3.1.53 \(\int \frac {(d+e x)^2}{x^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [53]

Optimal. Leaf size=209 \[ \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8} \]

[Out]

2/5*e^3*(e*x+d)/d^4/(-e^2*x^2+d^2)^(5/2)+1/15*e^3*(23*e*x+20*d)/d^6/(-e^2*x^2+d^2)^(3/2)-7*e^3*arctanh((-e^2*x
^2+d^2)^(1/2)/d)/d^8+2/15*e^3*(53*e*x+45*d)/d^8/(-e^2*x^2+d^2)^(1/2)-1/3*(-e^2*x^2+d^2)^(1/2)/d^6/x^3-e*(-e^2*
x^2+d^2)^(1/2)/d^7/x^2-14/3*e^2*(-e^2*x^2+d^2)^(1/2)/d^8/x

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Rubi [A]
time = 0.30, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1819, 1821, 821, 272, 65, 214} \begin {gather*} -\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(2*e^3*(d + e*x))/(5*d^4*(d^2 - e^2*x^2)^(5/2)) + (e^3*(20*d + 23*e*x))/(15*d^6*(d^2 - e^2*x^2)^(3/2)) + (2*e^
3*(45*d + 53*e*x))/(15*d^8*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(3*d^6*x^3) - (e*Sqrt[d^2 - e^2*x^2])/(d
^7*x^2) - (14*e^2*Sqrt[d^2 - e^2*x^2])/(3*d^8*x) - (7*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/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 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 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 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(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
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[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)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2-10 d e x-10 e^2 x^2-\frac {10 e^3 x^3}{d}-\frac {8 e^4 x^4}{d^2}}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2+30 d e x+45 e^2 x^2+\frac {60 e^3 x^3}{d}+\frac {46 e^4 x^4}{d^2}}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2-30 d e x-60 e^2 x^2-\frac {90 e^3 x^3}{d}}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}+\frac {\int \frac {90 d^3 e+210 d^2 e^2 x+270 d e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^8}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\int \frac {-420 d^4 e^2-630 d^3 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{10}}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {\left (7 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^7}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {\left (7 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^7}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {(7 e) \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 )}{d^7}\\ &=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 147, normalized size = 0.70 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (5 d^6+5 d^5 e x+40 d^4 e^2 x^2-246 d^3 e^3 x^3+122 d^2 e^4 x^4+247 d e^5 x^5-176 e^6 x^6\right )}{x^3 (-d+e x)^3 (d+e x)}+210 e^3 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{15 d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(5*d^6 + 5*d^5*e*x + 40*d^4*e^2*x^2 - 246*d^3*e^3*x^3 + 122*d^2*e^4*x^4 + 247*d*e^5*x^5
- 176*e^6*x^6))/(x^3*(-d + e*x)^3*(d + e*x)) + 210*e^3*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(15*d^
8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(380\) vs. \(2(185)=370\).
time = 0.09, size = 381, normalized size = 1.82

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (14 e^{2} x^{2}+3 d e x +d^{2}\right )}{3 d^{8} x^{3}}-\frac {e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{8 d^{8} \left (x +\frac {d}{e}\right )}-\frac {833 e^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{120 d^{8} \left (x -\frac {d}{e}\right )}-\frac {7 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{7} \sqrt {d^{2}}}+\frac {49 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{60 d^{7} \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{10 d^{6} \left (x -\frac {d}{e}\right )^{3}}\)	\(270\)
default	\(2 d e \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )}{2 d^{2}}\right )+e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )+d^{2} \left (-\frac {1}{3 d^{2} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {8 e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )}{3 d^{2}}\right )\)	\(381\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2*d*e*(-1/2/d^2/x^2/(-e^2*x^2+d^2)^(5/2)+7/2*e^2/d^2*(1/5/d^2/(-e^2*x^2+d^2)^(5/2)+1/d^2*(1/3/d^2/(-e^2*x^2+d^
2)^(3/2)+1/d^2*(1/d^2/(-e^2*x^2+d^2)^(1/2)-1/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^2*(-1/d^2/x/(-e^2*x^2+d^2)^(5/2)+6*e^2/d^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+
d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2))))+d^2*(-1/3/d^2/x^3/(-e^2*x^2+d^2)^(5/2)+8/3*e^2/d^2*(-1/d^2/x/(-e^
2*x^2+d^2)^(5/2)+6*e^2/d^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(
-e^2*x^2+d^2)^(1/2)))))

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Maxima [A]
time = 0.27, size = 226, normalized size = 1.08 \begin {gather*} \frac {22 \, x e^{4}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {7 \, e^{3}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} - \frac {11 \, e^{2}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x} - \frac {e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2}} - \frac {1}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{3}} + \frac {88 \, x e^{4}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {7 \, e^{3}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} - \frac {7 \, e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{8}} + \frac {176 \, x e^{4}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{8}} + \frac {7 \, e^{3}}{\sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

22/5*x*e^4/((-x^2*e^2 + d^2)^(5/2)*d^4) + 7/5*e^3/((-x^2*e^2 + d^2)^(5/2)*d^3) - 11/3*e^2/((-x^2*e^2 + d^2)^(5
/2)*d^2*x) - e/((-x^2*e^2 + d^2)^(5/2)*d*x^2) - 1/3/((-x^2*e^2 + d^2)^(5/2)*x^3) + 88/15*x*e^4/((-x^2*e^2 + d^
2)^(3/2)*d^6) + 7/3*e^3/((-x^2*e^2 + d^2)^(3/2)*d^5) - 7*e^3*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x
))/d^8 + 176/15*x*e^4/(sqrt(-x^2*e^2 + d^2)*d^8) + 7*e^3/(sqrt(-x^2*e^2 + d^2)*d^7)

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Fricas [A]
time = 1.86, size = 212, normalized size = 1.01 \begin {gather*} \frac {116 \, x^{7} e^{7} - 232 \, d x^{6} e^{6} + 232 \, d^{3} x^{4} e^{4} - 116 \, d^{4} x^{3} e^{3} + 105 \, {\left (x^{7} e^{7} - 2 \, d x^{6} e^{6} + 2 \, d^{3} x^{4} e^{4} - d^{4} x^{3} e^{3}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (176 \, x^{6} e^{6} - 247 \, d x^{5} e^{5} - 122 \, d^{2} x^{4} e^{4} + 246 \, d^{3} x^{3} e^{3} - 40 \, d^{4} x^{2} e^{2} - 5 \, d^{5} x e - 5 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{8} x^{7} e^{4} - 2 \, d^{9} x^{6} e^{3} + 2 \, d^{11} x^{4} e - d^{12} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(116*x^7*e^7 - 232*d*x^6*e^6 + 232*d^3*x^4*e^4 - 116*d^4*x^3*e^3 + 105*(x^7*e^7 - 2*d*x^6*e^6 + 2*d^3*x^4
*e^4 - d^4*x^3*e^3)*log(-(d - sqrt(-x^2*e^2 + d^2))/x) - (176*x^6*e^6 - 247*d*x^5*e^5 - 122*d^2*x^4*e^4 + 246*
d^3*x^3*e^3 - 40*d^4*x^2*e^2 - 5*d^5*x*e - 5*d^6)*sqrt(-x^2*e^2 + d^2))/(d^8*x^7*e^4 - 2*d^9*x^6*e^3 + 2*d^11*
x^4*e - d^12*x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**2/(x**4*(-(-d + e*x)*(d + e*x))**(7/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate((x*e + d)^2/((-x^2*e^2 + d^2)^(7/2)*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{x^4\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x)

[Out]

int((d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)), x)

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