3.1.2 \(\int x^4 (d+e x) (d^2-e^2 x^2)^{3/2} \, dx\) [2]

Optimal. Leaf size=201 \[ \frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]

[Out]

1/64*d^5*x*(-e^2*x^2+d^2)^(3/2)/e^4-4/63*d^2*x^2*(-e^2*x^2+d^2)^(5/2)/e^3-1/8*d*x^3*(-e^2*x^2+d^2)^(5/2)/e^2-1
/9*x^4*(-e^2*x^2+d^2)^(5/2)/e-1/5040*d^3*(315*e*x+128*d)*(-e^2*x^2+d^2)^(5/2)/e^5+3/128*d^9*arctan(e*x/(-e^2*x
^2+d^2)^(1/2))/e^5+3/128*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^4

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Rubi [A]
time = 0.10, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {847, 794, 201, 223, 209} \begin {gather*} \frac {3 d^9 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]

[Out]

(3*d^7*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4) - (4*d^2*x^2*(d^2 - e^2*x^2)^
(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8*e^2) - (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) - (d^3*(128*d + 31
5*e*x)*(d^2 - e^2*x^2)^(5/2))/(5040*e^5) + (3*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

Rule 201

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 209

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

Rule 223

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

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

Rubi steps

\begin {align*} \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x^3 \left (-4 d^2 e-9 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{9 e^2}\\ &=-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {\int x^2 \left (27 d^3 e^2+32 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{72 e^4}\\ &=-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x \left (-64 d^4 e^3-189 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{504 e^6}\\ &=-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {d^5 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^4}\\ &=\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 158, normalized size = 0.79 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-1024 d^8-945 d^7 e x-512 d^6 e^2 x^2-630 d^5 e^3 x^3-384 d^4 e^4 x^4+7560 d^3 e^5 x^5+6400 d^2 e^6 x^6-5040 d e^7 x^7-4480 e^8 x^8\right )}{40320 e^5}+\frac {3 d^9 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-1024*d^8 - 945*d^7*e*x - 512*d^6*e^2*x^2 - 630*d^5*e^3*x^3 - 384*d^4*e^4*x^4 + 7560*d^3
*e^5*x^5 + 6400*d^2*e^6*x^6 - 5040*d*e^7*x^7 - 4480*e^8*x^8))/(40320*e^5) + (3*d^9*Sqrt[-e^2]*Log[-(Sqrt[-e^2]
*x) + Sqrt[d^2 - e^2*x^2]])/(128*e^6)

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Maple [A]
time = 0.07, size = 215, normalized size = 1.07

method	result	size
risch	\(-\frac {\left (4480 e^{8} x^{8}+5040 d \,e^{7} x^{7}-6400 d^{2} e^{6} x^{6}-7560 d^{3} e^{5} x^{5}+384 d^{4} x^{4} e^{4}+630 d^{5} e^{3} x^{3}+512 d^{6} e^{2} x^{2}+945 d^{7} e x +1024 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{40320 e^{5}}+\frac {3 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\)	\(141\)
default	\(e \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{9 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )}{9 e^{2}}\right )+d \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )}{8 e^{2}}\right )\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

e*(-1/9*x^4*(-e^2*x^2+d^2)^(5/2)/e^2+4/9*d^2/e^2*(-1/7*x^2*(-e^2*x^2+d^2)^(5/2)/e^2-2/35*d^2/e^4*(-e^2*x^2+d^2
)^(5/2)))+d*(-1/8*x^3*(-e^2*x^2+d^2)^(5/2)/e^2+3/8*d^2/e^2*(-1/6*x*(-e^2*x^2+d^2)^(5/2)/e^2+1/6*d^2/e^2*(1/4*x
*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d^2/(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 = 0.51, size = 163, normalized size = 0.81 \begin {gather*} \frac {3}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} + \frac {3}{128} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7} x e^{\left (-4\right )} + \frac {1}{64} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x e^{\left (-4\right )} - \frac {1}{9} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{4} e^{\left (-1\right )} - \frac {1}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3} e^{\left (-2\right )} - \frac {4}{63} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{\left (-3\right )} - \frac {1}{16} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{\left (-4\right )} - \frac {8}{315} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")

