### 3.873 $$\int \frac{(c d^2-c e^2 x^2)^{3/2}}{(d+e x)^{3/2}} \, dx$$

Optimal. Leaf size=38 $-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}}$

[Out]

(-2*(c*d^2 - c*e^2*x^2)^(5/2))/(5*c*e*(d + e*x)^(5/2))

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Rubi [A]  time = 0.0134863, antiderivative size = 38, 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} $-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(c*d^2 - c*e^2*x^2)^(5/2))/(5*c*e*(d + e*x)^(5/2))

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

Mathematica [A]  time = 0.0504065, size = 43, normalized size = 1.13 $-\frac{2 c (d-e x)^2 \sqrt{c \left (d^2-e^2 x^2\right )}}{5 e \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])/(5*e*Sqrt[d + e*x])

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Maple [A]  time = 0.042, size = 36, normalized size = 1. \begin{align*} -{\frac{-2\,ex+2\,d}{5\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05618, size = 53, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (c^{\frac{3}{2}} e^{2} x^{2} - 2 \, c^{\frac{3}{2}} d e x + c^{\frac{3}{2}} d^{2}\right )} \sqrt{-e x + d}}{5 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(c^(3/2)*e^2*x^2 - 2*c^(3/2)*d*e*x + c^(3/2)*d^2)*sqrt(-e*x + d)/e

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Fricas [A]  time = 2.07715, size = 123, normalized size = 3.24 \begin{align*} -\frac{2 \,{\left (c e^{2} x^{2} - 2 \, c d e x + c d^{2}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{5 \,{\left (e^{2} x + d e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(c*e^2*x^2 - 2*c*d*e*x + c*d^2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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