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

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

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Rubi [A]  time = 0.14, 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 = {850, 819, 641, 217, 203} \begin {gather*} \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 {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 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 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 641

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 819

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 850

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 \operatorname {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 = 0.19, size = 115, normalized size = 0.78 \begin {gather*} -\frac {15 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\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 )}{(d-e x)^2 (d+e x)^3}}{15 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*((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
))/((d - e*x)^2*(d + e*x)^3) + 15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^7

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IntegrateAlgebraic [A]  time = 0.57, size = 137, normalized size = 0.93 \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 (e x-d)^2 (d+e x)^3}-\frac {d \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/e^8

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fricas [A]  time = 0.44, size = 258, normalized size = 1.74 \begin {gather*} -\frac {48 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} - 96 \, d^{3} e^{3} x^{3} - 96 \, d^{4} e^{2} x^{2} + 48 \, d^{5} e x + 48 \, d^{6} - 30 \, {\left (d e^{5} x^{5} + d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} - 2 \, d^{4} e^{2} x^{2} + d^{5} e x + d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{5} x^{5} + 38 \, d e^{4} x^{4} - 52 \, d^{2} e^{3} x^{3} - 87 \, d^{3} e^{2} x^{2} + 33 \, d^{4} e x + 48 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{12} x^{5} + d e^{11} x^{4} - 2 \, d^{2} e^{10} x^{3} - 2 \, d^{3} e^{9} x^{2} + d^{4} e^{8} x + 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*e^5*x^5 + 48*d^2*e^4*x^4 - 96*d^3*e^3*x^3 - 96*d^4*e^2*x^2 + 48*d^5*e*x + 48*d^6 - 30*(d*e^5*x^5 +
d^2*e^4*x^4 - 2*d^3*e^3*x^3 - 2*d^4*e^2*x^2 + d^5*e*x + d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*
e^5*x^5 + 38*d*e^4*x^4 - 52*d^2*e^3*x^3 - 87*d^3*e^2*x^2 + 33*d^4*e*x + 48*d^5)*sqrt(-e^2*x^2 + d^2))/(e^12*x^
5 + d*e^11*x^4 - 2*d^2*e^10*x^3 - 2*d^3*e^9*x^2 + d^4*e^8*x + d^5*e^7)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

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maple [B]  time = 0.01, size = 288, normalized size = 1.95 \begin {gather*} -\frac {x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}-\frac {d \,x^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4}}+\frac {5 d^{2} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}-\frac {2 d^{3} x}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6}}+\frac {4 d^{3} x}{15 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{6}}-\frac {d^{5}}{5 \left (x +\frac {d}{e}\right ) \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{8}}-\frac {3 d^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}+\frac {2 d x}{3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6}}+\frac {8 d x}{15 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{6}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{6}} \end {gather*}

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)

[Out]

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

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maxima [A]  time = 1.09, size = 259, normalized size = 1.75 \begin {gather*} -\frac {d^{5}}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{7}\right )}} - \frac {x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {5 \, d x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} + \frac {20 \, d^{2} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} + \frac {64 \, d^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}} + \frac {x^{2}}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{5}} - \frac {14 \, d^{4}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {52 \, d x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}} - \frac {d \arcsin \left (\frac {e x}{d}\right )}{e^{7}} + \frac {4 \, d^{2}}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, e^{7}} \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/((-e^2*x^2 + d^2)^(3/2)*e^8*x + (-e^2*x^2 + d^2)^(3/2)*d*e^7) - x^4/((-e^2*x^2 + d^2)^(3/2)*e^3) - 5*
d*x^3/((-e^2*x^2 + d^2)^(3/2)*e^4) + 20/3*d^2*x^2/((-e^2*x^2 + d^2)^(3/2)*e^5) + 64/15*d^3*x/((-e^2*x^2 + d^2)
^(3/2)*e^6) + 1/3*x^2/(sqrt(-e^2*x^2 + d^2)*e^5) - 14/3*d^4/((-e^2*x^2 + d^2)^(3/2)*e^7) - 52/15*d*x/(sqrt(-e^
2*x^2 + d^2)*e^6) - d*arcsin(e*x/d)/e^7 + 4/3*d^2/(sqrt(-e^2*x^2 + d^2)*e^7) + 1/3*sqrt(-e^2*x^2 + d^2)/e^7

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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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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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