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

Optimal. Leaf size=124 \[ \frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{5 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e} \]

[Out]

(5*d^5*x*Sqrt[d^2 - e^2*x^2])/16 + (5*d^3*x*(d^2 - e^2*x^2)^(3/2))/24 + (d*x*(d^2 - e^2*x^2)^(5/2))/6 + (d^2 -
e^2*x^2)^(7/2)/(7*e) + (5*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e)

________________________________________________________________________________________

Rubi [A]  time = 0.0385643, antiderivative size = 124, normalized size of antiderivative = 1., 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 = {665, 195, 217, 203} \[ \frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{5 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d^5*x*Sqrt[d^2 - e^2*x^2])/16 + (5*d^3*x*(d^2 - e^2*x^2)^(3/2))/24 + (d*x*(d^2 - e^2*x^2)^(5/2))/6 + (d^2 -
e^2*x^2)^(7/2)/(7*e) + (5*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e)

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]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 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])

Rubi steps

\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx &=\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{1}{6} \left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{1}{8} \left (5 d^5\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{1}{16} \left (5 d^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{1}{16} \left (5 d^7\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{5}{16} d^5 x \sqrt{d^2-e^2 x^2}+\frac{5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac{5 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e}\\ \end{align*}

Mathematica [A]  time = 0.0969637, size = 113, normalized size = 0.91 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+231 d^5 e x+48 d^6+56 d e^5 x^5-48 e^6 x^6\right )+105 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{336 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(48*d^6 + 231*d^5*e*x - 144*d^4*e^2*x^2 - 182*d^3*e^3*x^3 + 144*d^2*e^4*x^4 + 56*d*e^5*x^
5 - 48*e^6*x^6) + 105*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(336*e)

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 181, normalized size = 1.5 \begin{align*}{\frac{1}{7\,e} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{dx}{6} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{3}x}{24} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{5}x}{16}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{5\,{d}^{7}}{16}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7/e*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+1/6*d*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+5/24*d^3*(-(d/e+x)^2*e
^2+2*d*e*(d/e+x))^(3/2)*x+5/16*d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+5/16*d^7/(e^2)^(1/2)*arctan((e^2)^(1
/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))

________________________________________________________________________________________

Maxima [C]  time = 1.59661, size = 174, normalized size = 1.4 \begin{align*} -\frac{5 i \, d^{7} \arcsin \left (\frac{e x}{d} + 2\right )}{16 \, e} + \frac{5}{16} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x + \frac{5 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{8 \, e} + \frac{5}{24} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3} x + \frac{1}{6} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}{7 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*I*d^7*arcsin(e*x/d + 2)/e + 5/16*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^5*x + 5/8*sqrt(e^2*x^2 + 4*d*e*x + 3*
d^2)*d^6/e + 5/24*(-e^2*x^2 + d^2)^(3/2)*d^3*x + 1/6*(-e^2*x^2 + d^2)^(5/2)*d*x + 1/7*(-e^2*x^2 + d^2)^(7/2)/e

________________________________________________________________________________________

Fricas [A]  time = 2.15538, size = 257, normalized size = 2.07 \begin{align*} -\frac{210 \, d^{7} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (48 \, e^{6} x^{6} - 56 \, d e^{5} x^{5} - 144 \, d^{2} e^{4} x^{4} + 182 \, d^{3} e^{3} x^{3} + 144 \, d^{4} e^{2} x^{2} - 231 \, d^{5} e x - 48 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{336 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*(210*d^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (48*e^6*x^6 - 56*d*e^5*x^5 - 144*d^2*e^4*x^4 + 182
*d^3*e^3*x^3 + 144*d^4*e^2*x^2 - 231*d^5*e*x - 48*d^6)*sqrt(-e^2*x^2 + d^2))/e

________________________________________________________________________________________

Sympy [C]  time = 14.1586, size = 818, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**5*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(2*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, Tru
e)) - d**4*e*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - 2*d**3
*e**2*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 3*I*d*x**3/(8*s
qrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**4*
asin(e*x/d)/(8*e**3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2
*x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) + 2*d**2*e**3*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4)
- d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4,
True)) + d*e**4*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d
**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sq
rt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1
- e**2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e
**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2)), True)) - e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) -
4*d**4*x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 -
e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True))

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError