3.1.79 \(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=204 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}+e^8 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4} \]

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Rubi [A]  time = 0.30, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1807, 811, 844, 217, 203, 266, 63, 208} \begin {gather*} -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+e^8 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

-(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (e^4*(125*d + 64*e*x)*(d^2 - e^2*x^2)^(3/2))/(192*x^4
) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e
^2*x^2)^(7/2))/(7*x^7) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (125*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/128

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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])

Rule 208

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

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 844

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

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps

\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-24 d^4 e-25 d^3 e^2 x-8 d^2 e^3 x^2\right )}{x^8} \, dx}{8 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac {\int \frac {\left (175 d^5 e^2+56 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {\left (1750 d^7 e^4+672 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{672 d^6}\\ &=\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac {\int \frac {\left (10500 d^9 e^6+5376 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x^3} \, dx}{5376 d^8}\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {21000 d^{11} e^8+21504 d^{10} e^9 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{21504 d^{10}}\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {1}{128} \left (125 d e^8\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e^9 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {1}{256} \left (125 d e^8\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^9 \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{128} \left (125 d e^6\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}

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Mathematica [C]  time = 0.15, size = 245, normalized size = 1.20 \begin {gather*} -\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e^8 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )}{7 d^7}-\frac {d^4 e^3 \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{5 x^5 \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {-8 d^7 e^2+34 d^5 e^4 x^2-59 d^3 e^6 x^4+15 d e^8 x^6 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )+33 d e^8 x^6}{16 x^6 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

(-3*e*(d^2 - e^2*x^2)^(7/2))/(7*x^7) + (-8*d^7*e^2 + 34*d^5*e^4*x^2 - 59*d^3*e^6*x^4 + 33*d*e^8*x^6 + 15*d*e^8
*x^6*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(16*x^6*Sqrt[d^2 - e^2*x^2]) - (d^4*e^3*Sqrt[d^
2 - e^2*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, (e^2*x^2)/d^2])/(5*x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (e^8*(d^2 -
e^2*x^2)^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^7)

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IntegrateAlgebraic [A]  time = 0.83, size = 186, normalized size = 0.91 \begin {gather*} -\frac {125}{64} e^8 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\sqrt {-e^2} e^7 \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )+\frac {\sqrt {d^2-e^2 x^2} \left (-1680 d^7-5760 d^6 e x-1960 d^5 e^2 x^2+14592 d^4 e^3 x^3+17710 d^3 e^4 x^4-7424 d^2 e^5 x^5-27195 d e^6 x^6-14848 e^7 x^7\right )}{13440 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-1680*d^7 - 5760*d^6*e*x - 1960*d^5*e^2*x^2 + 14592*d^4*e^3*x^3 + 17710*d^3*e^4*x^4 - 74
24*d^2*e^5*x^5 - 27195*d*e^6*x^6 - 14848*e^7*x^7))/(13440*x^8) - (125*e^8*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2
- e^2*x^2]/d])/64 - e^7*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]]

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fricas [A]  time = 0.48, size = 163, normalized size = 0.80 \begin {gather*} \frac {26880 \, e^{8} x^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13125 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (14848 \, e^{7} x^{7} + 27195 \, d e^{6} x^{6} + 7424 \, d^{2} e^{5} x^{5} - 17710 \, d^{3} e^{4} x^{4} - 14592 \, d^{4} e^{3} x^{3} + 1960 \, d^{5} e^{2} x^{2} + 5760 \, d^{6} e x + 1680 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{13440 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="fricas")

[Out]

1/13440*(26880*e^8*x^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 13125*e^8*x^8*log(-(d - sqrt(-e^2*x^2 + d^2
))/x) - (14848*e^7*x^7 + 27195*d*e^6*x^6 + 7424*d^2*e^5*x^5 - 17710*d^3*e^4*x^4 - 14592*d^4*e^3*x^3 + 1960*d^5
*e^2*x^2 + 5760*d^6*e*x + 1680*d^7)*sqrt(-e^2*x^2 + d^2))/x^8

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giac [B]  time = 0.31, size = 538, normalized size = 2.64 \begin {gather*} -\arcsin \left (\frac {x e}{d}\right ) e^{8} \mathrm {sgn}\relax (d) + \frac {x^{8} {\left (\frac {720 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{16}}{x} + \frac {1120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{14}}{x^{2}} - \frac {3696 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{12}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{10}}{x^{4}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{8}}{x^{5}} + \frac {77280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{6}}{x^{6}} + \frac {122640 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{4}}{x^{7}} + 105 \, e^{18}\right )} e^{6}}{215040 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8}} - \frac {1}{215040} \, {\left (\frac {122640 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{86}}{x} + \frac {77280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{84}}{x^{2}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{82}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{80}}{x^{4}} - \frac {3696 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{78}}{x^{5}} + \frac {1120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{76}}{x^{6}} + \frac {720 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{74}}{x^{7}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{72}}{x^{8}}\right )} e^{\left (-80\right )} + \frac {125}{128} \, e^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="giac")

[Out]

-arcsin(x*e/d)*e^8*sgn(d) + 1/215040*x^8*(720*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^16/x + 1120*(d*e + sqrt(-x^2*e^
2 + d^2)*e)^2*e^14/x^2 - 3696*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^12/x^3 - 14280*(d*e + sqrt(-x^2*e^2 + d^2)*e)
^4*e^10/x^4 - 560*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^8/x^5 + 77280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^6/x^6 +
122640*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e^4/x^7 + 105*e^18)*e^6/(d*e + sqrt(-x^2*e^2 + d^2)*e)^8 - 1/215040*(1
22640*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^86/x + 77280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^84/x^2 - 560*(d*e + sqr
t(-x^2*e^2 + d^2)*e)^3*e^82/x^3 - 14280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^80/x^4 - 3696*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^5*e^78/x^5 + 1120*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^76/x^6 + 720*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e
^74/x^7 + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^72/x^8)*e^(-80) + 125/128*e^8*log(1/2*abs(-2*d*e - 2*sqrt(-x^
2*e^2 + d^2)*e)*e^(-2)/abs(x))

