3.7.99 \(\int \frac {1}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=205 \[ -\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

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Rubi [A]  time = 0.09, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {659, 192, 191} \begin {gather*} \frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 9/(143*d^2*e*(d + e*x)
^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^
2*x^2)^(5/2)) + (64*x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*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]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {9 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {72 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^2}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {48 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^4}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {192 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^6}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{715 d^8}\\ &=\frac {48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {128 x}{715 d^{10} \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 137, normalized size = 0.67 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e
^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7
)

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IntegrateAlgebraic [A]  time = 0.72, size = 137, normalized size = 0.67 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e
^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7
)

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fricas [A]  time = 1.19, size = 314, normalized size = 1.53 \begin {gather*} -\frac {180 \, e^{10} x^{10} + 720 \, d e^{9} x^{9} + 540 \, d^{2} e^{8} x^{8} - 1440 \, d^{3} e^{7} x^{7} - 2520 \, d^{4} e^{6} x^{6} + 2520 \, d^{6} e^{4} x^{4} + 1440 \, d^{7} e^{3} x^{3} - 540 \, d^{8} e^{2} x^{2} - 720 \, d^{9} e x - 180 \, d^{10} + {\left (128 \, e^{9} x^{9} + 512 \, d e^{8} x^{8} + 448 \, d^{2} e^{7} x^{7} - 768 \, d^{3} e^{6} x^{6} - 1552 \, d^{4} e^{5} x^{5} - 320 \, d^{5} e^{4} x^{4} + 1080 \, d^{6} e^{3} x^{3} + 800 \, d^{7} e^{2} x^{2} - 5 \, d^{8} e x - 180 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{715 \, {\left (d^{10} e^{11} x^{10} + 4 \, d^{11} e^{10} x^{9} + 3 \, d^{12} e^{9} x^{8} - 8 \, d^{13} e^{8} x^{7} - 14 \, d^{14} e^{7} x^{6} + 14 \, d^{16} e^{5} x^{4} + 8 \, d^{17} e^{4} x^{3} - 3 \, d^{18} e^{3} x^{2} - 4 \, d^{19} e^{2} x - d^{20} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/715*(180*e^10*x^10 + 720*d*e^9*x^9 + 540*d^2*e^8*x^8 - 1440*d^3*e^7*x^7 - 2520*d^4*e^6*x^6 + 2520*d^6*e^4*x
^4 + 1440*d^7*e^3*x^3 - 540*d^8*e^2*x^2 - 720*d^9*e*x - 180*d^10 + (128*e^9*x^9 + 512*d*e^8*x^8 + 448*d^2*e^7*
x^7 - 768*d^3*e^6*x^6 - 1552*d^4*e^5*x^5 - 320*d^5*e^4*x^4 + 1080*d^6*e^3*x^3 + 800*d^7*e^2*x^2 - 5*d^8*e*x -
180*d^9)*sqrt(-e^2*x^2 + d^2))/(d^10*e^11*x^10 + 4*d^11*e^10*x^9 + 3*d^12*e^9*x^8 - 8*d^13*e^8*x^7 - 14*d^14*e
^7*x^6 + 14*d^16*e^5*x^4 + 8*d^17*e^4*x^3 - 3*d^18*e^3*x^2 - 4*d^19*e^2*x - d^20*e)

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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(1/(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 = 0.05, size = 132, normalized size = 0.64 \begin {gather*} -\frac {\left (-e x +d \right ) \left (-128 e^{9} x^{9}-512 e^{8} x^{8} d -448 e^{7} x^{7} d^{2}+768 e^{6} x^{6} d^{3}+1552 e^{5} x^{5} d^{4}+320 e^{4} x^{4} d^{5}-1080 e^{3} x^{3} d^{6}-800 e^{2} x^{2} d^{7}+5 x \,d^{8} e +180 d^{9}\right )}{715 \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{10} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/715*(-e*x+d)*(-128*e^9*x^9-512*d*e^8*x^8-448*d^2*e^7*x^7+768*d^3*e^6*x^6+1552*d^4*e^5*x^5+320*d^5*e^4*x^4-1
080*d^6*e^3*x^3-800*d^7*e^2*x^2+5*d^8*e*x+180*d^9)/(e*x+d)^3/d^10/e/(-e^2*x^2+d^2)^(7/2)

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maxima [B]  time = 1.55, size = 393, normalized size = 1.92 \begin {gather*} -\frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {9}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} - \frac {8}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e\right )}} + \frac {48 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6}} + \frac {64 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8}} + \frac {128 \, x}{715 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)*d*e^5*x^4 + 4*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x^3 + 6*(-e^2*x^2 + d^2)^(5/2)*d^3*
e^3*x^2 + 4*(-e^2*x^2 + d^2)^(5/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^5*e) - 9/143/((-e^2*x^2 + d^2)^(5/2)*d
^2*e^4*x^3 + 3*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x^2 + 3*(-e^2*x^2 + d^2)^(5/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(5/2
)*d^5*e) - 8/143/((-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(
5/2)*d^5*e) - 8/143/((-e^2*x^2 + d^2)^(5/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^5*e) + 48/715*x/((-e^2*x^2 +
d^2)^(5/2)*d^6) + 64/715*x/((-e^2*x^2 + d^2)^(3/2)*d^8) + 128/715*x/(sqrt(-e^2*x^2 + d^2)*d^10)

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mupad [B]  time = 0.96, size = 242, normalized size = 1.18 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {64\,x}{715\,d^8}+\frac {189}{4576\,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 {1139\,x}{5720\,d^6}-\frac {427}{2288\,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^4\,e\,{\left (d+e\,x\right )}^7}-\frac {51\,\sqrt {d^2-e^2\,x^2}}{2288\,d^5\,e\,{\left (d+e\,x\right )}^6}-\frac {19\,\sqrt {d^2-e^2\,x^2}}{572\,d^6\,e\,{\left (d+e\,x\right )}^5}-\frac {189\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^4}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{715\,d^{10}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d^2 - e^2*x^2)^(7/2)*(d + e*x)^4),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*((64*x)/(715*d^8) + 189/(4576*d^7*e)))/((d + e*x)^2*(d - e*x)^2) + ((d^2 - e^2*x^2)^(1/
2)*((1139*x)/(5720*d^6) - 427/(2288*d^5*e)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(104*d^4*e*(d +
e*x)^7) - (51*(d^2 - e^2*x^2)^(1/2))/(2288*d^5*e*(d + e*x)^6) - (19*(d^2 - e^2*x^2)^(1/2))/(572*d^6*e*(d + e*
x)^5) - (189*(d^2 - e^2*x^2)^(1/2))/(4576*d^7*e*(d + e*x)^4) + (128*x*(d^2 - e^2*x^2)^(1/2))/(715*d^10*(d + e*
x)*(d - e*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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(1/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**4), x)

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