### 3.839 $$\int \frac{(d+e x)^3}{(d^2-e^2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=33 $\frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0099128, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {651} $\frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 651

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{(d+e x)^3}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0445357, size = 41, normalized size = 1.24 $\frac{(d+e x)^2}{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 36, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{4} \left ( -ex+d \right ) }{3\,de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.19949, size = 108, normalized size = 3.27 \begin{align*} \frac{e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{4 \, d x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{d^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x^2/(-e^2*x^2 + d^2)^(3/2) + 4/3*d*x/(-e^2*x^2 + d^2)^(3/2) + 1/3*d^2/((-e^2*x^2 + d^2)^(3/2)*e) - 1/3*x/(sq
rt(-e^2*x^2 + d^2)*d)

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Fricas [B]  time = 2.06081, size = 132, normalized size = 4. \begin{align*} \frac{e^{2} x^{2} - 2 \, d e x + d^{2} + \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}}{3 \,{\left (d e^{3} x^{2} - 2 \, d^{2} e^{2} x + d^{3} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30452, size = 76, normalized size = 2.3 \begin{align*} \frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (d^{2} e^{\left (-1\right )} +{\left (x{\left (\frac{x e^{2}}{d} + 3 \, e\right )} + 3 \, d\right )} x\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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