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]

(-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

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Rubi [A]  time = 0.0588687, antiderivative size = 138, 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 = {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 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]

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)^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*}

Mathematica [A]  time = 0.0871775, size = 87, normalized size = 0.63 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (381 d^2 e x+167 d^3+277 d e^2 x^2+15 e^3 x^3\right )}{15 e (d+e x)^3}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.051, size = 454, normalized size = 3.3 \begin{align*} -{\frac{1}{5\,{e}^{8}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-7}}+{\frac{2}{15\,{e}^{7}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-6}}-{\frac{2}{5\,{e}^{6}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-5}}-{\frac{8}{5\,{e}^{5}{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{8}{3\,{e}^{4}{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{16}{5\,{e}^{3}{d}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{16}{5\,e{d}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{56\,x}{15\,{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{14\,x}{3\,{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-7\,{\frac{x}{d}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-7\,{\frac{d}{\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ) } \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)^7,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.56656, size = 379, normalized size = 2.75 \begin{align*} -\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 )}} \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)^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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**7,x)

[Out]

Timed out

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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)^7,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError