3.1.75 \(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^5} \, dx\) [75]

Optimal. Leaf size=209 \[ -\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]

[Out]

15/8*d*e^3*(-e*x+2*d)*(-e^2*x^2+d^2)^(3/2)/x-3/8*e^2*(2*e*x+3*d)*(-e^2*x^2+d^2)^(5/2)/x^2-1/4*d*(-e^2*x^2+d^2)
^(7/2)/x^4-e*(-e^2*x^2+d^2)^(7/2)/x^3+45/8*d^4*e^4*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+45/8*d^4*e^4*arctanh((-e^2
*x^2+d^2)^(1/2)/d)-45/8*d^2*e^4*(-e*x+d)*(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1821, 827, 829, 858, 223, 209, 272, 65, 214} \begin {gather*} \frac {45}{8} d^4 e^4 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}+\frac {45}{8} d^4 e^4 \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^5,x]

[Out]

(-45*d^2*e^4*(d - e*x)*Sqrt[d^2 - e^2*x^2])/8 + (15*d*e^3*(2*d - e*x)*(d^2 - e^2*x^2)^(3/2))/(8*x) - (3*e^2*(3
*d + 2*e*x)*(d^2 - e^2*x^2)^(5/2))/(8*x^2) - (d*(d^2 - e^2*x^2)^(7/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(7/2))/x^3
+ (45*d^4*e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/8 + (45*d^4*e^4*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 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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^5} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-12 d^4 e-9 d^3 e^2 x-4 d^2 e^3 x^2\right )}{x^4} \, dx}{4 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {\int \frac {\left (27 d^5 e^2-36 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx}{12 d^4}\\ &=-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {\left (144 d^6 e^3+216 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx}{192 d^4}\\ &=\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {5 \int \frac {\left (-432 d^7 e^4+864 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{384 d^4}\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {864 d^9 e^6-864 d^8 e^7 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{768 d^4 e^2}\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{8} \left (45 d^5 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{8} \left (45 d^4 e^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{16} \left (45 d^5 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{8} \left (45 d^4 e^5\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{8} \left (45 d^5 e^2\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 187, normalized size = 0.89 \begin {gather*} \frac {1}{8} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^7-8 d^6 e x-3 d^5 e^2 x^2+48 d^4 e^3 x^3-48 d^3 e^4 x^4+3 d^2 e^5 x^5+8 d e^6 x^6+2 e^7 x^7\right )}{x^4}-90 d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )-45 d^4 e \left (-e^2\right )^{3/2} \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^5,x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-2*d^7 - 8*d^6*e*x - 3*d^5*e^2*x^2 + 48*d^4*e^3*x^3 - 48*d^3*e^4*x^4 + 3*d^2*e^5*x^5 +
8*d*e^6*x^6 + 2*e^7*x^7))/x^4 - 90*d^4*e^4*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d] - 45*d^4*e*(-e^2)^(
3/2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(597\) vs. \(2(183)=366\).
time = 0.07, size = 598, normalized size = 2.86

method	result	size
risch	\(-\frac {d^{4} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-48 e^{3} x^{3}+3 d \,e^{2} x^{2}+8 d^{2} e x +2 d^{3}\right )}{8 x^{4}}+\frac {e^{7} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4}+\frac {3 e^{5} d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{8}+\frac {45 e^{5} d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+e^{6} d \,x^{2} \sqrt {-e^{2} x^{2}+d^{2}}-6 e^{4} d^{3} \sqrt {-e^{2} x^{2}+d^{2}}+\frac {45 e^{4} d^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\)	\(223\)
default	\(d^{3} \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 )+3 d \,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 )+e^{3} \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} e \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 )\)	\(598\)

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^5,x,method=_RETURNVERBOSE)

[Out]

d^3*(-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))))))+3*d*e^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/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*(-1/3/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 = 0.56, size = 233, normalized size = 1.11 \begin {gather*} \frac {45}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{4} + \frac {45}{8} \, d^{4} e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {45}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} x e^{5} - \frac {45}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} e^{4} + \frac {15}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{5} - \frac {15}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} - \frac {9 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{8 \, d} + \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{x} - \frac {9 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{8 \, d x^{2}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{x^{3}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{4 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

