3.3.12 \(\int \frac {x^3}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [212]

Optimal. Leaf size=209 \[ -\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {64 x}{5005 d^7 e^3 \sqrt {d^2-e^2 x^2}} \]

[Out]

-24/5005*x/d^3/e^3/(-e^2*x^2+d^2)^(5/2)+1/13*d^2/e^4/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2)-30/143*d/e^4/(e*x+d)^3/(-e
^2*x^2+d^2)^(5/2)+21/143/e^4/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2)+4/1001/d/e^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2)-32/5005*
x/d^5/e^3/(-e^2*x^2+d^2)^(3/2)-64/5005*x/d^7/e^3/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1653, 807, 673, 198, 197} \begin {gather*} \frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {64 x}{5005 d^7 e^3 \sqrt {d^2-e^2 x^2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(-24*x)/(5005*d^3*e^3*(d^2 - e^2*x^2)^(5/2)) + d^2/(13*e^4*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - (30*d)/(143*e^
4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) + 21/(143*e^4*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) + 4/(1001*d*e^4*(d + e*x
)*(d^2 - e^2*x^2)^(5/2)) - (32*x)/(5005*d^5*e^3*(d^2 - e^2*x^2)^(3/2)) - (64*x)/(5005*d^7*e^3*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 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rule 807

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

Rule 1653

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^3}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {2 d^3 e^2-3 d^2 e^3 x-12 d e^4 x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 e^5}\\ &=-\frac {3 d}{14 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {-20 d^3 e^6+36 d^2 e^7 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{56 e^9}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 d}{14 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\left (9 d^2\right ) \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{182 e^3}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{7 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(36 d) \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1001 e^3}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^3}\\ &=\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {24 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1001 d e^3}\\ &=-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {96 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5005 d^3 e^3}\\ &=-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {64 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5005 d^5 e^3}\\ &=-\frac {24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {64 x}{5005 d^7 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.62, size = 137, normalized size = 0.66 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (90 d^9+360 d^8 e x+315 d^7 e^2 x^2-540 d^6 e^3 x^3+160 d^5 e^4 x^4+776 d^4 e^5 x^5+384 d^3 e^6 x^6-224 d^2 e^7 x^7-256 d e^8 x^8-64 e^9 x^9\right )}{5005 d^7 e^4 (d-e x)^3 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(90*d^9 + 360*d^8*e*x + 315*d^7*e^2*x^2 - 540*d^6*e^3*x^3 + 160*d^5*e^4*x^4 + 776*d^4*e^5
*x^5 + 384*d^3*e^6*x^6 - 224*d^2*e^7*x^7 - 256*d*e^8*x^8 - 64*e^9*x^9))/(5005*d^7*e^4*(d - e*x)^3*(d + e*x)^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1211\) vs. \(2(181)=362\).
time = 0.07, size = 1212, normalized size = 5.80

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (-64 e^{9} x^{9}-256 d \,e^{8} x^{8}-224 e^{7} x^{7} d^{2}+384 e^{6} x^{6} d^{3}+776 e^{5} x^{5} d^{4}+160 x^{4} d^{5} e^{4}-540 d^{6} e^{3} x^{3}+315 x^{2} d^{7} e^{2}+360 d^{8} x e +90 d^{9}\right )}{5005 \left (e x +d \right )^{3} d^{7} e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(132\)
trager	\(\frac {\left (-64 e^{9} x^{9}-256 d \,e^{8} x^{8}-224 e^{7} x^{7} d^{2}+384 e^{6} x^{6} d^{3}+776 e^{5} x^{5} d^{4}+160 x^{4} d^{5} e^{4}-540 d^{6} e^{3} x^{3}+315 x^{2} d^{7} e^{2}+360 d^{8} x e +90 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5005 d^{7} \left (e x +d \right )^{7} e^{4} \left (-e x +d \right )^{3}}\)	\(134\)
default	\(\text {Expression too large to display}\)	\(1212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-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))))+3*d^2/e^6*(-1/11/d/e/(x+d/e)^
3/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+8/11*e/d*(-1/9/d/e/(x+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+7/9*e
/d*(-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))))))-d^3/e^7*(-1/13/d/e/(x+d/e)^4/(
-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+9/13*e/d*(-1/11/d/e/(x+d/e)^3/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+8/11*e/
d*(-1/9/d/e/(x+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-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)))))))-3*d/e^5*(-1/9/d/e/(x+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (171) = 342\).
time = 0.31, size = 365, normalized size = 1.75 \begin {gather*} \frac {d^{2}}{13 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{4} e^{8} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3} e^{7} + 6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{6} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{5} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{4}\right )}} - \frac {30 \, d}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{3} e^{7} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2} e^{6} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x e^{5} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4}\right )}} + \frac {21}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{2} e^{6} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{5} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4}\right )}} + \frac {4}{1001 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{5} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4}\right )}} - \frac {24 \, x e^{\left (-3\right )}}{5005 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} - \frac {32 \, x e^{\left (-3\right )}}{5005 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} - \frac {64 \, x e^{\left (-3\right )}}{5005 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

