3.1.76 \(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^6} \, dx\) [76]

Optimal. Leaf size=216 \[ \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

1/24*d*e^3*(-52*e*x+25*d)*(-e^2*x^2+d^2)^(3/2)/x^2-1/60*e^2*(25*e*x+52*d)*(-e^2*x^2+d^2)^(5/2)/x^3-1/5*d*(-e^2
*x^2+d^2)^(7/2)/x^5-3/4*e*(-e^2*x^2+d^2)^(7/2)/x^4+13/2*d^3*e^5*arctan(e*x/(-e^2*x^2+d^2)^(1/2))-25/8*d^3*e^5*
arctanh((-e^2*x^2+d^2)^(1/2)/d)+1/8*d^2*e^4*(25*e*x+52*d)*(-e^2*x^2+d^2)^(1/2)/x

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Rubi [A]
time = 0.19, antiderivative size = 216, 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 = {1821, 827, 858, 223, 209, 272, 65, 214} \begin {gather*} \frac {13}{2} d^3 e^5 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}+\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {25}{8} d^3 e^5 \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^6,x]

[Out]

(d^2*e^4*(52*d + 25*e*x)*Sqrt[d^2 - e^2*x^2])/(8*x) + (d*e^3*(25*d - 52*e*x)*(d^2 - e^2*x^2)^(3/2))/(24*x^2) -
(e^2*(52*d + 25*e*x)*(d^2 - e^2*x^2)^(5/2))/(60*x^3) - (d*(d^2 - e^2*x^2)^(7/2))/(5*x^5) - (3*e*(d^2 - e^2*x^
2)^(7/2))/(4*x^4) + (13*d^3*e^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - (25*d^3*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]
/d])/8

Rule 65

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 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 214

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

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 272

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 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 858

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 1821

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^6} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-15 d^4 e-13 d^3 e^2 x-5 d^2 e^3 x^2\right )}{x^5} \, dx}{5 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (52 d^5 e^2-25 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx}{20 d^4}\\ &=-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (150 d^6 e^3+312 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx}{72 d^4}\\ &=\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (-1248 d^7 e^4+600 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{192 d^4}\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {-1200 d^8 e^5-2496 d^7 e^6 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{384 d^4}\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{8} \left (25 d^4 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{2} \left (13 d^3 e^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{16} \left (25 d^4 e^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{2} \left (13 d^3 e^6\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{8} \left (25 d^4 e^3\right ) \text {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 {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}

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Mathematica [A]
time = 0.69, size = 189, normalized size = 0.88 \begin {gather*} \frac {1}{120} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-24 d^7-90 d^6 e x-32 d^5 e^2 x^2+345 d^4 e^3 x^3+656 d^3 e^4 x^4+80 d^2 e^5 x^5+180 d e^6 x^6+40 e^7 x^7\right )}{x^5}+750 d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )+780 d^3 e^4 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^6,x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-24*d^7 - 90*d^6*e*x - 32*d^5*e^2*x^2 + 345*d^4*e^3*x^3 + 656*d^3*e^4*x^4 + 80*d^2*e^5*
x^5 + 180*d*e^6*x^6 + 40*e^7*x^7))/x^5 + 750*d^3*e^5*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d] + 780*d^3
*e^4*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/120

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(659\) vs. \(2(188)=376\).
time = 0.07, size = 660, normalized size = 3.06

method	result	size
risch	\(-\frac {d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-656 e^{4} x^{4}-345 d \,e^{3} x^{3}+32 d^{2} x^{2} e^{2}+90 d^{3} e x +24 d^{4}\right )}{120 x^{5}}+\frac {e^{7} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {2 e^{5} d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {3 e^{6} d x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {13 e^{6} d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {25 e^{5} d^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\)	\(210\)
default	\(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )+e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )}{d^{2}}\right )}{3 d^{2}}\right )\)	\(660\)

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^6,x,method=_RETURNVERBOSE)

[Out]

d^3*(-1/5/d^2/x^5*(-e^2*x^2+d^2)^(7/2)-2/5*e^2/d^2*(-1/3/d^2/x^3*(-e^2*x^2+d^2)^(7/2)-4/3*e^2/d^2*(-1/d^2/x*(-
e^2*x^2+d^2)^(7/2)-6*e^2/d^2*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^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))))))))+3*d^2*e*(-1/4/d^2/x^4*
(-e^2*x^2+d^2)^(7/2)-3/4*e^2/d^2*(-1/2/d^2/x^2*(-e^2*x^2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*
(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/
2))/x))))))+e^3*(-1/2/d^2/x^2*(-e^2*x^2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^
2)^(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)))))+3*d*e^
2*(-1/3/d^2/x^3*(-e^2*x^2+d^2)^(7/2)-4/3*e^2/d^2*(-1/d^2/x*(-e^2*x^2+d^2)^(7/2)-6*e^2/d^2*(1/6*x*(-e^2*x^2+d^2
)^(5/2)+5/6*d^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 = 0.48, size = 259, normalized size = 1.20 \begin {gather*} \frac {13}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{5} - \frac {25}{8} \, d^{3} e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {13}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} d x e^{6} + \frac {25}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e^{5} + \frac {13 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{6}}{3 \, d} + \frac {25}{24} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{8 \, d^{2}} + \frac {52 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{15 \, d x} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{2}} - \frac {13 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{15 \, d x^{3}} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{4 \, x^{4}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{5 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

