3.7.100 \(\int \frac {1}{(d+e x)^5 (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=238 \[ -\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{2145 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

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Rubi [A]  time = 0.11, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {256 x}{2145 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(32*x)/(715*d^7*(d^2 - e^2*x^2)^(5/2)) - 1/(15*d*e*(d + e*x)^5*(d^2 - e^2*x^2)^(5/2)) - 2/(39*d^2*e*(d + e*x)^
4*(d^2 - e^2*x^2)^(5/2)) - 6/(143*d^3*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 16/(429*d^4*e*(d + e*x)^2*(d^2 -
e^2*x^2)^(5/2)) - 16/(429*d^5*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (128*x)/(2145*d^9*(d^2 - e^2*x^2)^(3/2)) +
(256*x)/(2145*d^11*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)^5 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d}\\ &=-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {6 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d^2}\\ &=-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {48 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {112 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^4}\\ &=-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {32 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^5}\\ &=\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^7}\\ &=\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{2145 d^9}\\ &=\frac {32 x}{715 d^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{15 d e (d+e x)^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2}{39 d^2 e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6}{143 d^3 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^4 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16}{429 d^5 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{2145 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 x}{2145 d^{11} \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 148, normalized size = 0.62 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-503 d^{10}-370 d^9 e x+1590 d^8 e^2 x^2+3760 d^7 e^3 x^3+1520 d^6 e^4 x^4-3744 d^5 e^5 x^5-4640 d^4 e^6 x^6-640 d^3 e^7 x^7+1920 d^2 e^8 x^8+1280 d e^9 x^9+256 e^{10} x^{10}\right )}{2145 d^{11} e (d-e x)^3 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-503*d^10 - 370*d^9*e*x + 1590*d^8*e^2*x^2 + 3760*d^7*e^3*x^3 + 1520*d^6*e^4*x^4 - 3744*
d^5*e^5*x^5 - 4640*d^4*e^6*x^6 - 640*d^3*e^7*x^7 + 1920*d^2*e^8*x^8 + 1280*d*e^9*x^9 + 256*e^10*x^10))/(2145*d
^11*e*(d - e*x)^3*(d + e*x)^8)

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IntegrateAlgebraic [A]  time = 0.84, size = 148, normalized size = 0.62 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-503 d^{10}-370 d^9 e x+1590 d^8 e^2 x^2+3760 d^7 e^3 x^3+1520 d^6 e^4 x^4-3744 d^5 e^5 x^5-4640 d^4 e^6 x^6-640 d^3 e^7 x^7+1920 d^2 e^8 x^8+1280 d e^9 x^9+256 e^{10} x^{10}\right )}{2145 d^{11} e (d-e x)^3 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^5*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-503*d^10 - 370*d^9*e*x + 1590*d^8*e^2*x^2 + 3760*d^7*e^3*x^3 + 1520*d^6*e^4*x^4 - 3744*
d^5*e^5*x^5 - 4640*d^4*e^6*x^6 - 640*d^3*e^7*x^7 + 1920*d^2*e^8*x^8 + 1280*d*e^9*x^9 + 256*e^10*x^10))/(2145*d
^11*e*(d - e*x)^3*(d + e*x)^8)

