3.9.59 \(\int \frac {1}{(d+e x)^5 (d^2-e^2 x^2)^{7/2}} \, dx\) [859]

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

[Out]

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

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Rubi [A]
time = 0.08, 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 = {673, 198, 197} \begin {gather*} -\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}} \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 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]

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 = 0.66, 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs. \(2(206)=412\).
time = 0.48, size = 431, normalized size = 1.81

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-256 e^{10} x^{10}-1280 d \,e^{9} x^{9}-1920 d^{2} e^{8} x^{8}+640 e^{7} x^{7} d^{3}+4640 e^{6} x^{6} d^{4}+3744 d^{5} e^{5} x^{5}-1520 d^{6} e^{4} x^{4}-3760 e^{3} x^{3} d^{7}-1590 d^{8} e^{2} x^{2}+370 x \,d^{9} e +503 d^{10}\right )}{2145 \left (e x +d \right )^{4} d^{11} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(143\)
trager	\(-\frac {\left (-256 e^{10} x^{10}-1280 d \,e^{9} x^{9}-1920 d^{2} e^{8} x^{8}+640 e^{7} x^{7} d^{3}+4640 e^{6} x^{6} d^{4}+3744 d^{5} e^{5} x^{5}-1520 d^{6} e^{4} x^{4}-3760 e^{3} x^{3} d^{7}-1590 d^{8} e^{2} x^{2}+370 x \,d^{9} e +503 d^{10}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2145 d^{11} \left (e x +d \right )^{8} \left (-e x +d \right )^{3} e}\)	\(145\)
default	\(\frac {-\frac {1}{15 d e \left (x +\frac {d}{e}\right )^{5} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {2 e \left (-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{4} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {9 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}\right )}{13 d}\right )}{3 d}}{e^{5}}\)	\(431\)

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,method=_RETURNVERBOSE)

[Out]

1/e^5*(-1/15/d/e/(x+d/e)^5/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+2/3*e/d*(-1/13/d/e/(x+d/e)^4/(-e^2*(x+d/e)^2+2
*d*e*(x+d/e))^(5/2)+9/13*e/d*(-1/11/d/e/(x+d/e)^3/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+8/11*e/d*(-1/9/d/e/(x+d
/e)^2/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-1/7/d/e/(x+d/e)/(-e^2*(x+d/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/(-e^2*(x+d/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/(-e^2*(x+d/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)/(-e^2*(x+d/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. 506 vs. \(2 (198) = 396\).
time = 0.29, size = 506, normalized size = 2.13 \begin {gather*} -\frac {1}{15 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{5} e^{6} + 5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{4} e^{5} + 10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x^{3} e^{4} + 10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x^{2} e^{3} + 5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {2}{39 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{4} e^{5} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x^{3} e^{4} + 6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x^{2} e^{3} + 4 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {6}{143 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x^{3} e^{4} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x^{2} e^{3} + 3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {16}{429 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} - \frac {16}{429 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} e\right )}} + \frac {32 \, x}{715 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7}} + \frac {128 \, x}{2145 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9}} + \frac {256 \, x}{2145 \, \sqrt {-x^{2} e^{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/((-x^2*e^2 + d^2)^(5/2)*d*x^5*e^6 + 5*(-x^2*e^2 + d^2)^(5/2)*d^2*x^4*e^5 + 10*(-x^2*e^2 + d^2)^(5/2)*d^3
*x^3*e^4 + 10*(-x^2*e^2 + d^2)^(5/2)*d^4*x^2*e^3 + 5*(-x^2*e^2 + d^2)^(5/2)*d^5*x*e^2 + (-x^2*e^2 + d^2)^(5/2)
*d^6*e) - 2/39/((-x^2*e^2 + d^2)^(5/2)*d^2*x^4*e^5 + 4*(-x^2*e^2 + d^2)^(5/2)*d^3*x^3*e^4 + 6*(-x^2*e^2 + d^2)
^(5/2)*d^4*x^2*e^3 + 4*(-x^2*e^2 + d^2)^(5/2)*d^5*x*e^2 + (-x^2*e^2 + d^2)^(5/2)*d^6*e) - 6/143/((-x^2*e^2 + d
^2)^(5/2)*d^3*x^3*e^4 + 3*(-x^2*e^2 + d^2)^(5/2)*d^4*x^2*e^3 + 3*(-x^2*e^2 + d^2)^(5/2)*d^5*x*e^2 + (-x^2*e^2
+ d^2)^(5/2)*d^6*e) - 16/429/((-x^2*e^2 + d^2)^(5/2)*d^4*x^2*e^3 + 2*(-x^2*e^2 + d^2)^(5/2)*d^5*x*e^2 + (-x^2*
e^2 + d^2)^(5/2)*d^6*e) - 16/429/((-x^2*e^2 + d^2)^(5/2)*d^5*x*e^2 + (-x^2*e^2 + d^2)^(5/2)*d^6*e) + 32/715*x/
((-x^2*e^2 + d^2)^(5/2)*d^7) + 128/2145*x/((-x^2*e^2 + d^2)^(3/2)*d^9) + 256/2145*x/(sqrt(-x^2*e^2 + d^2)*d^11
)

