3.7.49 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^6} \, dx\)
Optimal. Leaf size=145 \[ -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A] time = 0.06, antiderivative size = 145, 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} \begin {gather*} -\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^6,x]
[Out]
(35*d*Sqrt[d^2 - e^2*x^2])/(2*e) + (35*(d^2 - e^2*x^2)^(3/2))/(6*e*(d + e*x)) + (14*(d^2 - e^2*x^2)^(5/2))/(3*
e*(d + e*x)^3) - (2*(d^2 - e^2*x^2)^(7/2))/(3*e*(d + e*x)^5) + (35*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(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)^6} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}-\frac {7}{3} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\\ &=\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {35}{3} \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx\\ &=\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {1}{2} (35 d) \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx\\ &=\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {1}{2} \left (35 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {1}{2} \left (35 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {35 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 87, normalized size = 0.60 \begin {gather*} \frac {105 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (164 d^3+229 d^2 e x+30 d e^2 x^2-3 e^3 x^3\right )}{(d+e x)^2}}{6 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^6,x]
[Out]
((Sqrt[d^2 - e^2*x^2]*(164*d^3 + 229*d^2*e*x + 30*d*e^2*x^2 - 3*e^3*x^3))/(d + e*x)^2 + 105*d^2*ArcTan[(e*x)/S
qrt[d^2 - e^2*x^2]])/(6*e)
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IntegrateAlgebraic [A] time = 0.48, size = 110, normalized size = 0.76 \begin {gather*} \frac {35 d^2 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{2 e^2}+\frac {\sqrt {d^2-e^2 x^2} \left (164 d^3+229 d^2 e x+30 d e^2 x^2-3 e^3 x^3\right )}{6 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^6,x]
[Out]
(Sqrt[d^2 - e^2*x^2]*(164*d^3 + 229*d^2*e*x + 30*d*e^2*x^2 - 3*e^3*x^3))/(6*e*(d + e*x)^2) + (35*d^2*Sqrt[-e^2
]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2*e^2)
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fricas [A] time = 0.43, size = 144, normalized size = 0.99 \begin {gather*} \frac {164 \, d^{2} e^{2} x^{2} + 328 \, d^{3} e x + 164 \, d^{4} - 210 \, {\left (d^{2} e^{2} x^{2} + 2 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (3 \, e^{3} x^{3} - 30 \, d e^{2} x^{2} - 229 \, d^{2} e x - 164 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^6,x, algorithm="fricas")
[Out]
1/6*(164*d^2*e^2*x^2 + 328*d^3*e*x + 164*d^4 - 210*(d^2*e^2*x^2 + 2*d^3*e*x + d^4)*arctan(-(d - sqrt(-e^2*x^2
+ d^2))/(e*x)) - (3*e^3*x^3 - 30*d*e^2*x^2 - 229*d^2*e*x - 164*d^3)*sqrt(-e^2*x^2 + d^2))/(e^3*x^2 + 2*d*e^2*x
+ d^2*e)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^6,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: (960*d^2*(-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)^2+960*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/
exp(2))^7*exp(1)^22*exp(2)^3+480*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^20*e
xp(2)^4+120*d^2*(-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)^5+1280*d^2*(-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)^2+3200*d^2*(-1/2*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^22*exp(2)^3+3200*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^7*exp(1)^20*exp(2)^4+1920*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*ex
p(1)^18*exp(2)^5+480*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^16*exp(2)^6+960*
d^2*(-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)^2+5152*d^2*(-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)^3+3760*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^6*exp(1)^20*exp(2)^4+2800*d^2*(-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)^5+2400*d^2*(-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)^6+600*d^2*(-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)^7+3200*d^2*(-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)^3-53120*d^2*(-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)^4-106560*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^6*exp(1)^18*exp(2)^5-72000*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*e
xp(1)^16*exp(2)^6-21120*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^14*exp(2)^7-2
400*d^2*(-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)^8+960*d^2*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^22*exp(2)^3+15280*d^2*(-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)^4-81056*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^5*exp(1)^18*exp(2)^5-153520*d^2*(-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)^6-101680*d^2*(-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)^7-31440*
d^2*(-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)^8-3840*d^2*(-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)^9+3200*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*ex
p(2))*exp(1))/x/exp(2))^3*exp(1)^20*exp(2)^4-181440*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/ex
p(2))^4*exp(1)^18*exp(2)^5-304160*d^2*(-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)^6-170000*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^14*exp(2)^7-38960*d^2
*(-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)^8-3480*d^2*(-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)^9-120*d^2*(-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)^10+480*d^2*(-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)^4+14320*d^2*(-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)^5
-311680*d^2*(-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)^6-362660*d^2*(-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)^7-53970*d^2*(-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)^8+82170*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^7*exp(1)^10*exp(2)^9+40065*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*e
xp(1)^8*exp(2)^10+5385*d^2*(-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)^11+12
80*d^2*(-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)^5-222400*d^2*(-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)^6-320400*d^2*(-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)^7-48880*d^2*(-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)^8+152400*d^2*(-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)^9+122160*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^8*exp(2)^10+386
40*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp(1)^6*exp(2)^11+4560*d^2*(-1/2*(-2*d*e
xp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*exp(1)^4*exp(2)^12+5800*d^2*(-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)^6-379920*d^2*(-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)^7-73730*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^1
2*exp(2)^8+449970*d^2*(-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)^9+374070*
