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

Optimal. Leaf size=160 \[ -\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{19 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5} \]

[Out]

-(d^3*(d - e*x)^4)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (19*d^2*(d - e*x)^3)/(15*e^5*(d^2 - e^2*x^2)^(3/2)) - (6*d*
(d - e*x)^2)/(e^5*Sqrt[d^2 - e^2*x^2]) - ((20*d - e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (19*d^2*ArcTan[(e*x)/Sqr
t[d^2 - e^2*x^2]])/(2*e^5)

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Rubi [A]  time = 0.415125, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {852, 1635, 780, 217, 203} \[ -\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{19 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(d - e*x)^4)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (19*d^2*(d - e*x)^3)/(15*e^5*(d^2 - e^2*x^2)^(3/2)) - (6*d*
(d - e*x)^2)/(e^5*Sqrt[d^2 - e^2*x^2]) - ((20*d - e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (19*d^2*ArcTan[(e*x)/Sqr
t[d^2 - e^2*x^2]])/(2*e^5)

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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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^4 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx &=\int \frac{x^4 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{(d-e x)^3 \left (\frac{4 d^4}{e^4}-\frac{5 d^3 x}{e^3}+\frac{5 d^2 x^2}{e^2}-\frac{5 d x^3}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{(d-e x)^2 \left (\frac{45 d^4}{e^4}-\frac{30 d^3 x}{e^3}+\frac{15 d^2 x^2}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{\left (\frac{135 d^4}{e^4}-\frac{15 d^3 x}{e^3}\right ) (d-e x)}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{\left (19 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^4}\\ &=-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{\left (19 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 )}{2 e^4}\\ &=-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{19 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5}\\ \end{align*}

Mathematica [A]  time = 0.203246, size = 98, normalized size = 0.61 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (713 d^2 e^2 x^2+1059 d^3 e x+448 d^4+75 d e^3 x^3-15 e^4 x^4\right )}{(d+e x)^3}+285 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{30 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(448*d^4 + 1059*d^3*e*x + 713*d^2*e^2*x^2 + 75*d*e^3*x^3 - 15*e^4*x^4))/(d + e*x)^3 + 2
85*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^5)

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Maple [A]  time = 0.1, size = 273, normalized size = 1.7 \begin{align*}{\frac{x}{2\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{2}}{2\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{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 ) ^{-4}}+{\frac{19\,{d}^{2}}{15\,{e}^{8}} \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}}-10\,{\frac{d}{{e}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-10\,{\frac{{d}^{2}}{{e}^{4}\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 ) }-6\,{\frac{d}{{e}^{7}} \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^4*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

1/2/e^4*x*(-e^2*x^2+d^2)^(1/2)+1/2/e^4*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/5*d^3/e^9/
(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)+19/15/e^8*d^2/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)-10
/e^5*d*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)-10/e^4*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))-6/e^7*d/(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^4*(-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 = 1.75078, size = 414, normalized size = 2.59 \begin{align*} -\frac{448 \, d^{2} e^{3} x^{3} + 1344 \, d^{3} e^{2} x^{2} + 1344 \, d^{4} e x + 448 \, d^{5} - 570 \,{\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x + d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{4} x^{4} - 75 \, d e^{3} x^{3} - 713 \, d^{2} e^{2} x^{2} - 1059 \, d^{3} e x - 448 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(448*d^2*e^3*x^3 + 1344*d^3*e^2*x^2 + 1344*d^4*e*x + 448*d^5 - 570*(d^2*e^3*x^3 + 3*d^3*e^2*x^2 + 3*d^4*
e*x + d^5)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (15*e^4*x^4 - 75*d*e^3*x^3 - 713*d^2*e^2*x^2 - 1059*d^3
*e*x - 448*d^4)*sqrt(-e^2*x^2 + d^2))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \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**4*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

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

[Out]

Exception raised: NotImplementedError