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

Optimal. Leaf size=138 \[ -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

[Out]

-14/3*(-e^2*x^2+d^2)^(3/2)/e/(e*x+d)^2+14/15*(-e^2*x^2+d^2)^(5/2)/e/(e*x+d)^4-2/5*(-e^2*x^2+d^2)^(7/2)/e/(e*x+
d)^6-7*d*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e-7*(-e^2*x^2+d^2)^(1/2)/e

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Rubi [A]  time = 0.06, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {663, 665, 217, 203} \[ -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[d^2 - e^2*x^2])/e - (14*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^2) + (14*(d^2 - e^2*x^2)^(5/2))/(15*e*(
d + e*x)^4) - (2*(d^2 - e^2*x^2)^(7/2))/(5*e*(d + e*x)^6) - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 663

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

Rule 665

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

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^7} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7}{5} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx\\ &=\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}+\frac {7}{3} \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\\ &=-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-7 \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx\\ &=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-(7 d) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-(7 d) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {14 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^2}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{15 e (d+e x)^4}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 87, normalized size = 0.63 \[ -\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {\sqrt {d^2-e^2 x^2} \left (167 d^3+381 d^2 e x+277 d e^2 x^2+15 e^3 x^3\right )}{15 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(Sqrt[d^2 - e^2*x^2]*(167*d^3 + 381*d^2*e*x + 277*d*e^2*x^2 + 15*e^3*x^3))/(e*(d + e*x)^3) - (7*d*ArcTan
[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

