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]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 1.66, size = 52, normalized size = 2.74 \[ -\frac {2 \, \sqrt {c x^{2} + b x + a} d}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

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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giac [B]  time = 0.20, size = 41, normalized size = 2.16 \[ -\frac {2 \, d^{2}}{3 \, {\left (c d x^{2} + b d x + a d\right )} \sqrt {\frac {c d x^{2} + b d x + a d}{d}}} \]

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]

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

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maple [A]  time = 0.04, size = 16, normalized size = 0.84 \[ -\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}} \]

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 = 1.37, size = 15, normalized size = 0.79 \[ -\frac {2 \, d}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]

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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mupad [B]  time = 0.58, size = 15, normalized size = 0.79 \[ -\frac {2\,d}{3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [B]  time = 1.68, size = 60, normalized size = 3.16 \[ - \frac {2 d}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} \]

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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