### 3.581 $$\int \frac{d+e x}{(a+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=51 $\frac{2 d x}{3 a^2 \sqrt{a+c x^2}}-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0114975, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {639, 191} $\frac{2 d x}{3 a^2 \sqrt{a+c x^2}}-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(a + c*x^2)^(5/2),x]

[Out]

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

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{(2 d) \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{2 d x}{3 a^2 \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0182051, size = 43, normalized size = 0.84 $\frac{-a^2 e+3 a c d x+2 c^2 d x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 39, normalized size = 0.8 \begin{align*} -{\frac{-2\,{c}^{2}d{x}^{3}-3\,dxac+{a}^{2}e}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17246, size = 65, normalized size = 1.27 \begin{align*} \frac{2 \, d x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{e}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.18817, size = 126, normalized size = 2.47 \begin{align*} \frac{{\left (2 \, c^{2} d x^{3} + 3 \, a c d x - a^{2} e\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 13.3911, size = 146, normalized size = 2.86 \begin{align*} d \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + e \left (\begin{cases} - \frac{1}{3 a c \sqrt{a + c x^{2}} + 3 c^{2} x^{2} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(3*a*x/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a)) + 2*c*x**3/(3*a**(7/2)*sqrt(1
+ c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a))) + e*Piecewise((-1/(3*a*c*sqrt(a + c*x**2) + 3*c**2*x**2*s
qrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(5/2)), True))

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Giac [A]  time = 1.6705, size = 51, normalized size = 1. \begin{align*} \frac{{\left (\frac{2 \, c d x^{2}}{a^{2}} + \frac{3 \, d}{a}\right )} x - \frac{e}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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