d^2*(-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)^10+120090*d^2*(-1/2*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^6*exp(2)^11+17880*d^2*(-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)^12+1200*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^9*exp(2)^14-127280*d^2*(-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)^7
-154640*d^2*(-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)^8+280800*d^2*(-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)^9+520800*d^2*(-1/2*(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^8*exp(2)^10+305760*d^2*(-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)^11+76320*d^2*(-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)^12+24*d^2*exp(1)^16*exp(2)^6+8520*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^8*exp(2)^14-212390*d^2*(-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)^8+2
77790*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^10*exp(2)^9+615920*d^2*(-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)^10+370800*d^2*(-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)^11+112560*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^6*exp(1)^4*exp(2)^12+80*d^2*exp(1)^14*exp(2)^7+20400*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))/x/exp(2))^7*exp(2)^14-37520*d^2*(-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)^9+393040*d^2*(-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)^10+49
5040*d^2*(-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)^11+201600*d^2*(-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)^12+202*d^2*exp(1)^12*exp(2)^8+34080*d^2*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(2)^14+231730*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^
2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^8*exp(2)^10+326310*d^2*(-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)^11+179040*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*ex
p(1)^4*exp(2)^12-3720*d^2*exp(1)^10*exp(2)^9+54000*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^5*exp(2)^14+227360*d^2*(-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)^11+1
92480*d^2*(-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)^12-6121*d^2*exp(1)^8*e
xp(2)^10+51120*d^2*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(2)^14+91920*d^2*(-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)^12-560*d^2*exp(1)^6*exp(2)^11+51600*d^2*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(2)^14+7560*d^2*exp(1)^4*exp(2)^12+34080*d^2*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)^14+8520*d^2*exp(2)^14+384*d^2*(-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)-8400*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2)
)*exp(1))*exp(2)^14/x/exp(2)-31320*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^4*exp(2)^12/x/exp(2)
-70215/2*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^6*exp(2)^11/x/exp(2)+2740*d^2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))*exp(1)^8*exp(2)^10/x/exp(2)+28685*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*
exp(1)^10*exp(2)^9/x/exp(2)+17400*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^12*exp(2)^8/x/exp(2)-
710*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^14*exp(2)^7/x/exp(2)-160*d^2*(-2*d*exp(1)-2*sqrt(d^
2-x^2*exp(2))*exp(1))*exp(1)^16*exp(2)^6/x/exp(2)-60*d^2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^18
*exp(2)^5/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))^5/(60*exp(1)^19+300*exp(1)^15*exp(2)^2-300*exp(1)^11*exp(2)^4-240*exp(1)^9*e
xp(2)^5-60*exp(1)^7*exp(2)^6+240*exp(1)^17*exp(2))+1/2*(-192*d^2*exp(1)^14*exp(2)^4-176*d^2*exp(1)^12*exp(2)^5
+536*d^2*exp(1)^10*exp(2)^6+1207*d^2*exp(1)^8*exp(2)^7+464*d^2*exp(1)^6*exp(2)^8-632*d^2*exp(1)^4*exp(2)^9-808
*d^2*exp(2)^11)*atan((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/sqr
t(-exp(1)^4+exp(2)^2)/(2*exp(1)^21+10*exp(1)^17*exp(2)^2-10*exp(1)^13*exp(2)^4-8*exp(1)^11*exp(2)^5-2*exp(1)^9
*exp(2)^6+8*exp(1)^19*exp(2))+35/2*d^2*sign(d)*asin(x*exp(2)/d/exp(1))/exp(1)+2*(-2*exp(1)*1/8/exp(1)*x+24*d*1
/8/exp(1))*sqrt(d^2-x^2*exp(2))
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maple [B] time = 0.05, size = 407, normalized size = 2.81 \begin {gather*} \frac {35 d^{2} \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 )}{2 \sqrt {e^{2}}}+\frac {35 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, x}{2}+\frac {35 \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^{2}}+\frac {28 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} x}{3 d^{4}}+\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{d^{5} 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}}}{3 \left (x +\frac {d}{e}\right )^{6} d \,e^{7}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{\left (x +\frac {d}{e}\right )^{5} d^{2} e^{6}}+\frac {4 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{\left (x +\frac {d}{e}\right )^{4} d^{3} e^{5}}+\frac {20 \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^{4} e^{4}}+\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}}}{\left (x +\frac {d}{e}\right )^{2} d^{5} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^6,x)
[Out]
-1/3/e^7/d/(x+d/e)^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+1/e^6/d^2/(x+d/e)^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/
2)+4/e^5/d^3/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+20/3/e^4/d^4/(x+d/e)^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2
)^(9/2)+8/e^3/d^5/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(9/2)+8/e/d^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)+28
/3/d^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)*x+35/3/d^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x+35/2*(2*(x+d/e)*d*
e-(x+d/e)^2*e^2)^(1/2)*x+35/2*d^2/(e^2)^(1/2)*arctan((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 = 2.95, size = 271, normalized size = 1.87 \begin {gather*} \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{2 \, {\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 {5}{2}} d}{2 \, {\left (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\right )}} - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{3 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {35 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e} + \frac {245 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{6 \, {\left (e^{2} x + d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^6,x, algorithm="maxima")
[Out]
1/2*(-e^2*x^2 + d^2)^(7/2)/(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
/2*(-e^2*x^2 + d^2)^(5/2)*d/(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) - 35/6*(-e^2*x^2 + d
^2)^(3/2)*d^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 35/3*sqrt(-e^2*x^2 + d^2)*d^3/(e^3*x^2 + 2*d*e^2
*x + d^2*e) + 35/2*d^2*arcsin(e*x/d)/e + 245/6*sqrt(-e^2*x^2 + d^2)*d^2/(e^2*x + d*e)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^6,x)
[Out]
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^6, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**6,x)
[Out]
Integral((-(-d + e*x)*(d + e*x))**(7/2)/(d + e*x)**6, x)
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