3.214 \(\int \frac {x}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=211 \[ -\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

64/2145*x/d^5/e/(-e^2*x^2+d^2)^(5/2)+1/13/e^2/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2)-4/143/d/e^2/(e*x+d)^3/(-e^2*x^2+d
^2)^(5/2)-32/1287/d^2/e^2/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2)-32/1287/d^3/e^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+256/6435
*x/d^7/e/(-e^2*x^2+d^2)^(3/2)+512/6435*x/d^9/e/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {793, 659, 192, 191} \[ \frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*x)/(2145*d^5*e*(d^2 - e^2*x^2)^(5/2)) + 1/(13*e^2*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 4/(143*d*e^2*(d + e
*x)^3*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^2*e^2*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^3*e^2*(d + e*x
)*(d^2 - e^2*x^2)^(5/2)) + (256*x)/(6435*d^7*e*(d^2 - e^2*x^2)^(3/2)) + (512*x)/(6435*d^9*e*Sqrt[d^2 - e^2*x^2
])

Rule 191

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 192

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 659

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 793

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]

Rubi steps

\begin {align*} \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 e}\\ &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {32 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d e}\\ &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {224 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d^2 e}\\ &=\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^3 e}\\ &=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^5 e}\\ &=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {512 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^7 e}\\ &=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 137, normalized size = 0.65 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-5 d^9-20 d^8 e x+3200 d^7 e^2 x^2+4320 d^6 e^3 x^3-1280 d^5 e^4 x^4-6208 d^4 e^5 x^5-3072 d^3 e^6 x^6+1792 d^2 e^7 x^7+2048 d e^8 x^8+512 e^9 x^9\right )}{6435 d^9 e^2 (d-e x)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5*d^9 - 20*d^8*e*x + 3200*d^7*e^2*x^2 + 4320*d^6*e^3*x^3 - 1280*d^5*e^4*x^4 - 6208*d^4*
e^5*x^5 - 3072*d^3*e^6*x^6 + 1792*d^2*e^7*x^7 + 2048*d*e^8*x^8 + 512*e^9*x^9))/(6435*d^9*e^2*(d - e*x)^3*(d +
e*x)^7)

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fricas [A]  time = 2.34, size = 316, normalized size = 1.50 \[ -\frac {5 \, e^{10} x^{10} + 20 \, d e^{9} x^{9} + 15 \, d^{2} e^{8} x^{8} - 40 \, d^{3} e^{7} x^{7} - 70 \, d^{4} e^{6} x^{6} + 70 \, d^{6} e^{4} x^{4} + 40 \, d^{7} e^{3} x^{3} - 15 \, d^{8} e^{2} x^{2} - 20 \, d^{9} e x - 5 \, d^{10} + {\left (512 \, e^{9} x^{9} + 2048 \, d e^{8} x^{8} + 1792 \, d^{2} e^{7} x^{7} - 3072 \, d^{3} e^{6} x^{6} - 6208 \, d^{4} e^{5} x^{5} - 1280 \, d^{5} e^{4} x^{4} + 4320 \, d^{6} e^{3} x^{3} + 3200 \, d^{7} e^{2} x^{2} - 20 \, d^{8} e x - 5 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{9} e^{12} x^{10} + 4 \, d^{10} e^{11} x^{9} + 3 \, d^{11} e^{10} x^{8} - 8 \, d^{12} e^{9} x^{7} - 14 \, d^{13} e^{8} x^{6} + 14 \, d^{15} e^{6} x^{4} + 8 \, d^{16} e^{5} x^{3} - 3 \, d^{17} e^{4} x^{2} - 4 \, d^{18} e^{3} x - d^{19} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6435*(5*e^10*x^10 + 20*d*e^9*x^9 + 15*d^2*e^8*x^8 - 40*d^3*e^7*x^7 - 70*d^4*e^6*x^6 + 70*d^6*e^4*x^4 + 40*d
^7*e^3*x^3 - 15*d^8*e^2*x^2 - 20*d^9*e*x - 5*d^10 + (512*e^9*x^9 + 2048*d*e^8*x^8 + 1792*d^2*e^7*x^7 - 3072*d^
3*e^6*x^6 - 6208*d^4*e^5*x^5 - 1280*d^5*e^4*x^4 + 4320*d^6*e^3*x^3 + 3200*d^7*e^2*x^2 - 20*d^8*e*x - 5*d^9)*sq
rt(-e^2*x^2 + d^2))/(d^9*e^12*x^10 + 4*d^10*e^11*x^9 + 3*d^11*e^10*x^8 - 8*d^12*e^9*x^7 - 14*d^13*e^8*x^6 + 14
*d^15*e^6*x^4 + 8*d^16*e^5*x^3 - 3*d^17*e^4*x^2 - 4*d^18*e^3*x - d^19*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),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 [A]  time = 0.01, size = 132, normalized size = 0.63 \[ -\frac {\left (-e x +d \right ) \left (-512 e^{9} x^{9}-2048 e^{8} x^{8} d -1792 e^{7} x^{7} d^{2}+3072 e^{6} x^{6} d^{3}+6208 e^{5} x^{5} d^{4}+1280 x^{4} d^{5} e^{4}-4320 x^{3} d^{6} e^{3}-3200 x^{2} d^{7} e^{2}+20 d^{8} x e +5 d^{9}\right )}{6435 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{9} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6435*(-e*x+d)*(-512*e^9*x^9-2048*d*e^8*x^8-1792*d^2*e^7*x^7+3072*d^3*e^6*x^6+6208*d^4*e^5*x^5+1280*d^5*e^4*
x^4-4320*d^6*e^3*x^3-3200*d^7*e^2*x^2+20*d^8*e*x+5*d^9)/(e*x+d)^3/d^9/e^2/(-e^2*x^2+d^2)^(7/2)

