### 3.596 $$\int \frac{a+c x^2}{(d+e x)^{5/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0222372, 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} $-\frac{2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}+\frac{4 c d}{e^3 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)/(d + e*x)^(5/2),x]

[Out]

(-2*(c*d^2 + a*e^2))/(3*e^3*(d + e*x)^(3/2)) + (4*c*d)/(e^3*Sqrt[d + e*x]) + (2*c*Sqrt[d + e*x])/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)^{5/2}} \, dx &=\int \left (\frac{c d^2+a e^2}{e^2 (d+e x)^{5/2}}-\frac{2 c d}{e^2 (d+e x)^{3/2}}+\frac{c}{e^2 \sqrt{d+e x}}\right ) \, dx\\ &=-\frac{2 \left (c d^2+a e^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 c d}{e^3 \sqrt{d+e x}}+\frac{2 c \sqrt{d+e x}}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.037179, size = 44, normalized size = 0.75 $\frac{2 \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-a e^2\right )}{3 e^3 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)/(d + e*x)^(5/2),x]

[Out]

(2*(-(a*e^2) + c*(8*d^2 + 12*d*e*x + 3*e^2*x^2)))/(3*e^3*(d + e*x)^(3/2))

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Maple [A]  time = 0.041, size = 40, normalized size = 0.7 \begin{align*} -{\frac{-6\,c{e}^{2}{x}^{2}-24\,cdex+2\,a{e}^{2}-16\,c{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(e*x+d)^(5/2),x)

[Out]

-2/3/(e*x+d)^(3/2)*(-3*c*e^2*x^2-12*c*d*e*x+a*e^2-8*c*d^2)/e^3

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Maxima [A]  time = 1.11347, size = 70, normalized size = 1.19 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} c}{e^{2}} + \frac{6 \,{\left (e x + d\right )} c d - c d^{2} - a e^{2}}{{\left (e x + d\right )}^{\frac{3}{2}} 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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.84766, size = 130, normalized size = 2.2 \begin{align*} \frac{2 \,{\left (3 \, c e^{2} x^{2} + 12 \, c d e x + 8 \, c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*c*e^2*x^2 + 12*c*d*e*x + 8*c*d^2 - a*e^2)*sqrt(e*x + d)/(e^5*x^2 + 2*d*e^4*x + d^2*e^3)

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Sympy [A]  time = 1.19617, size = 168, normalized size = 2.85 \begin{align*} \begin{cases} - \frac{2 a e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 c d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 c d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 c e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(e*x+d)**(5/2),x)

[Out]

Piecewise((-2*a*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 16*c*d**2/(3*d*e**3*sqrt(d + e*x) + 3
*e**4*x*sqrt(d + e*x)) + 24*c*d*e*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 6*c*e**2*x**2/(3*d*e**
3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)), Ne(e, 0)), ((a*x + c*x**3/3)/d**(5/2), True))

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Giac [A]  time = 1.31088, size = 65, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x e + d} c e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} c d - c d^{2} - a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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