13*d^2*(d^2 - e^2*x^2)^(9/2))/(90*e) - (13*d*(d + e*x)*(d^2 - e^2*
x^2)^(9/2))/(110*e) - ((d + e*x)^2*(d^2 - e^2*x^2)^(9/2))/(11*e) + (91*d^11*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])
/(256*e)

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{11} (13 d) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{10} \left (13 d^2\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{10} \left (13 d^3\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{80} \left (91 d^5\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{96} \left (91 d^7\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{128} \left (91 d^9\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{256} \left (91 d^{11}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{1}{256} \left (91 d^{11}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{91}{256} d^9 x \sqrt{d^2-e^2 x^2}+\frac{91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac{(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac{91 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}\\ \end{align*}

Mathematica [A]  time = 0.31498, size = 178, normalized size = 0.84 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (167680 d^8 e^2 x^2+12210 d^7 e^3 x^3-222720 d^6 e^4 x^4-142296 d^5 e^5 x^5+110080 d^4 e^6 x^6+131472 d^3 e^7 x^7+1280 d^2 e^8 x^8+81675 d^9 e x-44800 d^{10}-38016 d e^9 x^9-11520 e^{10} x^{10}\right )+45045 d^{10} \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{126720 e \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-44800*d^10 + 81675*d^9*e*x + 167680*d^8*e^2*x^2 + 12210*d^7*e^
3*x^3 - 222720*d^6*e^4*x^4 - 142296*d^5*e^5*x^5 + 110080*d^4*e^6*x^6 + 131472*d^3*e^7*x^7 + 1280*d^2*e^8*x^8 -
38016*d*e^9*x^9 - 11520*e^10*x^10) + 45045*d^10*ArcSin[(e*x)/d]))/(126720*e*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.079, size = 174, normalized size = 0.8 \begin{align*} -{\frac{e{x}^{2}}{11} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}-{\frac{35\,{d}^{2}}{99\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}-{\frac{3\,dx}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}+{\frac{13\,{d}^{3}x}{80} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{91\,{d}^{5}x}{480} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{91\,{d}^{7}x}{384} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{91\,{d}^{9}x}{256}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{91\,{d}^{11}}{256}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/11*e*x^2*(-e^2*x^2+d^2)^(9/2)-35/99*d^2*(-e^2*x^2+d^2)^(9/2)/e-3/10*d*x*(-e^2*x^2+d^2)^(9/2)+13/80*d^3*x*(-
e^2*x^2+d^2)^(7/2)+91/480*d^5*x*(-e^2*x^2+d^2)^(5/2)+91/384*d^7*x*(-e^2*x^2+d^2)^(3/2)+91/256*d^9*x*(-e^2*x^2+
d^2)^(1/2)+91/256*d^11/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

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Maxima [A]  time = 1.5995, size = 224, normalized size = 1.06 \begin{align*} \frac{91 \, d^{11} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}}} + \frac{91}{256} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{9} x + \frac{91}{384} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{7} x + \frac{91}{480} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{5} x + \frac{13}{80} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x - \frac{1}{11} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} e x^{2} - \frac{3}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d x - \frac{35 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d^{2}}{99 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

91/256*d^11*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 91/256*sqrt(-e^2*x^2 + d^2)*d^9*x + 91/384*(-e^2*x^2 + d^2
)^(3/2)*d^7*x + 91/480*(-e^2*x^2 + d^2)^(5/2)*d^5*x + 13/80*(-e^2*x^2 + d^2)^(7/2)*d^3*x - 1/11*(-e^2*x^2 + d^
2)^(9/2)*e*x^2 - 3/10*(-e^2*x^2 + d^2)^(9/2)*d*x - 35/99*(-e^2*x^2 + d^2)^(9/2)*d^2/e