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fricas [A]  time = 1.65, size = 369, normalized size = 1.55 \begin {gather*} -\frac {503 \, e^{11} x^{11} + 2515 \, d e^{10} x^{10} + 3521 \, d^{2} e^{9} x^{9} - 2515 \, d^{3} e^{8} x^{8} - 11066 \, d^{4} e^{7} x^{7} - 7042 \, d^{5} e^{6} x^{6} + 7042 \, d^{6} e^{5} x^{5} + 11066 \, d^{7} e^{4} x^{4} + 2515 \, d^{8} e^{3} x^{3} - 3521 \, d^{9} e^{2} x^{2} - 2515 \, d^{10} e x - 503 \, d^{11} + {\left (256 \, e^{10} x^{10} + 1280 \, d e^{9} x^{9} + 1920 \, d^{2} e^{8} x^{8} - 640 \, d^{3} e^{7} x^{7} - 4640 \, d^{4} e^{6} x^{6} - 3744 \, d^{5} e^{5} x^{5} + 1520 \, d^{6} e^{4} x^{4} + 3760 \, d^{7} e^{3} x^{3} + 1590 \, d^{8} e^{2} x^{2} - 370 \, d^{9} e x - 503 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2145 \, {\left (d^{11} e^{12} x^{11} + 5 \, d^{12} e^{11} x^{10} + 7 \, d^{13} e^{10} x^{9} - 5 \, d^{14} e^{9} x^{8} - 22 \, d^{15} e^{8} x^{7} - 14 \, d^{16} e^{7} x^{6} + 14 \, d^{17} e^{6} x^{5} + 22 \, d^{18} e^{5} x^{4} + 5 \, d^{19} e^{4} x^{3} - 7 \, d^{20} e^{3} x^{2} - 5 \, d^{21} e^{2} x - d^{22} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/2145*(503*e^11*x^11 + 2515*d*e^10*x^10 + 3521*d^2*e^9*x^9 - 2515*d^3*e^8*x^8 - 11066*d^4*e^7*x^7 - 7042*d^5
*e^6*x^6 + 7042*d^6*e^5*x^5 + 11066*d^7*e^4*x^4 + 2515*d^8*e^3*x^3 - 3521*d^9*e^2*x^2 - 2515*d^10*e*x - 503*d^
11 + (256*e^10*x^10 + 1280*d*e^9*x^9 + 1920*d^2*e^8*x^8 - 640*d^3*e^7*x^7 - 4640*d^4*e^6*x^6 - 3744*d^5*e^5*x^
5 + 1520*d^6*e^4*x^4 + 3760*d^7*e^3*x^3 + 1590*d^8*e^2*x^2 - 370*d^9*e*x - 503*d^10)*sqrt(-e^2*x^2 + d^2))/(d^
11*e^12*x^11 + 5*d^12*e^11*x^10 + 7*d^13*e^10*x^9 - 5*d^14*e^9*x^8 - 22*d^15*e^8*x^7 - 14*d^16*e^7*x^6 + 14*d^
17*e^6*x^5 + 22*d^18*e^5*x^4 + 5*d^19*e^4*x^3 - 7*d^20*e^3*x^2 - 5*d^21*e^2*x - d^22*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.05, size = 143, normalized size = 0.60 \begin {gather*} -\frac {\left (-e x +d \right ) \left (-256 e^{10} x^{10}-1280 e^{9} x^{9} d -1920 e^{8} x^{8} d^{2}+640 e^{7} x^{7} d^{3}+4640 e^{6} x^{6} d^{4}+3744 e^{5} x^{5} d^{5}-1520 e^{4} x^{4} d^{6}-3760 e^{3} x^{3} d^{7}-1590 e^{2} x^{2} d^{8}+370 x \,d^{9} e +503 d^{10}\right )}{2145 \left (e x +d \right )^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{11} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2145*(-e*x+d)*(-256*e^10*x^10-1280*d*e^9*x^9-1920*d^2*e^8*x^8+640*d^3*e^7*x^7+4640*d^4*e^6*x^6+3744*d^5*e^5
*x^5-1520*d^6*e^4*x^4-3760*d^7*e^3*x^3-1590*d^8*e^2*x^2+370*d^9*e*x+503*d^10)/(e*x+d)^4/d^11/e/(-e^2*x^2+d^2)^
(7/2)

