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

Optimal. Leaf size=224 \[ -\frac{337 d^5 \sqrt{d^2-e^2 x^2}}{15 e^5}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{239 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^5} \]

[Out]

-((d^3*(d - e*x)^4)/(e^5*Sqrt[d^2 - e^2*x^2])) - (337*d^5*Sqrt[d^2 - e^2*x^2])/(15*e^5) + (175*d^4*x*Sqrt[d^2
- e^2*x^2])/(16*e^4) - (71*d^3*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) + (47*d^2*x^3*Sqrt[d^2 - e^2*x^2])/(24*e^2) -
(4*d*x^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (x^5*Sqrt[d^2 - e^2*x^2])/6 - (239*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]
])/(16*e^5)

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Rubi [A]  time = 0.53395, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{337 d^5 \sqrt{d^2-e^2 x^2}}{15 e^5}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{239 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^3*(d - e*x)^4)/(e^5*Sqrt[d^2 - e^2*x^2])) - (337*d^5*Sqrt[d^2 - e^2*x^2])/(15*e^5) + (175*d^4*x*Sqrt[d^2
- e^2*x^2])/(16*e^4) - (71*d^3*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) + (47*d^2*x^3*Sqrt[d^2 - e^2*x^2])/(24*e^2) -
(4*d*x^4*Sqrt[d^2 - e^2*x^2])/(5*e) + (x^5*Sqrt[d^2 - e^2*x^2])/6 - (239*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]
])/(16*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 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^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac{x^4 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{(d-e x)^3 \left (\frac{4 d^4}{e^4}-\frac{d^3 x}{e^3}+\frac{d^2 x^2}{e^2}-\frac{d x^3}{e}\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{d}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{-\frac{24 d^7}{e^2}+\frac{78 d^6 x}{e}-96 d^5 x^2+66 d^4 e x^3-47 d^3 e^2 x^4+24 d^2 e^3 x^5}{\sqrt{d^2-e^2 x^2}} \, dx}{6 d e^2}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{120 d^7-390 d^6 e x+480 d^5 e^2 x^2-426 d^4 e^3 x^3+235 d^3 e^4 x^4}{\sqrt{d^2-e^2 x^2}} \, dx}{30 d e^4}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{-480 d^7 e^2+1560 d^6 e^3 x-2625 d^5 e^4 x^2+1704 d^4 e^5 x^3}{\sqrt{d^2-e^2 x^2}} \, dx}{120 d e^6}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{1440 d^7 e^4-8088 d^6 e^5 x+7875 d^5 e^6 x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{360 d e^8}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{-10755 d^7 e^6+16176 d^6 e^7 x}{\sqrt{d^2-e^2 x^2}} \, dx}{720 d e^{10}}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{337 d^5 \sqrt{d^2-e^2 x^2}}{15 e^5}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{\left (239 d^6\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^4}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{337 d^5 \sqrt{d^2-e^2 x^2}}{15 e^5}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{\left (239 d^6\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^4}\\ &=-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{337 d^5 \sqrt{d^2-e^2 x^2}}{15 e^5}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{239 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^5}\\ \end{align*}

Mathematica [A]  time = 0.189041, size = 125, normalized size = 0.56 \[ \frac{\sqrt{d^2-e^2 x^2} \left (769 d^4 e^2 x^2-426 d^3 e^3 x^3+278 d^2 e^4 x^4-2047 d^5 e x-5632 d^6-152 d e^5 x^5+40 e^6 x^6\right )-3585 d^6 (d+e x) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{240 e^5 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5632*d^6 - 2047*d^5*e*x + 769*d^4*e^2*x^2 - 426*d^3*e^3*x^3 + 278*d^2*e^4*x^4 - 152*d*e
^5*x^5 + 40*e^6*x^6) - 3585*d^6*(d + e*x)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^5*(d + e*x))

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Maple [B]  time = 0.071, size = 393, normalized size = 1.8 \begin{align*} -{\frac{{d}^{3}}{{e}^{9}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-7\,{\frac{{d}^{2}}{{e}^{8}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{22\,d}{3\,{e}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}+{\frac{5\,{d}^{4}x}{16\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{5\,{d}^{6}}{16\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{5\,{d}^{2}x}{24\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{122\,d}{15\,{e}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{x}{6\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{61\,{d}^{2}x}{6\,{e}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{61\,{d}^{4}x}{4\,{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{61\,{d}^{6}}{4\,{e}^{4}}\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(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x)

[Out]

-d^3/e^9/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)-7*d^2/e^8/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/
2)-22/3*d/e^7/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+5/16*d^4*x*(-e^2*x^2+d^2)^(1/2)/e^4+5/16/e^4*d^6/
(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+5/24/e^4*d^2*x*(-e^2*x^2+d^2)^(3/2)-122/15/e^5*d*(-(d/e
+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+1/6/e^4*x*(-e^2*x^2+d^2)^(5/2)-61/6/e^4*d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2
)*x-61/4/e^4*d^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-61/4/e^4*d^6/(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(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67713, size = 333, normalized size = 1.49 \begin{align*} -\frac{5632 \, d^{6} e x + 5632 \, d^{7} - 7170 \,{\left (d^{6} e x + d^{7}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (40 \, e^{6} x^{6} - 152 \, d e^{5} x^{5} + 278 \, d^{2} e^{4} x^{4} - 426 \, d^{3} e^{3} x^{3} + 769 \, d^{4} e^{2} x^{2} - 2047 \, d^{5} e x - 5632 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \,{\left (e^{6} x + d 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)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/240*(5632*d^6*e*x + 5632*d^7 - 7170*(d^6*e*x + d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (40*e^6*x^6
- 152*d*e^5*x^5 + 278*d^2*e^4*x^4 - 426*d^3*e^3*x^3 + 769*d^4*e^2*x^2 - 2047*d^5*e*x - 5632*d^6)*sqrt(-e^2*x^
2 + d^2))/(e^6*x + d*e^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}{\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)**(5/2)/(e*x+d)**4,x)

[Out]

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

[Out]

Exception raised: NotImplementedError