### 3.889 $$\int \frac{(d+e x)^{3/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=36 $\frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}$

[Out]

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

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Rubi [A]  time = 0.0130347, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.034, Rules used = {649} $\frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 649

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

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

Mathematica [A]  time = 0.0451778, size = 35, normalized size = 0.97 $\frac{2 \sqrt{d+e x}}{c e \sqrt{c \left (d^2-e^2 x^2\right )}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 36, normalized size = 1. \begin{align*} 2\,{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3/2}}{e \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{3/2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1656, size = 22, normalized size = 0.61 \begin{align*} \frac{2}{\sqrt{-e x + d} c^{\frac{3}{2}} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sqrt(-e*x + d)*c^(3/2)*e)

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Fricas [A]  time = 2.22249, size = 93, normalized size = 2.58 \begin{align*} -\frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

sage0*x