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

Optimal. Leaf size=252 \[ \frac{515 d^6 \sqrt{d^2-e^2 x^2}}{21 e^6}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}+\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{65 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^6} \]

[Out]

(d^4*(d - e*x)^4)/(e^6*Sqrt[d^2 - e^2*x^2]) + (515*d^6*Sqrt[d^2 - e^2*x^2])/(21*e^6) - (49*d^5*x*Sqrt[d^2 - e^
2*x^2])/(4*e^5) + (121*d^4*x^2*Sqrt[d^2 - e^2*x^2])/(21*e^4) - (17*d^3*x^3*Sqrt[d^2 - e^2*x^2])/(6*e^3) + (11*
d^2*x^4*Sqrt[d^2 - e^2*x^2])/(7*e^2) - (2*d*x^5*Sqrt[d^2 - e^2*x^2])/(3*e) + (x^6*Sqrt[d^2 - e^2*x^2])/7 + (65
*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^6)

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Rubi [A]  time = 0.662574, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{515 d^6 \sqrt{d^2-e^2 x^2}}{21 e^6}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}+\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{65 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^4*(d - e*x)^4)/(e^6*Sqrt[d^2 - e^2*x^2]) + (515*d^6*Sqrt[d^2 - e^2*x^2])/(21*e^6) - (49*d^5*x*Sqrt[d^2 - e^
2*x^2])/(4*e^5) + (121*d^4*x^2*Sqrt[d^2 - e^2*x^2])/(21*e^4) - (17*d^3*x^3*Sqrt[d^2 - e^2*x^2])/(6*e^3) + (11*
d^2*x^4*Sqrt[d^2 - e^2*x^2])/(7*e^2) - (2*d*x^5*Sqrt[d^2 - e^2*x^2])/(3*e) + (x^6*Sqrt[d^2 - e^2*x^2])/7 + (65
*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*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 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac{x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{(d-e x)^3 \left (-\frac{4 d^5}{e^5}+\frac{d^4 x}{e^4}-\frac{d^3 x^2}{e^3}+\frac{d^2 x^3}{e^2}-\frac{d x^4}{e}\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{d}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{\frac{28 d^8}{e^3}-\frac{91 d^7 x}{e^2}+\frac{112 d^6 x^2}{e}-77 d^5 x^3+56 d^4 e x^4-55 d^3 e^2 x^5+28 d^2 e^3 x^6}{\sqrt{d^2-e^2 x^2}} \, dx}{7 d e^2}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-\frac{168 d^8}{e}+546 d^7 x-672 d^6 e x^2+462 d^5 e^2 x^3-476 d^4 e^3 x^4+330 d^3 e^4 x^5}{\sqrt{d^2-e^2 x^2}} \, dx}{42 d e^4}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{840 d^8 e-2730 d^7 e^2 x+3360 d^6 e^3 x^2-3630 d^5 e^4 x^3+2380 d^4 e^5 x^4}{\sqrt{d^2-e^2 x^2}} \, dx}{210 d e^6}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-3360 d^8 e^3+10920 d^7 e^4 x-20580 d^6 e^5 x^2+14520 d^5 e^6 x^3}{\sqrt{d^2-e^2 x^2}} \, dx}{840 d e^8}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{10080 d^8 e^5-61800 d^7 e^6 x+61740 d^6 e^7 x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{2520 d e^{10}}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-81900 d^8 e^7+123600 d^7 e^8 x}{\sqrt{d^2-e^2 x^2}} \, dx}{5040 d e^{12}}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{515 d^6 \sqrt{d^2-e^2 x^2}}{21 e^6}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{\left (65 d^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{4 e^5}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{515 d^6 \sqrt{d^2-e^2 x^2}}{21 e^6}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{\left (65 d^7\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^5}\\ &=\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{515 d^6 \sqrt{d^2-e^2 x^2}}{21 e^6}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}+\frac{65 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^6}\\ \end{align*}

Mathematica [A]  time = 0.263833, size = 131, normalized size = 0.52 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-293 d^5 e^2 x^2+162 d^4 e^3 x^3-106 d^3 e^4 x^4+76 d^2 e^5 x^5+779 d^6 e x+2144 d^7-44 d e^6 x^6+12 e^7 x^7\right )}{d+e x}+1365 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{84 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(2144*d^7 + 779*d^6*e*x - 293*d^5*e^2*x^2 + 162*d^4*e^3*x^3 - 106*d^3*e^4*x^4 + 76*d^2*e
^5*x^5 - 44*d*e^6*x^6 + 12*e^7*x^7))/(d + e*x) + 1365*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(84*e^6)

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Maple [A]  time = 0.073, size = 416, normalized size = 1.7 \begin{align*} -{\frac{1}{7\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{28\,{d}^{2}}{3\,{e}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{d}^{3}x}{3\,{e}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{d}^{5}x}{2\,{e}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{35\,{d}^{7}}{2\,{e}^{5}}\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}}}}}-{\frac{2\,dx}{3\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{3}x}{6\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{5}x}{4\,{e}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{5\,{d}^{7}}{4\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{d}^{4}}{{e}^{10}} \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}}+8\,{\frac{{d}^{3}}{{e}^{9}} \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}^{2}}{3\,{e}^{8}} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/e^6*(-e^2*x^2+d^2)^(7/2)+28/3/e^6*d^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+35/3/e^5*d^3*(-(d/e+x)^2*e^2+2
*d*e*(d/e+x))^(3/2)*x+35/2/e^5*d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+35/2/e^5*d^7/(e^2)^(1/2)*arctan((e^2
)^(1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))-2/3/e^5*d*x*(-e^2*x^2+d^2)^(5/2)-5/6/e^5*d^3*x*(-e^2*x^2+d^2)^
(3/2)-5/4*d^5*x*(-e^2*x^2+d^2)^(1/2)/e^5-5/4/e^5*d^7/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+d^
4/e^10/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+8*d^3/e^9/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)
+22/3*d^2/e^8/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.66641, size = 351, normalized size = 1.39 \begin{align*} \frac{2144 \, d^{7} e x + 2144 \, d^{8} - 2730 \,{\left (d^{7} e x + d^{8}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (12 \, e^{7} x^{7} - 44 \, d e^{6} x^{6} + 76 \, d^{2} e^{5} x^{5} - 106 \, d^{3} e^{4} x^{4} + 162 \, d^{4} e^{3} x^{3} - 293 \, d^{5} e^{2} x^{2} + 779 \, d^{6} e x + 2144 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{84 \,{\left (e^{7} x + d 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)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/84*(2144*d^7*e*x + 2144*d^8 - 2730*(d^7*e*x + d^8)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (12*e^7*x^7 -
44*d*e^6*x^6 + 76*d^2*e^5*x^5 - 106*d^3*e^4*x^4 + 162*d^4*e^3*x^3 - 293*d^5*e^2*x^2 + 779*d^6*e*x + 2144*d^7)
*sqrt(-e^2*x^2 + d^2))/(e^7*x + d*e^6)

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

[Out]

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

[Out]

Exception raised: NotImplementedError