3.808 \(\int \frac{(d^2-e^2 x^2)^{7/2}}{(d+e x)^6} \, dx\)

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

[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
)

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Rubi [A]  time = 0.0587709, antiderivative size = 145, 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} \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac{35 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{35 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

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 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)^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*}

Mathematica [A]  time = 0.0972618, size = 87, normalized size = 0.6 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (229 d^2 e x+164 d^3+30 d e^2 x^2-3 e^3 x^3\right )}{(d+e x)^2}+105 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{6 e} \]

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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Maple [B]  time = 0.051, size = 407, normalized size = 2.8 \begin{align*} -{\frac{1}{3\,{e}^{7}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 ) ^{-6}}+{\frac{1}{{e}^{6}{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 ) ^{-5}}+4\,{\frac{1}{{e}^{5}{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-4}}+{\frac{20}{3\,{e}^{4}{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 ) ^{-3}}+8\,{\frac{1}{{e}^{3}{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}+8\,{\frac{1}{e{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2}}+{\frac{28\,x}{3\,{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{35\,x}{3\,{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{35\,x}{2}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{35\,{d}^{2}}{2}\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)^6,x)

[Out]

-1/3/e^7/d/(d/e+x)^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+1/e^6/d^2/(d/e+x)^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(
9/2)+4/e^5/d^3/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+20/3/e^4/d^4/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/
e+x))^(9/2)+8/e^3/d^5/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(9/2)+8/e/d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7
/2)+28/3/d^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+35/3/d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+35/2*(-(d/
e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+35/2*d^2/(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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.30443, size = 312, normalized size = 2.15 \begin{align*} \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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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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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)^6,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError