3.92 \(\int \frac{c+d x^2}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{2 c x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0096644, antiderivative size = 47, 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 = {378, 191} \[ \frac{2 c x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(a + b*x^2)^(5/2),x]

[Out]

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

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

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

Mathematica [A]  time = 0.0160048, size = 37, normalized size = 0.79 \[ \frac{x \left (3 a c+a d x^2+2 b c x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 34, normalized size = 0.7 \begin{align*}{\frac{x \left ( ad{x}^{2}+2\,bc{x}^{2}+3\,ac \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959582, size = 92, normalized size = 1.96 \begin{align*} \frac{2 \, c x}{3 \, \sqrt{b x^{2} + a} a^{2}} + \frac{c x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{d x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} + \frac{d x}{3 \, \sqrt{b x^{2} + a} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54471, size = 115, normalized size = 2.45 \begin{align*} \frac{{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 10.7922, size = 144, normalized size = 3.06 \begin{align*} c \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + \frac{d x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)) + 2*b*x**3/(3*a**(7/2)*sqrt(1
+ b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a))) + d*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b*x*
*2*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.11949, size = 54, normalized size = 1.15 \begin{align*} \frac{x{\left (\frac{3 \, c}{a} + \frac{{\left (2 \, b^{2} c + a b d\right )} x^{2}}{a^{2} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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