3.2.49 \(\int \frac {x^2}{(d+e x) (d^2-e^2 x^2)^{7/2}} \, dx\) [149]

Optimal. Leaf size=123 \[ -\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}} \]

[Out]

-1/7*x^2/d/e/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+2/35*(2*e*x+d)/d/e^3/(-e^2*x^2+d^2)^(5/2)-4/105*x/d^3/e^2/(-e^2*x^2+
d^2)^(3/2)-8/105*x/d^5/e^2/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {869, 792, 198, 197} \begin {gather*} -\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*x^2/(d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (2*(d + 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105
*d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 792

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a*(e*f + d*g) - (
c*d*f - a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 869

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

Rubi steps

\begin {align*} \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {x (2 d+4 e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d e}\\ &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d e^2}\\ &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{105 d^3 e^2}\\ &=-\frac {x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 104, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (6 d^6+6 d^5 e x-15 d^4 e^2 x^2+20 d^3 e^3 x^3+20 d^2 e^4 x^4-8 d e^5 x^5-8 e^6 x^6\right )}{105 d^5 e^3 (d-e x)^3 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(6*d^6 + 6*d^5*e*x - 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 20*d^2*e^4*x^4 - 8*d*e^5*x^5 - 8*e
^6*x^6))/(105*d^5*e^3*(d - e*x)^3*(d + e*x)^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(107)=214\).
time = 0.07, size = 319, normalized size = 2.59

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (-8 e^{6} x^{6}-8 d \,e^{5} x^{5}+20 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}-15 d^{4} e^{2} x^{2}+6 e \,d^{5} x +6 d^{6}\right )}{105 d^{5} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(92\)
trager	\(\frac {\left (-8 e^{6} x^{6}-8 d \,e^{5} x^{5}+20 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}-15 d^{4} e^{2} x^{2}+6 e \,d^{5} x +6 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{5} \left (e x +d \right )^{4} \left (-e x +d \right )^{3} e^{3}}\)	\(101\)
default	\(\frac {1}{5 e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{e^{2}}+\frac {d^{2} \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{e^{3}}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/e^3/(-e^2*x^2+d^2)^(5/2)-d/e^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)))+1/e^3*d^2*(-1/7/d/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+6/7*e/d*(-1/10*
(-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))^(5/2)+4/5/d^2*(-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 = 0.28, size = 121, normalized size = 0.98 \begin {gather*} -\frac {d}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x e^{4} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{3}\right )}} - \frac {x e^{\left (-2\right )}}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {e^{\left (-3\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, x e^{\left (-2\right )}}{105 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} - \frac {8 \, x e^{\left (-2\right )}}{105 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*d/((-x^2*e^2 + d^2)^(5/2)*x*e^4 + (-x^2*e^2 + d^2)^(5/2)*d*e^3) - 1/35*x*e^(-2)/((-x^2*e^2 + d^2)^(5/2)*d
) + 1/5*e^(-3)/(-x^2*e^2 + d^2)^(5/2) - 4/105*x*e^(-2)/((-x^2*e^2 + d^2)^(3/2)*d^3) - 8/105*x*e^(-2)/(sqrt(-x^
2*e^2 + d^2)*d^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (101) = 202\).
time = 2.29, size = 220, normalized size = 1.79 \begin {gather*} \frac {6 \, x^{7} e^{7} + 6 \, d x^{6} e^{6} - 18 \, d^{2} x^{5} e^{5} - 18 \, d^{3} x^{4} e^{4} + 18 \, d^{4} x^{3} e^{3} + 18 \, d^{5} x^{2} e^{2} - 6 \, d^{6} x e - 6 \, d^{7} + {\left (8 \, x^{6} e^{6} + 8 \, d x^{5} e^{5} - 20 \, d^{2} x^{4} e^{4} - 20 \, d^{3} x^{3} e^{3} + 15 \, d^{4} x^{2} e^{2} - 6 \, d^{5} x e - 6 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{105 \, {\left (d^{5} x^{7} e^{10} + d^{6} x^{6} e^{9} - 3 \, d^{7} x^{5} e^{8} - 3 \, d^{8} x^{4} e^{7} + 3 \, d^{9} x^{3} e^{6} + 3 \, d^{10} x^{2} e^{5} - d^{11} x e^{4} - d^{12} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(6*x^7*e^7 + 6*d*x^6*e^6 - 18*d^2*x^5*e^5 - 18*d^3*x^4*e^4 + 18*d^4*x^3*e^3 + 18*d^5*x^2*e^2 - 6*d^6*x*e
- 6*d^7 + (8*x^6*e^6 + 8*d*x^5*e^5 - 20*d^2*x^4*e^4 - 20*d^3*x^3*e^3 + 15*d^4*x^2*e^2 - 6*d^5*x*e - 6*d^6)*sq
rt(-x^2*e^2 + d^2))/(d^5*x^7*e^10 + d^6*x^6*e^9 - 3*d^7*x^5*e^8 - 3*d^8*x^4*e^7 + 3*d^9*x^3*e^6 + 3*d^10*x^2*e
^5 - d^11*x*e^4 - d^12*e^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 2.88, size = 161, normalized size = 1.31 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{56\,d^2\,e^3}-\frac {4\,x}{105\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {2}{35\,e^3}+\frac {3\,x}{70\,d\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{56\,d^2\,e^3\,{\left (d+e\,x\right )}^4}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{105\,d^5\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*(1/(56*d^2*e^3) - (4*x)/(105*d^3*e^2)))/((d + e*x)^2*(d - e*x)^2) + ((d^2 - e^2*x^2)^(1
/2)*(2/(35*e^3) + (3*x)/(70*d*e^2)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(56*d^2*e^3*(d + e*x)^4
) - (8*x*(d^2 - e^2*x^2)^(1/2))/(105*d^5*e^2*(d + e*x)*(d - e*x))

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