### 3.595 $$\int \frac{a+c x^2}{(d+e x)^{3/2}} \, dx$$

Optimal. Leaf size=59 $-\frac{2 \left (a e^2+c d^2\right )}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3}$

[Out]

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

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Rubi [A]  time = 0.0208537, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $-\frac{2 \left (a e^2+c d^2\right )}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)/(d + e*x)^(3/2),x]

[Out]

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

Rule 697

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

Rubi steps

\begin{align*} \int \frac{a+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}+\frac{c \sqrt{d+e x}}{e^2}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )}{e^3 \sqrt{d+e x}}-\frac{4 c d \sqrt{d+e x}}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.034477, size = 43, normalized size = 0.73 $\frac{2 \left (c \left (-8 d^2-4 d e x+e^2 x^2\right )-3 a e^2\right )}{3 e^3 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(-3*a*e^2 + c*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])

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Maple [A]  time = 0.041, size = 41, normalized size = 0.7 \begin{align*} -{\frac{-2\,c{e}^{2}{x}^{2}+8\,cdex+6\,a{e}^{2}+16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(e*x+d)^(1/2)*(-c*e^2*x^2+4*c*d*e*x+3*a*e^2+8*c*d^2)/e^3

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Maxima [A]  time = 1.12727, size = 73, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 6 \, \sqrt{e x + d} c d}{e^{2}} - \frac{3 \,{\left (c d^{2} + a e^{2}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((e*x + d)^(3/2)*c - 6*sqrt(e*x + d)*c*d)/e^2 - 3*(c*d^2 + a*e^2)/(sqrt(e*x + d)*e^2))/e

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Fricas [A]  time = 1.78122, size = 107, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (c e^{2} x^{2} - 4 \, c d e x - 8 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(c*e^2*x^2 - 4*c*d*e*x - 8*c*d^2 - 3*a*e^2)*sqrt(e*x + d)/(e^4*x + d*e^3)

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Sympy [A]  time = 5.83845, size = 58, normalized size = 0.98 \begin{align*} - \frac{4 c d \sqrt{d + e x}}{e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} - \frac{2 \left (a e^{2} + c d^{2}\right )}{e^{3} \sqrt{d + e x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c*d*sqrt(d + e*x)/e**3 + 2*c*(d + e*x)**(3/2)/(3*e**3) - 2*(a*e**2 + c*d**2)/(e**3*sqrt(d + e*x))

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Giac [A]  time = 1.72146, size = 73, normalized size = 1.24 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((x*e + d)^(3/2)*c*e^6 - 6*sqrt(x*e + d)*c*d*e^6)*e^(-9) - 2*(c*d^2 + a*e^2)*e^(-3)/sqrt(x*e + d)