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maple [B]  time = 0.06, size = 402, normalized size = 1.97 \begin {gather*} \frac {125 d \,e^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}-\frac {e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{9} x}{d^{2}}-\frac {125 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8}}{128 d}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{9} x}{3 d^{4}}-\frac {125 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8}}{384 d^{3}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{9} x}{15 d^{6}}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8}}{128 d^{5}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{7}}{15 d^{6} x}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{128 d^{5} x^{2}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{15 d^{4} x^{3}}+\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{192 d^{3} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{5 d^{2} x^{5}}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{48 d \,x^{6}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{7 x^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x)

[Out]

-1/8*d*(-e^2*x^2+d^2)^(7/2)/x^8-25/48/d*e^2/x^6*(-e^2*x^2+d^2)^(7/2)+25/192/d^3*e^4/x^4*(-e^2*x^2+d^2)^(7/2)-2
5/128/d^5*e^6/x^2*(-e^2*x^2+d^2)^(7/2)-25/128/d^5*e^8*(-e^2*x^2+d^2)^(5/2)-125/384/d^3*e^8*(-e^2*x^2+d^2)^(3/2
)-125/128/d*e^8*(-e^2*x^2+d^2)^(1/2)+125/128*d*e^8/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x
)-3/7*e*(-e^2*x^2+d^2)^(7/2)/x^7-1/5*e^3/d^2/x^5*(-e^2*x^2+d^2)^(7/2)+2/15*e^5/d^4/x^3*(-e^2*x^2+d^2)^(7/2)-8/
15*e^7/d^6/x*(-e^2*x^2+d^2)^(7/2)-8/15*e^9/d^6*x*(-e^2*x^2+d^2)^(5/2)-2/3*e^9/d^4*x*(-e^2*x^2+d^2)^(3/2)-e^9/d
^2*x*(-e^2*x^2+d^2)^(1/2)-e^9/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)

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maxima [A]  time = 1.01, size = 352, normalized size = 1.73 \begin {gather*} -e^{8} \arcsin \left (\frac {e x}{d}\right ) + \frac {125}{128} \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{9} x}{d^{2}} - \frac {125 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9} x}{3 \, d^{4}} - \frac {125 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{384 \, d^{3}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}}{128 \, d^{5}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{15 \, d^{4} x} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{128 \, d^{5} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{15 \, d^{4} x^{3}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{192 \, d^{3} x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{5 \, d^{2} x^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{48 \, d x^{6}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{7 \, x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="maxima")

[Out]

-e^8*arcsin(e*x/d) + 125/128*e^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - sqrt(-e^2*x^2 + d^2)*e^
9*x/d^2 - 125/128*sqrt(-e^2*x^2 + d^2)*e^8/d - 2/3*(-e^2*x^2 + d^2)^(3/2)*e^9*x/d^4 - 125/384*(-e^2*x^2 + d^2)
^(3/2)*e^8/d^3 - 25/128*(-e^2*x^2 + d^2)^(5/2)*e^8/d^5 - 8/15*(-e^2*x^2 + d^2)^(5/2)*e^7/(d^4*x) - 25/128*(-e^
2*x^2 + d^2)^(7/2)*e^6/(d^5*x^2) + 2/15*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^3) + 25/192*(-e^2*x^2 + d^2)^(7/2)*e
^4/(d^3*x^4) - 1/5*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^5) - 25/48*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^6) - 3/7*(-e^2
*x^2 + d^2)^(7/2)*e/x^7 - 1/8*(-e^2*x^2 + d^2)^(7/2)*d/x^8

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^9,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^9, x)

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sympy [C]  time = 31.40, size = 1719, normalized size = 8.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**9,x)

[Out]

d**7*Piecewise((-d**2/(8*e*x**9*sqrt(d**2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(192*d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(384*d**4*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 5*e**7/(128*d
**6*x*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(8*e
*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*x**5*sqrt(
-d**2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d**4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*x*sqrt(-d*
*2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) + 3*d**6*e*Piecewise((-e*sqrt(d**2/(e**2*x**2
) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4
*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2
) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(10
5*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) + d**5*e**2*Piecewise((-d**2/(6*e*x**7*
sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d**2*x**3*sqrt(d**2/(e**2*x*
*2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/(16*d**5), Abs(d**2/(e**2*x**2))
> 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(
48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**6*asin(d/(e*
x))/(16*d**5), True)) - 5*d**4*e**3*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**
7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6*x**6*sqrt(-1 + e**2*x
**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e*
*2*x**7), Abs(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*d*e**2
*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5
*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), True)) - 5*d
**3*e**4*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e*
*3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(
4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sqrt(-d*
*2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + d**2*e**5*Piecewise((-e*sqrt(d**2/(e**2*x**2) -
1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x*
*2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True)) + 3*d*e**6*Piecewise((-d**2/(2*e*x**3*
sqrt(d**2/(e**2*x**2) - 1)) + e/(2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x
**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(2*x) - I*e**2*asin(d/(e*x))/(2*d), True)) + e**7*Piecewise((I*d
/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2
) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True))

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