### 3.827 $$\int \frac{(d+e x)^4}{\sqrt{d^2-e^2 x^2}} \, dx$$

Optimal. Leaf size=149 $-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}$

[Out]

(-35*d^3*Sqrt[d^2 - e^2*x^2])/(8*e) - (35*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(24*e) - (7*d*(d + e*x)^2*Sqrt[d^
2 - e^2*x^2])/(12*e) - ((d + e*x)^3*Sqrt[d^2 - e^2*x^2])/(4*e) + (35*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8
*e)

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Rubi [A]  time = 0.0618129, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {671, 641, 217, 203} $-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-35*d^3*Sqrt[d^2 - e^2*x^2])/(8*e) - (35*d^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(24*e) - (7*d*(d + e*x)^2*Sqrt[d^
2 - e^2*x^2])/(12*e) - ((d + e*x)^3*Sqrt[d^2 - e^2*x^2])/(4*e) + (35*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8
*e)

Rule 671

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

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{(d+e x)^4}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{4} (7 d) \int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{12} \left (35 d^2\right ) \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{8} \left (35 d^3\right ) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{8} \left (35 d^4\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{8} \left (35 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}\\ \end{align*}

Mathematica [A]  time = 0.0835836, size = 81, normalized size = 0.54 $\frac{105 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (81 d^2 e x+160 d^3+32 d e^2 x^2+6 e^3 x^3\right )}{24 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(160*d^3 + 81*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 105*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^
2*x^2]])/(24*e)

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Maple [A]  time = 0.056, size = 119, normalized size = 0.8 \begin{align*} -{\frac{{e}^{2}{x}^{3}}{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{27\,{d}^{2}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{35\,{d}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{4\,de{x}^{2}}{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{20\,{d}^{3}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/4*e^2*x^3*(-e^2*x^2+d^2)^(1/2)-27/8*d^2*x*(-e^2*x^2+d^2)^(1/2)+35/8*d^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-
e^2*x^2+d^2)^(1/2))-4/3*e*d*x^2*(-e^2*x^2+d^2)^(1/2)-20/3*d^3*(-e^2*x^2+d^2)^(1/2)/e

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Maxima [A]  time = 1.6912, size = 150, normalized size = 1.01 \begin{align*} -\frac{1}{4} \, \sqrt{-e^{2} x^{2} + d^{2}} e^{2} x^{3} - \frac{4}{3} \, \sqrt{-e^{2} x^{2} + d^{2}} d e x^{2} + \frac{35 \, d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}}} - \frac{27}{8} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x - \frac{20 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}}{3 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-e^2*x^2 + d^2)*e^2*x^3 - 4/3*sqrt(-e^2*x^2 + d^2)*d*e*x^2 + 35/8*d^4*arcsin(e^2*x/sqrt(d^2*e^2))/sq
rt(e^2) - 27/8*sqrt(-e^2*x^2 + d^2)*d^2*x - 20/3*sqrt(-e^2*x^2 + d^2)*d^3/e

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Fricas [A]  time = 2.19204, size = 181, normalized size = 1.21 \begin{align*} -\frac{210 \, d^{4} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (6 \, e^{3} x^{3} + 32 \, d e^{2} x^{2} + 81 \, d^{2} e x + 160 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/24*(210*d^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (6*e^3*x^3 + 32*d*e^2*x^2 + 81*d^2*e*x + 160*d^3)*s
qrt(-e^2*x^2 + d^2))/e

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Sympy [A]  time = 8.19789, size = 549, normalized size = 3.68 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**4*Piecewise((sqrt(d**2/e**2)*asin(x*sqrt(e**2/d**2))/sqrt(d**2), (d**2 > 0) & (e**2 > 0)), (sqrt(-d**2/e**2
)*asinh(x*sqrt(-e**2/d**2))/sqrt(d**2), (d**2 > 0) & (e**2 < 0)), (sqrt(d**2/e**2)*acosh(x*sqrt(e**2/d**2))/sq
rt(-d**2), (d**2 < 0) & (e**2 < 0))) + 4*d**3*e*Piecewise((x**2/(2*sqrt(d**2)), Eq(e**2, 0)), (-sqrt(d**2 - e*
*2*x**2)/e**2, True)) + 6*d**2*e**2*Piecewise((-I*d**2*acosh(e*x/d)/(2*e**3) - I*d*x*sqrt(-1 + e**2*x**2/d**2)
/(2*e**2), Abs(e**2*x**2)/Abs(d**2) > 1), (d**2*asin(e*x/d)/(2*e**3) - d*x/(2*e**2*sqrt(1 - e**2*x**2/d**2)) +
x**3/(2*d*sqrt(1 - e**2*x**2/d**2)), True)) + 4*d*e**3*Piecewise((-2*d**2*sqrt(d**2 - e**2*x**2)/(3*e**4) - x
**2*sqrt(d**2 - e**2*x**2)/(3*e**2), Ne(e, 0)), (x**4/(4*sqrt(d**2)), True)) + e**4*Piecewise((-3*I*d**4*acosh
(e*x/d)/(8*e**5) + 3*I*d**3*x/(8*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d*x**3/(8*e**2*sqrt(-1 + e**2*x**2/d**2))
- I*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (3*d**4*asin(e*x/d)/(8*e**5) - 3*d**
3*x/(8*e**4*sqrt(1 - e**2*x**2/d**2)) + d*x**3/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + x**5/(4*d*sqrt(1 - e**2*x**
2/d**2)), True))

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Giac [A]  time = 1.29989, size = 85, normalized size = 0.57 \begin{align*} \frac{35}{8} \, d^{4} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{24} \,{\left (160 \, d^{3} e^{\left (-1\right )} +{\left (81 \, d^{2} + 2 \,{\left (3 \, x e^{2} + 16 \, d e\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

35/8*d^4*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/24*(160*d^3*e^(-1) + (81*d^2 + 2*(3*x*e^2 + 16*d*e)*x)*x)*sqrt(-x^2*e
^2 + d^2)