3.2.37 \(\int \frac {x^6}{(d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\) [137]

Optimal. Leaf size=148 \[ \frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d-8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7} \]

[Out]

1/5*x^5*(-e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^3*(-6*e*x+5*d)/e^4/(-e^2*x^2+d^2)^(3/2)-d*arctan(e*x/(-e^2*x^
2+d^2)^(1/2))/e^7+1/5*x*(-8*e*x+5*d)/e^6/(-e^2*x^2+d^2)^(1/2)-16/5*(-e^2*x^2+d^2)^(1/2)/e^7

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Rubi [A]
time = 0.09, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {864, 833, 655, 223, 209} \begin {gather*} -\frac {d \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7}+\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}+\frac {x (5 d-8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^5*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^3*(5*d - 6*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x*(5*d -
8*e*x))/(5*e^6*Sqrt[d^2 - e^2*x^2]) - (16*Sqrt[d^2 - e^2*x^2])/(5*e^7) - (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]
)/e^7

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 655

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

Rule 833

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 + 1)*((a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 864

Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + c*(x/e))*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))

Rubi steps

\begin {align*} \int \frac {x^6}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^6 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^4 \left (5 d^3-6 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^2 \left (15 d^5-24 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d-8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^7-48 d^6 e x}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d-8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^6}\\ &=\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d-8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}\\ &=\frac {x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d-8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 136, normalized size = 0.92 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-48 d^5-33 d^4 e x+87 d^3 e^2 x^2+52 d^2 e^3 x^3-38 d e^4 x^4-15 e^5 x^5\right )}{15 e^7 (-d+e x)^2 (d+e x)^3}+\frac {d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^6 \sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-48*d^5 - 33*d^4*e*x + 87*d^3*e^2*x^2 + 52*d^2*e^3*x^3 - 38*d*e^4*x^4 - 15*e^5*x^5))/(15
*e^7*(-d + e*x)^2*(d + e*x)^3) + (d*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(e^6*Sqrt[-e^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(532\) vs. \(2(130)=260\).
time = 0.08, size = 533, normalized size = 3.60

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{7}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6} \sqrt {e^{2}}}-\frac {493 d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{240 e^{8} \left (x +\frac {d}{e}\right )}+\frac {25 d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{48 e^{8} \left (x -\frac {d}{e}\right )}+\frac {d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{24 e^{9} \left (x -\frac {d}{e}\right )^{2}}+\frac {23 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{60 e^{9} \left (x +\frac {d}{e}\right )^{2}}-\frac {d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 e^{10} \left (x +\frac {d}{e}\right )^{3}}\)	\(281\)
default	\(\frac {-\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 d^{2} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{2}}}{e}-\frac {d \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )}{e^{2}}+\frac {d^{2} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{3}}-\frac {d^{3} \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )}{e^{4}}+\frac {d^{4}}{3 e^{7} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{5} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6}}+\frac {d^{6} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{7}}\)	\(533\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 237, normalized size = 1.60 \begin {gather*} -\frac {d^{5}}{5 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{8} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{7}\right )}} - \frac {x^{4} e^{\left (-3\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {5 \, d x^{3} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {20 \, d^{2} x^{2} e^{\left (-5\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {64 \, d^{3} x e^{\left (-6\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {14 \, d^{4} e^{\left (-7\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-7\right )} + \frac {x^{2} e^{\left (-5\right )}}{3 \, \sqrt {-x^{2} e^{2} + d^{2}}} - \frac {52 \, d x e^{\left (-6\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}}} + \frac {4 \, d^{2} e^{\left (-7\right )}}{3 \, \sqrt {-x^{2} e^{2} + d^{2}}} + \frac {1}{3} \, \sqrt {-x^{2} e^{2} + d^{2}} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/5*d^5/((-x^2*e^2 + d^2)^(3/2)*x*e^8 + (-x^2*e^2 + d^2)^(3/2)*d*e^7) - x^4*e^(-3)/(-x^2*e^2 + d^2)^(3/2) - 5
*d*x^3*e^(-4)/(-x^2*e^2 + d^2)^(3/2) + 20/3*d^2*x^2*e^(-5)/(-x^2*e^2 + d^2)^(3/2) + 64/15*d^3*x*e^(-6)/(-x^2*e
^2 + d^2)^(3/2) - 14/3*d^4*e^(-7)/(-x^2*e^2 + d^2)^(3/2) - d*arcsin(x*e/d)*e^(-7) + 1/3*x^2*e^(-5)/sqrt(-x^2*e
^2 + d^2) - 52/15*d*x*e^(-6)/sqrt(-x^2*e^2 + d^2) + 4/3*d^2*e^(-7)/sqrt(-x^2*e^2 + d^2) + 1/3*sqrt(-x^2*e^2 +
d^2)*e^(-7)

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Fricas [A]
time = 3.25, size = 240, normalized size = 1.62 \begin {gather*} -\frac {48 \, d x^{5} e^{5} + 48 \, d^{2} x^{4} e^{4} - 96 \, d^{3} x^{3} e^{3} - 96 \, d^{4} x^{2} e^{2} + 48 \, d^{5} x e + 48 \, d^{6} - 30 \, {\left (d x^{5} e^{5} + d^{2} x^{4} e^{4} - 2 \, d^{3} x^{3} e^{3} - 2 \, d^{4} x^{2} e^{2} + d^{5} x e + d^{6}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (15 \, x^{5} e^{5} + 38 \, d x^{4} e^{4} - 52 \, d^{2} x^{3} e^{3} - 87 \, d^{3} x^{2} e^{2} + 33 \, d^{4} x e + 48 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{5} e^{12} + d x^{4} e^{11} - 2 \, d^{2} x^{3} e^{10} - 2 \, d^{3} x^{2} e^{9} + d^{4} x e^{8} + d^{5} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/15*(48*d*x^5*e^5 + 48*d^2*x^4*e^4 - 96*d^3*x^3*e^3 - 96*d^4*x^2*e^2 + 48*d^5*x*e + 48*d^6 - 30*(d*x^5*e^5 +
d^2*x^4*e^4 - 2*d^3*x^3*e^3 - 2*d^4*x^2*e^2 + d^5*x*e + d^6)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (
15*x^5*e^5 + 38*d*x^4*e^4 - 52*d^2*x^3*e^3 - 87*d^3*x^2*e^2 + 33*d^4*x*e + 48*d^5)*sqrt(-x^2*e^2 + d^2))/(x^5*
e^12 + d*x^4*e^11 - 2*d^2*x^3*e^10 - 2*d^3*x^2*e^9 + d^4*x*e^8 + d^5*e^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**6/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), 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(x^6/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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