1/13*d^2/((-x^2*e^2 + d^2)^(5/2)*x^4*e^8 + 4*(-x^2*e^2 + d^2)^(5/2)*d*x^3*e^7 + 6*(-x^2*e^2 + d^2)^(5/2)*d^2*x
^2*e^6 + 4*(-x^2*e^2 + d^2)^(5/2)*d^3*x*e^5 + (-x^2*e^2 + d^2)^(5/2)*d^4*e^4) - 30/143*d/((-x^2*e^2 + d^2)^(5/
2)*x^3*e^7 + 3*(-x^2*e^2 + d^2)^(5/2)*d*x^2*e^6 + 3*(-x^2*e^2 + d^2)^(5/2)*d^2*x*e^5 + (-x^2*e^2 + d^2)^(5/2)*
d^3*e^4) + 21/143/((-x^2*e^2 + d^2)^(5/2)*x^2*e^6 + 2*(-x^2*e^2 + d^2)^(5/2)*d*x*e^5 + (-x^2*e^2 + d^2)^(5/2)*
d^2*e^4) + 4/1001/((-x^2*e^2 + d^2)^(5/2)*d*x*e^5 + (-x^2*e^2 + d^2)^(5/2)*d^2*e^4) - 24/5005*x*e^(-3)/((-x^2*
e^2 + d^2)^(5/2)*d^3) - 32/5005*x*e^(-3)/((-x^2*e^2 + d^2)^(3/2)*d^5) - 64/5005*x*e^(-3)/(sqrt(-x^2*e^2 + d^2)
*d^7)

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Fricas [A]
time = 4.41, size = 291, normalized size = 1.39 \begin {gather*} \frac {90 \, x^{10} e^{10} + 360 \, d x^{9} e^{9} + 270 \, d^{2} x^{8} e^{8} - 720 \, d^{3} x^{7} e^{7} - 1260 \, d^{4} x^{6} e^{6} + 1260 \, d^{6} x^{4} e^{4} + 720 \, d^{7} x^{3} e^{3} - 270 \, d^{8} x^{2} e^{2} - 360 \, d^{9} x e - 90 \, d^{10} + {\left (64 \, x^{9} e^{9} + 256 \, d x^{8} e^{8} + 224 \, d^{2} x^{7} e^{7} - 384 \, d^{3} x^{6} e^{6} - 776 \, d^{4} x^{5} e^{5} - 160 \, d^{5} x^{4} e^{4} + 540 \, d^{6} x^{3} e^{3} - 315 \, d^{7} x^{2} e^{2} - 360 \, d^{8} x e - 90 \, d^{9}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5005 \, {\left (d^{7} x^{10} e^{14} + 4 \, d^{8} x^{9} e^{13} + 3 \, d^{9} x^{8} e^{12} - 8 \, d^{10} x^{7} e^{11} - 14 \, d^{11} x^{6} e^{10} + 14 \, d^{13} x^{4} e^{8} + 8 \, d^{14} x^{3} e^{7} - 3 \, d^{15} x^{2} e^{6} - 4 \, d^{16} x e^{5} - d^{17} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/5005*(90*x^10*e^10 + 360*d*x^9*e^9 + 270*d^2*x^8*e^8 - 720*d^3*x^7*e^7 - 1260*d^4*x^6*e^6 + 1260*d^6*x^4*e^4
+ 720*d^7*x^3*e^3 - 270*d^8*x^2*e^2 - 360*d^9*x*e - 90*d^10 + (64*x^9*e^9 + 256*d*x^8*e^8 + 224*d^2*x^7*e^7 -
384*d^3*x^6*e^6 - 776*d^4*x^5*e^5 - 160*d^5*x^4*e^4 + 540*d^6*x^3*e^3 - 315*d^7*x^2*e^2 - 360*d^8*x*e - 90*d^
9)*sqrt(-x^2*e^2 + d^2))/(d^7*x^10*e^14 + 4*d^8*x^9*e^13 + 3*d^9*x^8*e^12 - 8*d^10*x^7*e^11 - 14*d^11*x^6*e^10
+ 14*d^13*x^4*e^8 + 8*d^14*x^3*e^7 - 3*d^15*x^2*e^6 - 4*d^16*x*e^5 - d^17*e^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x**3/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**4), 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^3/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate(x^3/((-x^2*e^2 + d^2)^(7/2)*(x*e + d)^4), x)

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Mupad [B]
time = 3.22, size = 252, normalized size = 1.21 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {107}{4004\,d^2\,e^4}-\frac {1139\,x}{80080\,d^3\,e^3}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {23}{32032\,d^4\,e^4}+\frac {32\,x}{5005\,d^5\,e^3}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}}{104\,d\,e^4\,{\left (d+e\,x\right )}^7}-\frac {27\,\sqrt {d^2-e^2\,x^2}}{2288\,d^2\,e^4\,{\left (d+e\,x\right )}^6}-\frac {15\,\sqrt {d^2-e^2\,x^2}}{2288\,d^3\,e^4\,{\left (d+e\,x\right )}^5}+\frac {23\,\sqrt {d^2-e^2\,x^2}}{32032\,d^4\,e^4\,{\left (d+e\,x\right )}^4}-\frac {64\,x\,\sqrt {d^2-e^2\,x^2}}{5005\,d^7\,e^3\,\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^3/((d^2 - e^2*x^2)^(7/2)*(d + e*x)^4),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*(107/(4004*d^2*e^4) - (1139*x)/(80080*d^3*e^3)))/((d + e*x)^3*(d - e*x)^3) - ((d^2 - e^
2*x^2)^(1/2)*(23/(32032*d^4*e^4) + (32*x)/(5005*d^5*e^3)))/((d + e*x)^2*(d - e*x)^2) + (d^2 - e^2*x^2)^(1/2)/(
104*d*e^4*(d + e*x)^7) - (27*(d^2 - e^2*x^2)^(1/2))/(2288*d^2*e^4*(d + e*x)^6) - (15*(d^2 - e^2*x^2)^(1/2))/(2
288*d^3*e^4*(d + e*x)^5) + (23*(d^2 - e^2*x^2)^(1/2))/(32032*d^4*e^4*(d + e*x)^4) - (64*x*(d^2 - e^2*x^2)^(1/2
))/(5005*d^7*e^3*(d + e*x)*(d - e*x))

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