[Out]

3/128*d^9*arcsin(x*e/d)*e^(-5) + 3/128*sqrt(-x^2*e^2 + d^2)*d^7*x*e^(-4) + 1/64*(-x^2*e^2 + d^2)^(3/2)*d^5*x*e
^(-4) - 1/9*(-x^2*e^2 + d^2)^(5/2)*x^4*e^(-1) - 1/8*(-x^2*e^2 + d^2)^(5/2)*d*x^3*e^(-2) - 4/63*(-x^2*e^2 + d^2
)^(5/2)*d^2*x^2*e^(-3) - 1/16*(-x^2*e^2 + d^2)^(5/2)*d^3*x*e^(-4) - 8/315*(-x^2*e^2 + d^2)^(5/2)*d^4*e^(-5)

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Fricas [A]
time = 2.58, size = 128, normalized size = 0.64 \begin {gather*} -\frac {1}{40320} \, {\left (1890 \, d^{9} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (4480 \, x^{8} e^{8} + 5040 \, d x^{7} e^{7} - 6400 \, d^{2} x^{6} e^{6} - 7560 \, d^{3} x^{5} e^{5} + 384 \, d^{4} x^{4} e^{4} + 630 \, d^{5} x^{3} e^{3} + 512 \, d^{6} x^{2} e^{2} + 945 \, d^{7} x e + 1024 \, d^{8}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")

[Out]

-1/40320*(1890*d^9*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (4480*x^8*e^8 + 5040*d*x^7*e^7 - 6400*d^2*x^
6*e^6 - 7560*d^3*x^5*e^5 + 384*d^4*x^4*e^4 + 630*d^5*x^3*e^3 + 512*d^6*x^2*e^2 + 945*d^7*x*e + 1024*d^8)*sqrt(
-x^2*e^2 + d^2))*e^(-5)

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Sympy [C] Result contains complex when optimal does not.
time = 25.94, size = 830, normalized size = 4.13 \begin {gather*} d^{3} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} - \frac {16 d^{8} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac {8 d^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac {2 d^{4} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac {x^{8} \sqrt {d^{2} - e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)*(-e**2*x**2+d**2)**(3/2),x)

[Out]

d**3*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/(4
8*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/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*sq
rt(1 - e**2*x**2/d**2)), True)) + d**2*e*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sq
rt(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)) - d*e**2*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*s
qrt(-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/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)) - e**3*Piecewise((-16*d**8*sq
rt(d**2 - e**2*x**2)/(315*e**8) - 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d**4*x**4*sqrt(d**2 - e**2
*x**2)/(105*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2 - e**2*x**2)/9, Ne(e, 0)), (x*
*8*sqrt(d**2)/8, True))

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Giac [A]
time = 2.41, size = 117, normalized size = 0.58 \begin {gather*} \frac {3}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{40320} \, {\left (1024 \, d^{8} e^{\left (-5\right )} + {\left (945 \, d^{7} e^{\left (-4\right )} + 2 \, {\left (256 \, d^{6} e^{\left (-3\right )} + {\left (315 \, d^{5} e^{\left (-2\right )} + 4 \, {\left (48 \, d^{4} e^{\left (-1\right )} - 5 \, {\left (189 \, d^{3} + 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, x e^{3} + 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")

[Out]

3/128*d^9*arcsin(x*e/d)*e^(-5)*sgn(d) - 1/40320*(1024*d^8*e^(-5) + (945*d^7*e^(-4) + 2*(256*d^6*e^(-3) + (315*
d^5*e^(-2) + 4*(48*d^4*e^(-1) - 5*(189*d^3 + 2*(80*d^2*e - 7*(8*x*e^3 + 9*d*e^2)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^
2*e^2 + d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d^2 - e^2*x^2)^(3/2)*(d + e*x),x)

[Out]

int(x^4*(d^2 - e^2*x^2)^(3/2)*(d + e*x), x)

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