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Fricas [A]  time = 2.28852, size = 405, normalized size = 1.91 \begin{align*} -\frac{90090 \, d^{11} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (11520 \, e^{10} x^{10} + 38016 \, d e^{9} x^{9} - 1280 \, d^{2} e^{8} x^{8} - 131472 \, d^{3} e^{7} x^{7} - 110080 \, d^{4} e^{6} x^{6} + 142296 \, d^{5} e^{5} x^{5} + 222720 \, d^{6} e^{4} x^{4} - 12210 \, d^{7} e^{3} x^{3} - 167680 \, d^{8} e^{2} x^{2} - 81675 \, d^{9} e x + 44800 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{126720 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/126720*(90090*d^11*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (11520*e^10*x^10 + 38016*d*e^9*x^9 - 1280*d^
2*e^8*x^8 - 131472*d^3*e^7*x^7 - 110080*d^4*e^6*x^6 + 142296*d^5*e^5*x^5 + 222720*d^6*e^4*x^4 - 12210*d^7*e^3*
x^3 - 167680*d^8*e^2*x^2 - 81675*d^9*e*x + 44800*d^10)*sqrt(-e^2*x^2 + d^2))/e

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Sympy [C]  time = 34.6203, size = 1503, normalized size = 7.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(7/2),x)

[Out]

d**9*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(2*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, Tru
e)) + 3*d**8*e*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - 8*d*
*6*e**3*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**
4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) - 6*d**5*e**4*Piecewise((-I*d**6*acosh(e*x/d
)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) -
5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d
**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1
- e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2)), True)
) + 6*d**4*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqrt(d**2 - e**2*x**2)/(105
*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**
2)/6, True)) + 8*d**3*e**6*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(128*e**6*sqrt(-1 + e**2*
x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/(192*e**2*sqrt(-1 + e**2*x**2/d
**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2
)/Abs(d**2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2/d**2)) + 5*d**5*x**3/
(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2)) + 7*d*x**7/(48*sqrt(1 - e*
*2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True)) - 3*d*e**8*Piecewise((-7*I*d**10*acosh(e*x/d
)/(256*e**9) + 7*I*d**9*x/(256*e**8*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**7*x**3/(768*e**6*sqrt(-1 + e**2*x**2/d
**2)) - 7*I*d**5*x**5/(1920*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**7/(480*e**2*sqrt(-1 + e**2*x**2/d**2))
- 9*I*d*x**9/(80*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**11/(10*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/A
bs(d**2) > 1), (7*d**10*asin(e*x/d)/(256*e**9) - 7*d**9*x/(256*e**8*sqrt(1 - e**2*x**2/d**2)) + 7*d**7*x**3/(7
68*e**6*sqrt(1 - e**2*x**2/d**2)) + 7*d**5*x**5/(1920*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**7/(480*e**2*sqr
t(1 - e**2*x**2/d**2)) + 9*d*x**9/(80*sqrt(1 - e**2*x**2/d**2)) - e**2*x**11/(10*d*sqrt(1 - e**2*x**2/d**2)),
True)) - e**9*Piecewise((-128*d**10*sqrt(d**2 - e**2*x**2)/(3465*e**10) - 64*d**8*x**2*sqrt(d**2 - e**2*x**2)/
(3465*e**8) - 16*d**6*x**4*sqrt(d**2 - e**2*x**2)/(1155*e**6) - 8*d**4*x**6*sqrt(d**2 - e**2*x**2)/(693*e**4)
- d**2*x**8*sqrt(d**2 - e**2*x**2)/(99*e**2) + x**10*sqrt(d**2 - e**2*x**2)/11, Ne(e, 0)), (x**10*sqrt(d**2)/1
0, True))

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Giac [A]  time = 1.26169, size = 186, normalized size = 0.88 \begin{align*} \frac{91}{256} \, d^{11} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{126720} \,{\left (44800 \, d^{10} e^{\left (-1\right )} -{\left (81675 \, d^{9} + 2 \,{\left (83840 \, d^{8} e +{\left (6105 \, d^{7} e^{2} - 4 \,{\left (27840 \, d^{6} e^{3} +{\left (17787 \, d^{5} e^{4} - 2 \,{\left (6880 \, d^{4} e^{5} +{\left (8217 \, d^{3} e^{6} + 8 \,{\left (10 \, d^{2} e^{7} - 9 \,{\left (10 \, x e^{9} + 33 \, d e^{8}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

91/256*d^11*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/126720*(44800*d^10*e^(-1) - (81675*d^9 + 2*(83840*d^8*e + (6105*d^
7*e^2 - 4*(27840*d^6*e^3 + (17787*d^5*e^4 - 2*(6880*d^4*e^5 + (8217*d^3*e^6 + 8*(10*d^2*e^7 - 9*(10*x*e^9 + 33
*d*e^8)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)