3.810 \(\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=143 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.041401, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {663, 217, 203} \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]

[Out]

(2*Sqrt[d^2 - e^2*x^2])/(e*(d + e*x)) - (2*(d^2 - e^2*x^2)^(3/2))/(3*e*(d + e*x)^3) + (2*(d^2 - e^2*x^2)^(5/2)
)/(5*e*(d + e*x)^5) - (2*(d^2 - e^2*x^2)^(7/2))/(7*e*(d + e*x)^7) + 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 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)^8} \, dx &=-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx\\ &=\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx\\ &=-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}-\int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^2} \, dx\\ &=\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{2 \sqrt{d^2-e^2 x^2}}{e (d+e x)}-\frac{2 \left (d^2-e^2 x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.0950117, size = 85, normalized size = 0.59 \[ \frac{8 \sqrt{d^2-e^2 x^2} \left (76 d^2 e x+19 d^3+71 d e^2 x^2+44 e^3 x^3\right )}{105 e (d+e x)^4}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^8,x]

[Out]

(8*Sqrt[d^2 - e^2*x^2]*(19*d^3 + 76*d^2*e*x + 71*d*e^2*x^2 + 44*e^3*x^3))/(105*e*(d + e*x)^4) + ArcTan[(e*x)/S
qrt[d^2 - e^2*x^2]]/e

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Maple [B]  time = 0.051, size = 496, normalized size = 3.5 \begin{align*} \text{result too large to display} \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)^8,x)

[Out]

-1/7/e^9/d/(d/e+x)^8*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+1/35/e^8/d^2/(d/e+x)^7*(-(d/e+x)^2*e^2+2*d*e*(d/e+x)
)^(9/2)-2/105/e^7/d^3/(d/e+x)^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+2/35/e^6/d^4/(d/e+x)^5*(-(d/e+x)^2*e^2+2*
d*e*(d/e+x))^(9/2)+8/35/e^5/d^5/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+8/21/e^4/d^6/(d/e+x)^3*(-(d/e+x
)^2*e^2+2*d*e*(d/e+x))^(9/2)+16/35/e^3/d^7/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+16/35/e/d^7*(-(d/e+x
)^2*e^2+2*d*e*(d/e+x))^(7/2)+8/15/d^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+2/3/d^4*(-(d/e+x)^2*e^2+2*d*e*(d/
e+x))^(3/2)*x+1/d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+1/(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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.5588, size = 432, normalized size = 3.02 \begin{align*} \frac{2 \,{\left (76 \, e^{4} x^{4} + 304 \, d e^{3} x^{3} + 456 \, d^{2} e^{2} x^{2} + 304 \, d^{3} e x + 76 \, d^{4} - 105 \,{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \,{\left (44 \, e^{3} x^{3} + 71 \, d e^{2} x^{2} + 76 \, d^{2} e x + 19 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}}{105 \,{\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 )}} \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)^8,x, algorithm="fricas")

[Out]

2/105*(76*e^4*x^4 + 304*d*e^3*x^3 + 456*d^2*e^2*x^2 + 304*d^3*e*x + 76*d^4 - 105*(e^4*x^4 + 4*d*e^3*x^3 + 6*d^
2*e^2*x^2 + 4*d^3*e*x + d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 4*(44*e^3*x^3 + 71*d*e^2*x^2 + 76*d^2
*e*x + 19*d^3)*sqrt(-e^2*x^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)

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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)**8,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)^8,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError