3.120 \(\int \frac{x^3}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4} \]

[Out]

(x^2*(d - e*x))/(e^2*Sqrt[d^2 - e^2*x^2]) + ((4*d - 3*e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^4) + (3*d^2*ArcTan[(e*x)/
Sqrt[d^2 - e^2*x^2]])/(2*e^4)

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Rubi [A]  time = 0.0635089, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {850, 819, 780, 217, 203} \[ \frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(d - e*x))/(e^2*Sqrt[d^2 - e^2*x^2]) + ((4*d - 3*e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^4) + (3*d^2*ArcTan[(e*x)/
Sqrt[d^2 - e^2*x^2]])/(2*e^4)

Rule 850

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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 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}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=\int \frac{x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{x \left (2 d^3-3 d^2 e x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{\left (3 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^3}\\ &=\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3}\\ &=\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0673267, size = 80, normalized size = 0.88 \[ \frac{\sqrt{d^2-e^2 x^2} \left (4 d^2+d e x-e^2 x^2\right )+3 d^2 (d+e x) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(4*d^2 + d*e*x - e^2*x^2) + 3*d^2*(d + e*x)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^4*(d
+ e*x))

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Maple [A]  time = 0.058, size = 120, normalized size = 1.3 \begin{align*} -{\frac{x}{2\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,{d}^{2}}{2\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d}{{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{2}}{{e}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/2/e^3*x*(-e^2*x^2+d^2)^(1/2)+3/2*d^2/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+d/e^4*(-e^2
*x^2+d^2)^(1/2)+d^2/e^5/(d/e+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^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56663, size = 205, normalized size = 2.25 \begin{align*} \frac{4 \, d^{2} e x + 4 \, d^{3} - 6 \,{\left (d^{2} e x + d^{3}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (e^{2} x^{2} - d e x - 4 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*(4*d^2*e*x + 4*d^3 - 6*(d^2*e*x + d^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (e^2*x^2 - d*e*x - 4*d^
2)*sqrt(-e^2*x^2 + d^2))/(e^5*x + d*e^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(e*x+d)/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(x**3/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)), 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^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError