3.1255 \(\int \frac{b d+2 c d x}{(a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d)/(3*(a + b*x + c*x^2)^(3/2))

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Rubi [A]  time = 0.0062324, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {629} \[ -\frac{2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(3*(a + b*x + c*x^2)^(3/2))

Rule 629

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

Rubi steps

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

Mathematica [A]  time = 0.0067048, size = 18, normalized size = 0.95 \[ -\frac{2 d}{3 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(3*(a + x*(b + c*x))^(3/2))

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Maple [A]  time = 0.043, size = 16, normalized size = 0.8 \begin{align*} -{\frac{2\,d}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14394, size = 20, normalized size = 1.05 \begin{align*} -\frac{2 \, d}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*d/(c*x^2 + b*x + a)^(3/2)

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Fricas [B]  time = 4.77405, size = 119, normalized size = 6.26 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x + a} d}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.45053, size = 60, normalized size = 3.16 \begin{align*} - \frac{2 d}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15215, size = 86, normalized size = 4.53 \begin{align*} -\frac{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d}{3 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b^4*d - 8*a*b^2*c*d + 16*a^2*c^2*d)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^2 + b*x + a)^(3/2))