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Fricas [A]
time = 6.04, size = 341, normalized size = 1.43 \begin {gather*} -\frac {503 \, x^{11} e^{11} + 2515 \, d x^{10} e^{10} + 3521 \, d^{2} x^{9} e^{9} - 2515 \, d^{3} x^{8} e^{8} - 11066 \, d^{4} x^{7} e^{7} - 7042 \, d^{5} x^{6} e^{6} + 7042 \, d^{6} x^{5} e^{5} + 11066 \, d^{7} x^{4} e^{4} + 2515 \, d^{8} x^{3} e^{3} - 3521 \, d^{9} x^{2} e^{2} - 2515 \, d^{10} x e - 503 \, d^{11} + {\left (256 \, x^{10} e^{10} + 1280 \, d x^{9} e^{9} + 1920 \, d^{2} x^{8} e^{8} - 640 \, d^{3} x^{7} e^{7} - 4640 \, d^{4} x^{6} e^{6} - 3744 \, d^{5} x^{5} e^{5} + 1520 \, d^{6} x^{4} e^{4} + 3760 \, d^{7} x^{3} e^{3} + 1590 \, d^{8} x^{2} e^{2} - 370 \, d^{9} x e - 503 \, d^{10}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{2145 \, {\left (d^{11} x^{11} e^{12} + 5 \, d^{12} x^{10} e^{11} + 7 \, d^{13} x^{9} e^{10} - 5 \, d^{14} x^{8} e^{9} - 22 \, d^{15} x^{7} e^{8} - 14 \, d^{16} x^{6} e^{7} + 14 \, d^{17} x^{5} e^{6} + 22 \, d^{18} x^{4} e^{5} + 5 \, d^{19} x^{3} e^{4} - 7 \, d^{20} x^{2} e^{3} - 5 \, d^{21} x e^{2} - 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*x^11*e^11 + 2515*d*x^10*e^10 + 3521*d^2*x^9*e^9 - 2515*d^3*x^8*e^8 - 11066*d^4*x^7*e^7 - 7042*d^5
*x^6*e^6 + 7042*d^6*x^5*e^5 + 11066*d^7*x^4*e^4 + 2515*d^8*x^3*e^3 - 3521*d^9*x^2*e^2 - 2515*d^10*x*e - 503*d^
11 + (256*x^10*e^10 + 1280*d*x^9*e^9 + 1920*d^2*x^8*e^8 - 640*d^3*x^7*e^7 - 4640*d^4*x^6*e^6 - 3744*d^5*x^5*e^
5 + 1520*d^6*x^4*e^4 + 3760*d^7*x^3*e^3 + 1590*d^8*x^2*e^2 - 370*d^9*x*e - 503*d^10)*sqrt(-x^2*e^2 + d^2))/(d^
11*x^11*e^12 + 5*d^12*x^10*e^11 + 7*d^13*x^9*e^10 - 5*d^14*x^8*e^9 - 22*d^15*x^7*e^8 - 14*d^16*x^6*e^7 + 14*d^
17*x^5*e^6 + 22*d^18*x^4*e^5 + 5*d^19*x^3*e^4 - 7*d^20*x^2*e^3 - 5*d^21*x*e^2 - d^22*e)

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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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Giac [C] Result contains complex when optimal does not.
time = 2.57, size = 369, normalized size = 1.55 \begin {gather*} \frac {1}{2196480} \, {\left ({\left (\frac {143 \, {\left (675 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{2} + \frac {100 \, d}{x e + d} - 47\right )} e^{\left (-10\right )}}{d^{11} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (143 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {15}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 1650 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 8775 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 28600 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 64350 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 108108 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 150150 \, d^{154} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14} + 257400 \, d^{154} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{140} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{14}\right )} e^{\left (-150\right )}}{d^{165} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{15}}\right )} e^{10} + \frac {262144 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{11}}\right )} e^{\left (-1\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="giac")

[Out]

1/2196480*((143*(675*(2*d/(x*e + d) - 1)^2 + 100*d/(x*e + d) - 47)*e^(-10)/(d^11*(2*d/(x*e + d) - 1)^(5/2)*sgn
(1/(x*e + d))) - (143*d^154*(2*d/(x*e + d) - 1)^(15/2)*e^140*sgn(1/(x*e + d))^14 + 1650*d^154*(2*d/(x*e + d) -
1)^(13/2)*e^140*sgn(1/(x*e + d))^14 + 8775*d^154*(2*d/(x*e + d) - 1)^(11/2)*e^140*sgn(1/(x*e + d))^14 + 28600
*d^154*(2*d/(x*e + d) - 1)^(9/2)*e^140*sgn(1/(x*e + d))^14 + 64350*d^154*(2*d/(x*e + d) - 1)^(7/2)*e^140*sgn(1
/(x*e + d))^14 + 108108*d^154*(2*d/(x*e + d) - 1)^(5/2)*e^140*sgn(1/(x*e + d))^14 + 150150*d^154*(2*d/(x*e + d
) - 1)^(3/2)*e^140*sgn(1/(x*e + d))^14 + 257400*d^154*sqrt(2*d/(x*e + d) - 1)*e^140*sgn(1/(x*e + d))^14)*e^(-1
50)/(d^165*sgn(1/(x*e + d))^15))*e^10 + 262144*I*sgn(1/(x*e + d))/d^11)*e^(-1)

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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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