### 3.580 $$\int \frac{(d+e x)^2}{(a+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=58 $\frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}}$

[Out]

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

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Rubi [A]  time = 0.0202496, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {729, 637} $\frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2/(a + c*x^2)^(5/2),x]

[Out]

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

Rule 729

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^m*(2*c*x)*(a + c*x^2)^(
p + 1))/(4*a*c*(p + 1)), x] - Dist[(m*(2*c*d))/(4*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x],
x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]

Rule 637

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

Rubi steps

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

Mathematica [A]  time = 0.0801214, size = 57, normalized size = 0.98 $\frac{-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 55, normalized size = 1. \begin{align*} -{\frac{-ac{e}^{2}{x}^{3}-2\,{c}^{2}{d}^{2}{x}^{3}-3\,{d}^{2}xac+2\,de{a}^{2}}{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)^2/(c*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.17244, size = 124, normalized size = 2.14 \begin{align*} \frac{2 \, d^{2} x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d^{2} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{e^{2} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{e^{2} x}{3 \, \sqrt{c x^{2} + a} a c} - \frac{2 \, d 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)^2/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.16358, size = 153, normalized size = 2.64 \begin{align*} \frac{{\left (3 \, a c d^{2} x - 2 \, a^{2} d e +{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{3}\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)^2/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/3*(3*a*c*d^2*x - 2*a^2*d*e + (2*c^2*d^2 + a*c*e^2)*x^3)*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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3501, size = 74, normalized size = 1.28 \begin{align*} \frac{{\left (\frac{3 \, d^{2}}{a} + \frac{{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}}{a^{2} c}\right )} x - \frac{2 \, d 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)^2/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

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