3.159 \(\int \frac{x^3 (d^2-e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=171 \[ -\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4} \]

[Out]

-(d^5*x*Sqrt[d^2 - e^2*x^2])/(8*e^3) - (11*d^2*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^2) + (d*x^3*(d^2 - e^2*x^2)^(3
/2))/(3*e) - (x^4*(d^2 - e^2*x^2)^(3/2))/7 - (d^3*(88*d - 105*e*x)*(d^2 - e^2*x^2)^(3/2))/(420*e^4) - (d^7*Arc
Tan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^4)

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Rubi [A]  time = 0.215467, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]

[Out]

-(d^5*x*Sqrt[d^2 - e^2*x^2])/(8*e^3) - (11*d^2*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^2) + (d*x^3*(d^2 - e^2*x^2)^(3
/2))/(3*e) - (x^4*(d^2 - e^2*x^2)^(3/2))/7 - (d^3*(88*d - 105*e*x)*(d^2 - e^2*x^2)^(3/2))/(420*e^4) - (d^7*Arc
Tan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^4)

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^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int x^3 (d-e x)^2 \sqrt{d^2-e^2 x^2} \, dx\\ &=-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x^3 \left (-11 d^2 e^2+14 d e^3 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{7 e^2}\\ &=\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int x^2 \left (-42 d^3 e^3+66 d^2 e^4 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{42 e^4}\\ &=-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int x \left (-132 d^4 e^4+210 d^3 e^5 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{210 e^6}\\ &=-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac{d^5 \int \sqrt{d^2-e^2 x^2} \, dx}{4 e^3}\\ &=-\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac{d^7 \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^3}\\ &=-\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac{d^7 \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}\\ &=-\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac{d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac{d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4}\\ \end{align*}

Mathematica [A]  time = 0.13492, size = 113, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-88 d^4 e^2 x^2+70 d^3 e^3 x^3+144 d^2 e^4 x^4+105 d^5 e x-176 d^6-280 d e^5 x^5+120 e^6 x^6\right )-105 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{840 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-176*d^6 + 105*d^5*e*x - 88*d^4*e^2*x^2 + 70*d^3*e^3*x^3 + 144*d^2*e^4*x^4 - 280*d*e^5*x
^5 + 120*e^6*x^6) - 105*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(840*e^4)

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Maple [B]  time = 0.065, size = 327, normalized size = 1.9 \begin{align*} -{\frac{1}{7\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{dx}{3\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{3}x}{12\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{5}x}{8\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{5\,{d}^{7}}{8\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{4\,{d}^{2}}{15\,{e}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}x}{3\,{e}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}x}{2\,{e}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{{d}^{7}}{2\,{e}^{3}}\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}^{2}}{3\,{e}^{6}} \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^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^2,x)

[Out]

-1/7/e^4*(-e^2*x^2+d^2)^(7/2)-1/3*d/e^3*x*(-e^2*x^2+d^2)^(5/2)-5/12/e^3*d^3*x*(-e^2*x^2+d^2)^(3/2)-5/8*d^5*x*(
-e^2*x^2+d^2)^(1/2)/e^3-5/8/e^3*d^7/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+4/15/e^4*d^2*(-(d/e
+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+1/3/e^3*d^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+1/2/e^3*d^5*(-(d/e+x)^2*e^2+
2*d*e*(d/e+x))^(1/2)*x+1/2/e^3*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))-1/3*
d^2/e^6/(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^3*(-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 = 1.52085, size = 259, normalized size = 1.51 \begin{align*} \frac{210 \, d^{7} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (120 \, e^{6} x^{6} - 280 \, d e^{5} x^{5} + 144 \, d^{2} e^{4} x^{4} + 70 \, d^{3} e^{3} x^{3} - 88 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x - 176 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{840 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/840*(210*d^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (120*e^6*x^6 - 280*d*e^5*x^5 + 144*d^2*e^4*x^4 + 70
*d^3*e^3*x^3 - 88*d^4*e^2*x^2 + 105*d^5*e*x - 176*d^6)*sqrt(-e^2*x^2 + d^2))/e^4

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Sympy [A]  time = 19.365, size = 452, normalized size = 2.64 \begin{align*} d^{2} \left (\begin{cases} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\\frac{x^{4} \sqrt{d^{2}}}{4} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} - \frac{i d^{6} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{16 e^{5}} + \frac{i d^{5} x}{16 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d^{3} x^{3}}{48 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{5 i d x^{5}}{24 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{7}}{6 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{6} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{16 e^{5}} - \frac{d^{5} x}{16 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d^{3} x^{3}}{48 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{5 d x^{5}}{24 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{7}}{6 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{8 d^{6} \sqrt{d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac{4 d^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac{d^{2} x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac{x^{6} \sqrt{d^{2} - e^{2} x^{2}}}{7} & \text{for}\: e \neq 0 \\\frac{x^{6} \sqrt{d^{2}}}{6} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-e**2*x**2+d**2)**(5/2)/(e*x+d)**2,x)

[Out]

d**2*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*s
qrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) - 2*d*e*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/(48*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)) + e**2*
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))

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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^3*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^2,x, algorithm="giac")

[Out]

sage0*x