3.9.8 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^6} \, dx\) [808]

Optimal. Leaf size=145 \[ \frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {35 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

35/6*(-e^2*x^2+d^2)^(3/2)/e/(e*x+d)+14/3*(-e^2*x^2+d^2)^(5/2)/e/(e*x+d)^3-2/3*(-e^2*x^2+d^2)^(7/2)/e/(e*x+d)^5
+35/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e+35/2*d*(-e^2*x^2+d^2)^(1/2)/e

________________________________________________________________________________________

Rubi [A]
time = 0.05, 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 = {677, 679, 223, 209} \begin {gather*} \frac {35 d^2 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}-\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} \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 209

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

Rule 223

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 677

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 679

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 ) \text {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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.42, size = 107, normalized size = 0.74 \begin {gather*} \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}-\frac {35 d^2 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 \sqrt {-e^2}} \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))/(6*e*(d + e*x)^2) - (35*d^2*Log[-(Sqr
t[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2*Sqrt[-e^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(506\) vs. \(2(125)=250\).
time = 0.51, size = 507, normalized size = 3.50

method	result	size
risch	\(\frac {\left (-e x +12 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e}+\frac {35 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {16 d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 e^{3} \left (x +\frac {d}{e}\right )^{2}}+\frac {80 d^{2} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 e^{2} \left (x +\frac {d}{e}\right )}\)	\(151\)
default	\(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{6}}-\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{5}}-\frac {4 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}\right )}{d}\right )}{d}}{e^{6}}\)	\(507\)

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,method=_RETURNVERBOSE)

[Out]

1/e^6*(-1/3/d/e/(x+d/e)^6*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)-e/d*(-1/d/e/(x+d/e)^5*(-e^2*(x+d/e)^2+2*d*e*(x+
d/e))^(9/2)-4*e/d*(1/d/e/(x+d/e)^4*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+5*e/d*(1/3/d/e/(x+d/e)^3*(-e^2*(x+d/e)
^2+2*d*e*(x+d/e))^(9/2)+2*e/d*(1/5/d/e/(x+d/e)^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(9/2)+7/5*e/d*(1/7*(-e^2*(x+d/
e)^2+2*d*e*(x+d/e))^(7/2)+d*e*(-1/12*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(5/2)+5/6*d^2*(
-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/e)+2*d*e)/e^2*
(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(
1/2)))))))))))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (119) = 238\).
time = 0.51, size = 256, normalized size = 1.77 \begin {gather*} \frac {35}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{2 \, {\left (x^{5} e^{6} + 5 \, d x^{4} e^{5} + 10 \, d^{2} x^{3} e^{4} + 10 \, d^{3} x^{2} e^{3} + 5 \, d^{4} x e^{2} + d^{5} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\left (x^{4} e^{5} + 4 \, d x^{3} e^{4} + 6 \, d^{2} x^{2} e^{3} + 4 \, d^{3} x e^{2} + d^{4} e\right )}} - \frac {35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {35 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{3 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} + \frac {245 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}}{6 \, {\left (x e^{2} + 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]

35/2*d^2*arcsin(x*e/d)*e^(-1) + 1/2*(-x^2*e^2 + d^2)^(7/2)/(x^5*e^6 + 5*d*x^4*e^5 + 10*d^2*x^3*e^4 + 10*d^3*x^
2*e^3 + 5*d^4*x*e^2 + d^5*e) + 7/2*(-x^2*e^2 + d^2)^(5/2)*d/(x^4*e^5 + 4*d*x^3*e^4 + 6*d^2*x^2*e^3 + 4*d^3*x*e
^2 + d^4*e) - 35/6*(-x^2*e^2 + d^2)^(3/2)*d^2/(x^3*e^4 + 3*d*x^2*e^3 + 3*d^2*x*e^2 + d^3*e) - 35/3*sqrt(-x^2*e
^2 + d^2)*d^3/(x^2*e^3 + 2*d*x*e^2 + d^2*e) + 245/6*sqrt(-x^2*e^2 + d^2)*d^2/(x*e^2 + d*e)

________________________________________________________________________________________

Fricas [A]
time = 2.27, size = 139, normalized size = 0.96 \begin {gather*} \frac {164 \, d^{2} x^{2} e^{2} + 328 \, d^{3} x e + 164 \, d^{4} - 210 \, {\left (d^{2} x^{2} e^{2} + 2 \, d^{3} x e + d^{4}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (3 \, x^{3} e^{3} - 30 \, d x^{2} e^{2} - 229 \, d^{2} x e - 164 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + 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*x^2*e^2 + 328*d^3*x*e + 164*d^4 - 210*(d^2*x^2*e^2 + 2*d^3*x*e + d^4)*arctan(-(d - sqrt(-x^2*e^2
+ d^2))*e^(-1)/x) - (3*x^3*e^3 - 30*d*x^2*e^2 - 229*d^2*x*e - 164*d^3)*sqrt(-x^2*e^2 + d^2))/(x^2*e^3 + 2*d*x*
e^2 + d^2*e)

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [A]
time = 1.23, size = 147, normalized size = 1.01 \begin {gather*} \frac {35}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (12 \, d e^{\left (-1\right )} - x\right )} - \frac {32 \, {\left (\frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{\left (-2\right )}}{x} + \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{\left (-4\right )}}{x^{2}} + 4 \, d^{2}\right )} e^{\left (-1\right )}}{3 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{3}} \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]

35/2*d^2*arcsin(x*e/d)*e^(-1)*sgn(d) + 1/2*sqrt(-x^2*e^2 + d^2)*(12*d*e^(-1) - x) - 32/3*(9*(d*e + sqrt(-x^2*e
^2 + d^2)*e)*d^2*e^(-2)/x + 3*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*e^(-4)/x^2 + 4*d^2)*e^(-1)/((d*e + sqrt(-x^
2*e^2 + d^2)*e)*e^(-2)/x + 1)^3

________________________________________________________________________________________

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)

________________________________________________________________________________________