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fricas [A]  time = 0.73, size = 172, normalized size = 1.25 \[ -\frac {167 \, d e^{3} x^{3} + 501 \, d^{2} e^{2} x^{2} + 501 \, d^{3} e x + 167 \, d^{4} - 210 \, {\left (d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{3} x^{3} + 277 \, d e^{2} x^{2} + 381 \, d^{2} e x + 167 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(167*d*e^3*x^3 + 501*d^2*e^2*x^2 + 501*d^3*e*x + 167*d^4 - 210*(d*e^3*x^3 + 3*d^2*e^2*x^2 + 3*d^3*e*x +
d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^3*x^3 + 277*d*e^2*x^2 + 381*d^2*e*x + 167*d^3)*sqrt(-e^
2*x^2 + d^2))/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-3840*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^30*exp(2)^2-4800*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^8*exp(1)^28*exp(2)^3-3200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^26*exp
(2)^4-1200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^24*exp(2)^5-240*d*(-1/2*(-2
*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^22*exp(2)^6-4608*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^30*exp(2)^2-13824*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^7*exp(1)^28*exp(2)^3-17280*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^26*
exp(2)^4-11520*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^24*exp(2)^5-4800*d*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^22*exp(2)^6-960*d*(-1/2*(-2*d*exp(1)-2*sqrt(d
^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^20*exp(2)^7-3840*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)/x/exp(2))^5*exp(1)^30*exp(2)^2-14272*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^
28*exp(2)^3-23616*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^26*exp(2)^4-16560*d*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^24*exp(2)^5-9040*d*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^22*exp(2)^6-4800*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^10*exp(1)^20*exp(2)^7-960*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(
1)^18*exp(2)^8-13824*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^28*exp(2)^3-108544
*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^26*exp(2)^4-164352*d*(-1/2*(-2*d*exp(1
)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^24*exp(2)^5-102720*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^8*exp(1)^22*exp(2)^6-34880*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^9*exp(1)^20*exp(2)^7-5760*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^18*exp(2)^
8-4800*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^28*exp(2)^3-23616*d*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^26*exp(2)^4-7136*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))/x/exp(2))^6*exp(1)^24*exp(2)^5+129312*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/ex
p(2))^7*exp(1)^22*exp(2)^6+176160*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^20*ex
p(2)^7+101120*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^18*exp(2)^8+31680*d*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^16*exp(2)^9+4320*d*(-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^14*exp(2)^10-17280*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^4*exp(1)^26*exp(2)^4-348672*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp
(1)^24*exp(2)^5-320448*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^22*exp(2)^6+1906
56*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^20*exp(2)^7+390000*d*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^18*exp(2)^8+229920*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*e
xp(2))*exp(1))/x/exp(2))^9*exp(1)^16*exp(2)^9+63720*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(
2))^10*exp(1)^14*exp(2)^10+6840*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^12*exp
(2)^11-3200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^26*exp(2)^4-28080*d*(-1/2*(
-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^24*exp(2)^5+6432*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2
-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^22*exp(2)^6+703608*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^6*exp(1)^20*exp(2)^7+1276104*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)
^18*exp(2)^8+1034070*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^16*exp(2)^9+453300
*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^14*exp(2)^10+102465*d*(-1/2*(-2*d*exp(
1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^12*exp(2)^11+9315*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^11*exp(1)^10*exp(2)^12-11520*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^3*exp(1)^24*exp(2)^5-486720*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^22*exp
(2)^6+244416*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^20*exp(2)^7+1683520*d*(-1/
2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^18*exp(2)^8+1741440*d*(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^16*exp(2)^9+873240*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*e
xp(1))/x/exp(2))^8*exp(1)^14*exp(2)^10+239240*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*
exp(1)^12*exp(2)^11+33960*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^10*exp(2)^12
+2040*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^8*exp(2)^13-1200*d*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^24*exp(2)^5-16720*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))/x/exp(2))^3*exp(1)^22*exp(2)^6+51480*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^4*exp(1)^20*exp(2)^7+2034984*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^18*ex
p(2)^8+3505280*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^16*exp(2)^9+2412120*d*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^14*exp(2)^10+727125*d*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^12*exp(2)^11+40825*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))
*exp(1))/x/exp(2))^9*exp(1)^10*exp(2)^12-23430*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^1
0*exp(1)^8*exp(2)^13-3570*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^11*exp(1)^6*exp(2)^14-
3840*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^22*exp(2)^6-343680*d*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^20*exp(2)^7+1005600*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^
2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^18*exp(2)^8+2777280*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^5*exp(1)^16*exp(2)^9+2370640*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^1
4*exp(2)^10+883440*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^12*exp(2)^11+32040*d
*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^10*exp(2)^12-92760*d*(-1/2*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^8*exp(2)^13-31200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^10*exp(1)^6*exp(2)^14-3360*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^1
1*exp(1)^4*exp(2)^15-4200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^20*exp(2)^7+6
5640*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^18*exp(2)^8+2729580*d*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^16*exp(2)^9+3564960*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^14*exp(2)^10+1142210*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^6*exp(1)^12*exp(2)^11-504930*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)
^10*exp(2)^12-450150*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^8*exp(2)^13-121830
*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^6*exp(2)^14-15000*d*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(1)^4*exp(2)^15-840*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^11*exp(2)^17-133824*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)
^18*exp(2)^8+933120*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^16*exp(2)^9+2158320
*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^14*exp(2)^10+1214160*d*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^12*exp(2)^11-465840*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))/x/exp(2))^6*exp(1)^10*exp(2)^12-849360*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^7*exp(1)^8*exp(2)^13-394080*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^6*ex
p(2)^14-79200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^4*exp(2)^15-40*d*exp(1)^2
0*exp(2)^7-7200*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*exp(2)^17+33264*d*(-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^16*exp(2)^9+1947560*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^14*exp(2)^10+1342290*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)/x/exp(2))^4*exp(1)^12*exp(2)^11-919470*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1
)^10*exp(2)^12-1286860*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^8*exp(2)^13-5661
00*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^6*exp(2)^14-134520*d*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^4*exp(2)^15-144*d*exp(1)^18*exp(2)^8-19800*d*(-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(2)^17+410568*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))/x/exp(2))^2*exp(1)^14*exp(2)^10+991320*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^3*exp(1)^12*exp(2)^11-178800*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^10*exp(2)
^12-1358640*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^8*exp(2)^13-990400*d*(-1/2*
(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^6*exp(2)^14-283200*d*(-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^4*exp(2)^15-146*d*exp(1)^16*exp(2)^9-36000*d*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(2)^17+779757*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^2*exp(1)^12*exp(2)^11-131585*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^1
0*exp(2)^12-1126140*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^8*exp(2)^13-849900*
d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^6*exp(2)^14-319600*d*(-1/2*(-2*d*exp(1)
-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^4*exp(2)^15-2312*d*exp(1)^14*exp(2)^10-70800*d*(-1/2*(-2*d*
exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(2)^17+293256*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))
*exp(1))/x/exp(2))^2*exp(1)^10*exp(2)^12-550440*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^
3*exp(1)^8*exp(2)^13-918720*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^6*exp(2)^14
-408000*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^4*exp(2)^15+1097*d*exp(1)^12*ex
p(2)^11-72000*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(2)^17-251406*d*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^8*exp(2)^13-470970*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^3*exp(1)^6*exp(2)^14-301680*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^4*exp(1)^4*exp(2)^15+8136*d*exp(1)^10*exp(2)^12-102000*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(
1))/x/exp(2))^5*exp(2)^17-286560*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^6*exp(
2)^14-266400*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^4*exp(2)^15+14594*d*exp(1)
^8*exp(2)^13-72000*d*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^17-107640*d*(-1/2*(-
2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^4*exp(2)^15+4640*d*exp(1)^6*exp(2)^14-66600*d*(-1
/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^17-6040*d*exp(1)^4*exp(2)^15-36000*d*(-1/2*(
-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^17-7200*d*exp(2)^17-1280*d*(-1/2*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^32*exp(2)+8220*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*e
xp(2)^17/x/exp(2)+32880*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^15/x/exp(2)+34455*d*(-2*
d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^14/x/exp(2)-26820*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))*exp(1)^8*exp(2)^13/x/exp(2)-165813/2*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^10*exp(2)^
12/x/exp(2)-45396*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^12*exp(2)^11/x/exp(2)-4422*d*(-2*d*exp(
1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^14*exp(2)^10/x/exp(2)+13872*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))*exp(1)^16*exp(2)^9/x/exp(2)+396*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^18*exp(2)^8/x/exp(2)
+384*d*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^20*exp(2)^7/x/exp(2)+120*d*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))*exp(1)^22*exp(2)^6/x/exp(2))/((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2
*exp(2)-(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))^6/(-120*exp(1)^23-480*exp(1)^19*exp(2)^2+480*exp
(1)^17*exp(2)^3+1200*exp(1)^15*exp(2)^4+480*exp(1)^13*exp(2)^5-480*exp(1)^11*exp(2)^6-480*exp(1)^9*exp(2)^7-12
0*exp(1)^7*exp(2)^8-480*exp(1)^21*exp(2))+1/2*(64*d*exp(1)^18*exp(2)^4-240*d*exp(1)^16*exp(2)^5-840*d*exp(1)^1
4*exp(2)^6-1133*d*exp(1)^12*exp(2)^7+248*d*exp(1)^10*exp(2)^8+1342*d*exp(1)^8*exp(2)^9+672*d*exp(1)^6*exp(2)^1
0-392*d*exp(1)^4*exp(2)^11-560*d*exp(2)^13)*atan((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))/s
qrt(-exp(1)^4+exp(2)^2))/sqrt(-exp(1)^4+exp(2)^2)/(4*exp(1)^25+16*exp(1)^21*exp(2)^2-16*exp(1)^19*exp(2)^3-40*
exp(1)^17*exp(2)^4-16*exp(1)^15*exp(2)^5+16*exp(1)^13*exp(2)^6+16*exp(1)^11*exp(2)^7+4*exp(1)^9*exp(2)^8+16*ex
p(1)^23*exp(2))-7*d*sign(d)*asin(x*exp(2)/d/exp(1))/exp(1)-4*1/4/exp(1)*sqrt(d^2-x^2*exp(2))