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maxima [B]  time = 1.51, size = 539, normalized size = 2.26 \begin {gather*} -\frac {1}{15 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{6} x^{5} + 5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x^{4} + 10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x^{3} + 10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {2}{39 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {6}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {16}{429 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {16}{429 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} + \frac {32 \, x}{715 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7}} + \frac {128 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9}} + \frac {256 \, x}{2145 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/15/((-e^2*x^2 + d^2)^(5/2)*d*e^6*x^5 + 5*(-e^2*x^2 + d^2)^(5/2)*d^2*e^5*x^4 + 10*(-e^2*x^2 + d^2)^(5/2)*d^3
*e^4*x^3 + 10*(-e^2*x^2 + d^2)^(5/2)*d^4*e^3*x^2 + 5*(-e^2*x^2 + d^2)^(5/2)*d^5*e^2*x + (-e^2*x^2 + d^2)^(5/2)
*d^6*e) - 2/39/((-e^2*x^2 + d^2)^(5/2)*d^2*e^5*x^4 + 4*(-e^2*x^2 + d^2)^(5/2)*d^3*e^4*x^3 + 6*(-e^2*x^2 + d^2)
^(5/2)*d^4*e^3*x^2 + 4*(-e^2*x^2 + d^2)^(5/2)*d^5*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^6*e) - 6/143/((-e^2*x^2 + d
^2)^(5/2)*d^3*e^4*x^3 + 3*(-e^2*x^2 + d^2)^(5/2)*d^4*e^3*x^2 + 3*(-e^2*x^2 + d^2)^(5/2)*d^5*e^2*x + (-e^2*x^2
+ d^2)^(5/2)*d^6*e) - 16/429/((-e^2*x^2 + d^2)^(5/2)*d^4*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^5*e^2*x + (-e^2*
x^2 + d^2)^(5/2)*d^6*e) - 16/429/((-e^2*x^2 + d^2)^(5/2)*d^5*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^6*e) + 32/715*x/
((-e^2*x^2 + d^2)^(5/2)*d^7) + 128/2145*x/((-e^2*x^2 + d^2)^(3/2)*d^9) + 256/2145*x/(sqrt(-e^2*x^2 + d^2)*d^11
)

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mupad [B]  time = 1.14, size = 271, normalized size = 1.14 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {128\,x}{2145\,d^9}+\frac {647}{18304\,d^8\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1757\,x}{11440\,d^7}-\frac {3371}{22880\,d^6\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{120\,d^4\,e\,{\left (d+e\,x\right )}^8}-\frac {59\,\sqrt {d^2-e^2\,x^2}}{3120\,d^5\,e\,{\left (d+e\,x\right )}^7}-\frac {313\,\sqrt {d^2-e^2\,x^2}}{11440\,d^6\,e\,{\left (d+e\,x\right )}^6}-\frac {149\,\sqrt {d^2-e^2\,x^2}}{4576\,d^7\,e\,{\left (d+e\,x\right )}^5}-\frac {647\,\sqrt {d^2-e^2\,x^2}}{18304\,d^8\,e\,{\left (d+e\,x\right )}^4}+\frac {256\,x\,\sqrt {d^2-e^2\,x^2}}{2145\,d^{11}\,\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)^5),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*((128*x)/(2145*d^9) + 647/(18304*d^8*e)))/((d + e*x)^2*(d - e*x)^2) + ((d^2 - e^2*x^2)^
(1/2)*((1757*x)/(11440*d^7) - 3371/(22880*d^6*e)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(120*d^4*
e*(d + e*x)^8) - (59*(d^2 - e^2*x^2)^(1/2))/(3120*d^5*e*(d + e*x)^7) - (313*(d^2 - e^2*x^2)^(1/2))/(11440*d^6*
e*(d + e*x)^6) - (149*(d^2 - e^2*x^2)^(1/2))/(4576*d^7*e*(d + e*x)^5) - (647*(d^2 - e^2*x^2)^(1/2))/(18304*d^8
*e*(d + e*x)^4) + (256*x*(d^2 - e^2*x^2)^(1/2))/(2145*d^11*(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 )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**5), x)

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