45/8*d^4*arcsin(x*e/d)*e^4 + 45/8*d^4*e^4*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x)) + 45/8*sqrt(-x^2
*e^2 + d^2)*d^2*x*e^5 - 45/8*sqrt(-x^2*e^2 + d^2)*d^3*e^4 + 15/4*(-x^2*e^2 + d^2)^(3/2)*x*e^5 - 15/8*(-x^2*e^2
+ d^2)^(3/2)*d*e^4 - 9/8*(-x^2*e^2 + d^2)^(5/2)*e^4/d + 3*(-x^2*e^2 + d^2)^(5/2)*e^3/x - 9/8*(-x^2*e^2 + d^2)
^(7/2)*e^2/(d*x^2) - (-x^2*e^2 + d^2)^(7/2)*e/x^3 - 1/4*(-x^2*e^2 + d^2)^(7/2)*d/x^4

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Fricas [A]
time = 1.48, size = 168, normalized size = 0.80 \begin {gather*} -\frac {90 \, d^{4} x^{4} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{4} + 45 \, d^{4} x^{4} e^{4} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + 48 \, d^{4} x^{4} e^{4} - {\left (2 \, x^{7} e^{7} + 8 \, d x^{6} e^{6} + 3 \, d^{2} x^{5} e^{5} - 48 \, d^{3} x^{4} e^{4} + 48 \, d^{4} x^{3} e^{3} - 3 \, d^{5} x^{2} e^{2} - 8 \, d^{6} x e - 2 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{8 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

-1/8*(90*d^4*x^4*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x)*e^4 + 45*d^4*x^4*e^4*log(-(d - sqrt(-x^2*e^2 + d
^2))/x) + 48*d^4*x^4*e^4 - (2*x^7*e^7 + 8*d*x^6*e^6 + 3*d^2*x^5*e^5 - 48*d^3*x^4*e^4 + 48*d^4*x^3*e^3 - 3*d^5*
x^2*e^2 - 8*d^6*x*e - 2*d^7)*sqrt(-x^2*e^2 + d^2))/x^4

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Sympy [C] Result contains complex when optimal does not.
time = 9.47, size = 1028, normalized size = 4.92 \begin {gather*} d^{7} \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 ) + 3 d^{6} e \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 ) + d^{5} e^{2} \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^{4} e^{3} \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 ) - 5 d^{3} e^{4} \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 ) + d^{2} e^{5} \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 ) + 3 d e^{6} \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 ) + e^{7} \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{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**5,x)

[Out]

d**7*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)) + 3*d**6*e*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)) + d**5*e**2*Piecewise((-e*sqrt(d**2/(e**2*x
**2) - 1)/(2*x) + e**2*acosh(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**4*e**3*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)) -
5*d**3*e**4*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)) + d**2*e**5*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, True)) + 3*d*e**6*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)
**(3/2)/(3*e**2), True)) + e**7*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**
2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x*
*2/d**2) > 1), (d**4*asin(e*x/d)/(8*e**3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e*
*2*x**2/d**2)) - e**2*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (176) = 352\).
time = 1.38, size = 365, normalized size = 1.75 \begin {gather*} \frac {45}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{4} \mathrm {sgn}\left (d\right ) + \frac {45}{8} \, d^{4} e^{4} \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 {23 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{2}}{8 \, x} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-2\right )}}{8 \, x^{3}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{\left (-4\right )}}{64 \, x^{4}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4}}{8 \, x^{2}} + \frac {{\left (d^{4} e^{4} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{2}}{x} - \frac {184 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-2\right )}}{x^{3}} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4}}{x^{2}}\right )} x^{4} e^{8}}{64 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}} - \frac {1}{8} \, {\left (48 \, d^{3} e^{4} - {\left (3 \, d^{2} e^{5} + 2 \, {\left (x e^{7} + 4 \, d e^{6}\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((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^5,x, algorithm="giac")

[Out]

45/8*d^4*arcsin(x*e/d)*e^4*sgn(d) + 45/8*d^4*e^4*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))
+ 23/8*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^4*e^2/x - 1/8*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*e^(-2)/x^3 - 1/64*
(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^4*e^(-4)/x^4 - 1/8*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4/x^2 + 1/64*(d^4*e^4
+ 8*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^4*e^2/x - 184*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*e^(-2)/x^3 + 8*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^2*d^4/x^2)*x^4*e^8/(d*e + sqrt(-x^2*e^2 + d^2)*e)^4 - 1/8*(48*d^3*e^4 - (3*d^2*e^5 +
2*(x*e^7 + 4*d*e^6)*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 \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^5} \,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^5,x)

[Out]

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

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