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maple [B]  time = 0.05, size = 454, normalized size = 3.29 \[ -\frac {7 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}}-\frac {7 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, x}{d}-\frac {14 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} x}{3 d^{3}}-\frac {56 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} x}{15 d^{5}}-\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{5 d^{6} e}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{5 \left (x +\frac {d}{e}\right )^{7} d \,e^{8}}+\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{15 \left (x +\frac {d}{e}\right )^{6} d^{2} e^{7}}-\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{5 \left (x +\frac {d}{e}\right )^{5} d^{3} e^{6}}-\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{5 \left (x +\frac {d}{e}\right )^{4} d^{4} e^{5}}-\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{3 \left (x +\frac {d}{e}\right )^{3} d^{5} e^{4}}-\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{5 \left (x +\frac {d}{e}\right )^{2} d^{6} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/e^8/d/(x+d/e)^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+2/15/e^7/d^2/(x+d/e)^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^
(9/2)-2/5/e^6/d^3/(x+d/e)^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)-8/5/e^5/d^4/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2
*e^2)^(9/2)-8/3/e^4/d^5/(x+d/e)^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)-16/5/e^3/d^6/(x+d/e)^2*(2*(x+d/e)*d*e-(x
+d/e)^2*e^2)^(9/2)-16/5/e/d^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)-56/15/d^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2
)*x-14/3/d^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x-7/d*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x-7*d/(e^2)^(1/2)*a
rctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x)

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maxima [B]  time = 3.07, size = 401, normalized size = 2.91 \[ \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e} + \frac {42 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{5 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {49 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{15 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {7 \, d \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {266 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{15 \, {\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-e^2*x^2 + d^2)^(7/2)/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e^2*x
+ d^6*e) - 7/5*(-e^2*x^2 + d^2)^(5/2)*d/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*
x + d^5*e) - 7*(-e^2*x^2 + d^2)^(3/2)*d^2/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e) + 42/5
*sqrt(-e^2*x^2 + d^2)*d^3/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) + 7/3*(-e^2*x^2 + d^2)^(3/2)*d/(e^4*x^
3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) + 49/15*sqrt(-e^2*x^2 + d^2)*d^2/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 7*d*ar
csin(e*x/d)/e - 266/15*sqrt(-e^2*x^2 + d^2)*d/(e^2*x + d*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^7} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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