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maxima [B]  time = 0.49, size = 405, normalized size = 1.92 \[ \frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {4}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} + \frac {64 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e} + \frac {256 \, x}{6435 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} e} + \frac {512 \, x}{6435 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13/((-e^2*x^2 + d^2)^(5/2)*e^6*x^4 + 4*(-e^2*x^2 + d^2)^(5/2)*d*e^5*x^3 + 6*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x
^2 + 4*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x + (-e^2*x^2 + d^2)^(5/2)*d^4*e^2) - 4/143/((-e^2*x^2 + d^2)^(5/2)*d*e^
5*x^3 + 3*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x^2 + 3*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x + (-e^2*x^2 + d^2)^(5/2)*d^4
*e^2) - 32/1287/((-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x + (-e^2*x^2 + d^2)^(5
/2)*d^4*e^2) - 32/1287/((-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x + (-e^2*x^2 + d^2)^(5/2)*d^4*e^2) + 64/2145*x/((-e^2*
x^2 + d^2)^(5/2)*d^5*e) + 256/6435*x/((-e^2*x^2 + d^2)^(3/2)*d^7*e) + 512/6435*x/(sqrt(-e^2*x^2 + d^2)*d^9*e)

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mupad [B]  time = 3.19, size = 252, normalized size = 1.19 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {41}{41184\,d^6\,e^2}+\frac {256\,x}{6435\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {47}{1716\,d^4\,e^2}-\frac {1369\,x}{34320\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^3\,e^2\,{\left (d+e\,x\right )}^7}+\frac {25\,\sqrt {d^2-e^2\,x^2}}{2288\,d^4\,e^2\,{\left (d+e\,x\right )}^6}+\frac {125\,\sqrt {d^2-e^2\,x^2}}{20592\,d^5\,e^2\,{\left (d+e\,x\right )}^5}-\frac {41\,\sqrt {d^2-e^2\,x^2}}{41184\,d^6\,e^2\,{\left (d+e\,x\right )}^4}+\frac {512\,x\,\sqrt {d^2-e^2\,x^2}}{6435\,d^9\,e\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*(41/(41184*d^6*e^2) + (256*x)/(6435*d^7*e)))/((d + e*x)^2*(d - e*x)^2) - ((d^2 - e^2*x^
2)^(1/2)*(47/(1716*d^4*e^2) - (1369*x)/(34320*d^5*e)))/((d + e*x)^3*(d - e*x)^3) + (d^2 - e^2*x^2)^(1/2)/(104*
d^3*e^2*(d + e*x)^7) + (25*(d^2 - e^2*x^2)^(1/2))/(2288*d^4*e^2*(d + e*x)^6) + (125*(d^2 - e^2*x^2)^(1/2))/(20
592*d^5*e^2*(d + e*x)^5) - (41*(d^2 - e^2*x^2)^(1/2))/(41184*d^6*e^2*(d + e*x)^4) + (512*x*(d^2 - e^2*x^2)^(1/
2))/(6435*d^9*e*(d + e*x)*(d - e*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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