3.188 \(\int \frac{x^5 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=204 \[ \frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}+\frac{18 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

[Out]

(d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(3/2)) + (10*d^2*
(d - e*x)^2)/(e^6*Sqrt[d^2 - e^2*x^2]) + (59*d^2*Sqrt[d^2 - e^2*x^2])/(3*e^6) - (2*d*x*Sqrt[d^2 - e^2*x^2])/e^
5 + (x^2*Sqrt[d^2 - e^2*x^2])/(3*e^4) + (18*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

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Rubi [A]  time = 0.593206, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {852, 1635, 1815, 641, 217, 203} \[ \frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}+\frac{18 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

(d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(3/2)) + (10*d^2*
(d - e*x)^2)/(e^6*Sqrt[d^2 - e^2*x^2]) + (59*d^2*Sqrt[d^2 - e^2*x^2])/(3*e^6) - (2*d*x*Sqrt[d^2 - e^2*x^2])/e^
5 + (x^2*Sqrt[d^2 - e^2*x^2])/(3*e^4) + (18*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

Rule 852

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1635

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*
a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q
+ f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p +
1/2, 0] && GtQ[m, 0]

Rule 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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{x^5 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx &=\int \frac{x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{(d-e x)^3 \left (-\frac{4 d^5}{e^5}+\frac{5 d^4 x}{e^4}-\frac{5 d^3 x^2}{e^3}+\frac{5 d^2 x^3}{e^2}-\frac{5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{(d-e x)^2 \left (-\frac{60 d^5}{e^5}+\frac{45 d^4 x}{e^4}-\frac{30 d^3 x^2}{e^3}+\frac{15 d^2 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{(d-e x) \left (-\frac{240 d^5}{e^5}+\frac{45 d^4 x}{e^4}-\frac{15 d^3 x^2}{e^3}\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{\int \frac{\frac{720 d^6}{e^3}-\frac{885 d^5 x}{e^2}+\frac{180 d^4 x^2}{e}}{\sqrt{d^2-e^2 x^2}} \, dx}{45 d^3 e^2}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}-\frac{\int \frac{-\frac{1620 d^6}{e}+1770 d^5 x}{\sqrt{d^2-e^2 x^2}} \, dx}{90 d^3 e^4}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{\left (18 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{\left (18 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac{d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{18 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.2061, size = 109, normalized size = 0.53 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (674 d^3 e^2 x^2+70 d^2 e^3 x^3+1002 d^4 e x+424 d^5-15 d e^4 x^4+5 e^5 x^5\right )}{(d+e x)^3}+270 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(424*d^5 + 1002*d^4*e*x + 674*d^3*e^2*x^2 + 70*d^2*e^3*x^3 - 15*d*e^4*x^4 + 5*e^5*x^5))/
(d + e*x)^3 + 270*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6)

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Maple [A]  time = 0.069, size = 297, normalized size = 1.5 \begin{align*} -{\frac{1}{3\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-2\,{\frac{dx\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}{{e}^{5}}}-2\,{\frac{{d}^{3}}{{e}^{5}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}} \right ) }+{\frac{{d}^{4}}{5\,{e}^{10}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{8\,{d}^{3}}{5\,{e}^{9}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+20\,{\frac{{d}^{2}}{{e}^{6}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+20\,{\frac{{d}^{3}}{{e}^{5}\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 ) }+10\,{\frac{{d}^{2}}{{e}^{8}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{3/2} \left ({\frac{d}{e}}+x \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/3/e^6*(-e^2*x^2+d^2)^(3/2)-2*d*x*(-e^2*x^2+d^2)^(1/2)/e^5-2/e^5*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*
x^2+d^2)^(1/2))+1/5*d^4/e^10/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)-8/5/e^9*d^3/(d/e+x)^3*(-(d/e+x)^2*
e^2+2*d*e*(d/e+x))^(3/2)+20/e^6*d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)+20/e^5*d^3/(e^2)^(1/2)*arctan((e^2)^(
1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))+10/e^8*d^2/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)

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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(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89485, size = 435, normalized size = 2.13 \begin{align*} \frac{424 \, d^{3} e^{3} x^{3} + 1272 \, d^{4} e^{2} x^{2} + 1272 \, d^{5} e x + 424 \, d^{6} - 540 \,{\left (d^{3} e^{3} x^{3} + 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x + d^{6}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (5 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} + 70 \, d^{2} e^{3} x^{3} + 674 \, d^{3} e^{2} x^{2} + 1002 \, d^{4} e x + 424 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/15*(424*d^3*e^3*x^3 + 1272*d^4*e^2*x^2 + 1272*d^5*e*x + 424*d^6 - 540*(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3*d^5*e
*x + d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (5*e^5*x^5 - 15*d*e^4*x^4 + 70*d^2*e^3*x^3 + 674*d^3*e^2
*x^2 + 1002*d^4*e*x + 424*d^5)*sqrt(-e^2*x^2 + d^2))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(x**5*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, 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(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError