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

Optimal. Leaf size=200 \[ \frac{13 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{13 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5} \]

[Out]

(13*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (8*d^3*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^3) - (13*d^2*x^3*(d^2 - e^2
*x^2)^(3/2))/(48*e^2) + (2*d*x^4*(d^2 - e^2*x^2)^(3/2))/(7*e) - (x^5*(d^2 - e^2*x^2)^(3/2))/8 + (d^4*(1024*d -
1365*e*x)*(d^2 - e^2*x^2)^(3/2))/(6720*e^5) + (13*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

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Rubi [A]  time = 0.269624, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {852, 1809, 833, 780, 195, 217, 203} \[ \frac{13 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{13 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (8*d^3*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^3) - (13*d^2*x^3*(d^2 - e^2
*x^2)^(3/2))/(48*e^2) + (2*d*x^4*(d^2 - e^2*x^2)^(3/2))/(7*e) - (x^5*(d^2 - e^2*x^2)^(3/2))/8 + (d^4*(1024*d -
1365*e*x)*(d^2 - e^2*x^2)^(3/2))/(6720*e^5) + (13*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*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 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[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{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int x^4 (d-e x)^2 \sqrt{d^2-e^2 x^2} \, dx\\ &=-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x^4 \left (-13 d^2 e^2+16 d e^3 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{8 e^2}\\ &=\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int x^3 \left (-64 d^3 e^3+91 d^2 e^4 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{56 e^4}\\ &=-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x^2 \left (-273 d^4 e^4+384 d^3 e^5 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{336 e^6}\\ &=\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int x \left (-768 d^5 e^5+1365 d^4 e^6 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{1680 e^8}\\ &=\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac{\left (13 d^6\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac{13 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac{\left (13 d^8\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac{13 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac{\left (13 d^8\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4}\\ &=\frac{13 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac{13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac{2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac{1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac{13 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5}\\ \end{align*}

Mathematica [A]  time = 0.171357, size = 124, normalized size = 0.62 \[ \frac{\sqrt{d^2-e^2 x^2} \left (1024 d^5 e^2 x^2-910 d^4 e^3 x^3+768 d^3 e^4 x^4+1960 d^2 e^5 x^5-1365 d^6 e x+2048 d^7-3840 d e^6 x^6+1680 e^7 x^7\right )+1365 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{13440 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(2048*d^7 - 1365*d^6*e*x + 1024*d^5*e^2*x^2 - 910*d^4*e^3*x^3 + 768*d^3*e^4*x^4 + 1960*d^
2*e^5*x^5 - 3840*d*e^6*x^6 + 1680*e^7*x^7) + 1365*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(13440*e^5)

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Maple [B]  time = 0.068, size = 350, normalized size = 1.8 \begin{align*} -{\frac{x}{8\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{25\,{d}^{2}x}{48\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{125\,{d}^{4}x}{192\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{125\,{d}^{6}x}{128\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{125\,{d}^{8}}{128\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{2\,d}{7\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{d}^{3}}{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{7\,{d}^{4}x}{12\,{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{7\,{d}^{6}x}{8\,{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{7\,{d}^{8}}{8\,{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}}}}}+{\frac{{d}^{3}}{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}} \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)^2,x)

[Out]

-1/8/e^4*x*(-e^2*x^2+d^2)^(7/2)+25/48*d^2/e^4*x*(-e^2*x^2+d^2)^(5/2)+125/192/e^4*d^4*x*(-e^2*x^2+d^2)^(3/2)+12
5/128*d^6*x*(-e^2*x^2+d^2)^(1/2)/e^4+125/128/e^4*d^8/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+2/
7*d/e^5*(-e^2*x^2+d^2)^(7/2)-7/15/e^5*d^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)-7/12/e^4*d^4*(-(d/e+x)^2*e^2+2*
d*e*(d/e+x))^(3/2)*x-7/8/e^4*d^6*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-7/8/e^4*d^8/(e^2)^(1/2)*arctan((e^2)^(
1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))+1/3*d^3/e^7/(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^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60569, size = 300, normalized size = 1.5 \begin{align*} -\frac{2730 \, d^{8} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (1680 \, e^{7} x^{7} - 3840 \, d e^{6} x^{6} + 1960 \, d^{2} e^{5} x^{5} + 768 \, d^{3} e^{4} x^{4} - 910 \, d^{4} e^{3} x^{3} + 1024 \, d^{5} e^{2} x^{2} - 1365 \, d^{6} e x + 2048 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{13440 \, e^{5}} \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)^2,x, algorithm="fricas")

[Out]

-1/13440*(2730*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (1680*e^7*x^7 - 3840*d*e^6*x^6 + 1960*d^2*e^5*x
^5 + 768*d^3*e^4*x^4 - 910*d^4*e^3*x^3 + 1024*d^5*e^2*x^2 - 1365*d^6*e*x + 2048*d^7)*sqrt(-e^2*x^2 + d^2))/e^5

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Sympy [C]  time = 27.2561, size = 694, normalized size = 3.47 \begin{align*} \text{result too large to display} \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)**2,x)

[Out]

d**2*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(4
8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**
2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2*x**2
/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6
*d*sqrt(1 - e**2*x**2/d**2)), True)) - 2*d*e*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**
2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x**2)
/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True)) + e**2*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(12
8*e**6*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/(192*e**2
*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x**2
/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2
/d**2)) + 5*d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2)) + 7*
d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \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)^2,x, algorithm="giac")

[Out]

sage0*x