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

Optimal. Leaf size=130 \[ \frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]

[Out]

(10*d*x*Sqrt[d^2 - e^2*x^2])/e + (20*(d^2 - e^2*x^2)^(3/2))/(3*e^2) + (8*(d^2 - e^2*x^2)^(5/2))/(e^2*(d + e*x)
^2) + (d^2 - e^2*x^2)^(7/2)/(e^2*(d + e*x)^4) + (10*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2

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Rubi [A]  time = 0.0641239, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {793, 663, 665, 195, 217, 203} \[ \frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*d*x*Sqrt[d^2 - e^2*x^2])/e + (20*(d^2 - e^2*x^2)^(3/2))/(3*e^2) + (8*(d^2 - e^2*x^2)^(5/2))/(e^2*(d + e*x)
^2) + (d^2 - e^2*x^2)^(7/2)/(e^2*(d + e*x)^4) + (10*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2

Rule 793

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

Rule 663

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

Rule 665

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

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 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{4 \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx}{e}\\ &=\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{20 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{(20 d) \int \sqrt{d^2-e^2 x^2} \, dx}{e}\\ &=\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{\left (10 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e}\\ &=\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{\left (10 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e}\\ &=\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac{8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.112286, size = 83, normalized size = 0.64 \[ \frac{1}{3} \sqrt{d^2-e^2 x^2} \left (\frac{24 d^3}{e^2 (d+e x)}+\frac{23 d^2}{e^2}-\frac{6 d x}{e}+x^2\right )+\frac{10 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*((23*d^2)/e^2 - (6*d*x)/e + x^2 + (24*d^3)/(e^2*(d + e*x))))/3 + (10*d^3*ArcTan[(e*x)/Sqr
t[d^2 - e^2*x^2]])/e^2

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Maple [B]  time = 0.06, size = 290, normalized size = 2.2 \begin{align*}{\frac{1}{{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 ) ^{-4}}+4\,{\frac{1}{d{e}^{5}} \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{16}{3\,{e}^{4}{d}^{2}} \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{16}{3\,{d}^{2}{e}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{20\,x}{3\,de} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+10\,{\frac{dx}{e}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+10\,{\frac{{d}^{3}}{e\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^6/(d/e+x)^4*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+4/d/e^5/(d/e+x)^3*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+16
/3/d^2/e^4/(d/e+x)^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(7/2)+16/3/d^2/e^2*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+20
/3/d/e*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x+10*d/e*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+10*d^3/e/(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*(-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 = 1.67388, size = 236, normalized size = 1.82 \begin{align*} \frac{47 \, d^{3} e x + 47 \, d^{4} - 60 \,{\left (d^{3} e x + d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (e^{3} x^{3} - 5 \, d e^{2} x^{2} + 17 \, d^{2} e x + 47 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (e^{3} x + d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(47*d^3*e*x + 47*d^4 - 60*(d^3*e*x + d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (e^3*x^3 - 5*d*e^2*x
^2 + 17*d^2*e*x + 47*d^3)*sqrt(-e^2*x^2 + d^2))/(e^3*x + d*e^2)

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

[Out]

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

[Out]

Exception raised: NotImplementedError