13/2*d^3*arcsin(x*e/d)*e^5 - 25/8*d^3*e^5*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x)) + 13/2*sqrt(-x^2
*e^2 + d^2)*d*x*e^6 + 25/8*sqrt(-x^2*e^2 + d^2)*d^2*e^5 + 13/3*(-x^2*e^2 + d^2)^(3/2)*x*e^6/d + 25/24*(-x^2*e^
2 + d^2)^(3/2)*e^5 + 5/8*(-x^2*e^2 + d^2)^(5/2)*e^5/d^2 + 52/15*(-x^2*e^2 + d^2)^(5/2)*e^4/(d*x) + 5/8*(-x^2*e
^2 + d^2)^(7/2)*e^3/(d^2*x^2) - 13/15*(-x^2*e^2 + d^2)^(7/2)*e^2/(d*x^3) - 3/4*(-x^2*e^2 + d^2)^(7/2)*e/x^4 -
1/5*(-x^2*e^2 + d^2)^(7/2)*d/x^5

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Fricas [A]
time = 1.64, size = 168, normalized size = 0.78 \begin {gather*} -\frac {1560 \, d^{3} x^{5} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{5} - 375 \, d^{3} x^{5} e^{5} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - 80 \, d^{3} x^{5} e^{5} - {\left (40 \, x^{7} e^{7} + 180 \, d x^{6} e^{6} + 80 \, d^{2} x^{5} e^{5} + 656 \, d^{3} x^{4} e^{4} + 345 \, d^{4} x^{3} e^{3} - 32 \, d^{5} x^{2} e^{2} - 90 \, d^{6} x e - 24 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{120 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/120*(1560*d^3*x^5*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x)*e^5 - 375*d^3*x^5*e^5*log(-(d - sqrt(-x^2*e^
2 + d^2))/x) - 80*d^3*x^5*e^5 - (40*x^7*e^7 + 180*d*x^6*e^6 + 80*d^2*x^5*e^5 + 656*d^3*x^4*e^4 + 345*d^4*x^3*e
^3 - 32*d^5*x^2*e^2 - 90*d^6*x*e - 24*d^7)*sqrt(-x^2*e^2 + d^2))/x^5

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Sympy [C] Result contains complex when optimal does not.
time = 8.13, size = 1178, normalized size = 5.45 \begin {gather*} d^{7} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) + 3 d^{6} e \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + d^{5} e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) - 5 d^{4} e^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) - 5 d^{3} e^{4} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + d^{2} e^{5} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + 3 d e^{6} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) + e^{7} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\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**6,x)

[Out]

d**7*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)) + 3*d**6*e*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**5*e**2*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)) - 5*d**4*e**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(2*x) + e**2*acos
h(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e**2*x**2) + 1)) - I*e/(2*x*sqrt(-
d**2/(e**2*x**2) + 1)) - I*e**2*asin(d/(e*x))/(2*d), True)) - 5*d**3*e**4*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)) + d**2*e**5*Piecewise((d**2/(
e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1
), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True))
+ 3*d*e**6*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/d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, Tr
ue)) + e**7*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (181) = 362\).
time = 1.63, size = 425, normalized size = 1.97 \begin {gather*} \frac {13}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{5} \mathrm {sgn}\left (d\right ) - \frac {25}{8} \, d^{3} e^{5} \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 ) + \frac {43 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{3}}{16 \, x} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e}{8 \, x^{2}} - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-1\right )}}{96 \, x^{3}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-3\right )}}{64 \, x^{4}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{3} e^{\left (-5\right )}}{160 \, x^{5}} + \frac {{\left (6 \, d^{3} e^{5} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{3}}{x} + \frac {50 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-1\right )}}{x^{3}} - \frac {2580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-3\right )}}{x^{4}}\right )} x^{5} e^{10}}{960 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5}} + \frac {1}{6} \, {\left (4 \, d^{2} e^{5} + {\left (2 \, x e^{7} + 9 \, d e^{6}\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \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^6,x, algorithm="giac")

[Out]

13/2*d^3*arcsin(x*e/d)*e^5*sgn(d) - 25/8*d^3*e^5*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))
+ 43/16*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*e^3/x + 5/8*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^3*e/x^2 - 5/96*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)^3*d^3*e^(-1)/x^3 - 3/64*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^3*e^(-3)/x^4 - 1/160*(d*
e + sqrt(-x^2*e^2 + d^2)*e)^5*d^3*e^(-5)/x^5 + 1/960*(6*d^3*e^5 + 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*e^3/x
+ 50*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^3*e/x^2 - 600*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^3*e^(-1)/x^3 - 2580*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^3*e^(-3)/x^4)*x^5*e^10/(d*e + sqrt(-x^2*e^2 + d^2)*e)^5 + 1/6*(4*d^2*e^5 + (
2*x*e^7 + 9*d*e^6)*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 \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^6} \,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^6,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^6, x)

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