3.1268 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac{1}{12 c^2 d^3 (b d+2 c d x)^{3/2}} \]

[Out]

(b^2 - 4*a*c)/(28*c^2*d*(b*d + 2*c*d*x)^(7/2)) - 1/(12*c^2*d^3*(b*d + 2*c*d*x)^(3/2))

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Rubi [A]  time = 0.023352, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ \frac{b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac{1}{12 c^2 d^3 (b d+2 c d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(b^2 - 4*a*c)/(28*c^2*d*(b*d + 2*c*d*x)^(7/2)) - 1/(12*c^2*d^3*(b*d + 2*c*d*x)^(3/2))

Rule 683

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

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c (b d+2 c d x)^{9/2}}+\frac{1}{4 c d^2 (b d+2 c d x)^{5/2}}\right ) \, dx\\ &=\frac{b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac{1}{12 c^2 d^3 (b d+2 c d x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0372111, size = 51, normalized size = 0.93 \[ -\frac{\left (c \left (3 a+7 c x^2\right )+b^2+7 b c x\right ) \sqrt{d (b+2 c x)}}{21 c^2 d^5 (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(9/2),x]

[Out]

-(Sqrt[d*(b + 2*c*x)]*(b^2 + 7*b*c*x + c*(3*a + 7*c*x^2)))/(21*c^2*d^5*(b + 2*c*x)^4)

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Maple [A]  time = 0.042, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( 7\,{c}^{2}{x}^{2}+7\,bcx+3\,ac+{b}^{2} \right ) }{21\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/(2*c*d*x+b*d)^(9/2),x)

[Out]

-1/21*(2*c*x+b)*(7*c^2*x^2+7*b*c*x+3*a*c+b^2)/c^2/(2*c*d*x+b*d)^(9/2)

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Maxima [A]  time = 1.0138, size = 62, normalized size = 1.13 \begin{align*} \frac{3 \,{\left (b^{2} - 4 \, a c\right )} d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{2}}{84 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^(9/2),x, algorithm="maxima")

[Out]

1/84*(3*(b^2 - 4*a*c)*d^2 - 7*(2*c*d*x + b*d)^2)/((2*c*d*x + b*d)^(7/2)*c^2*d^3)

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Fricas [B]  time = 2.00594, size = 205, normalized size = 3.73 \begin{align*} -\frac{{\left (7 \, c^{2} x^{2} + 7 \, b c x + b^{2} + 3 \, a c\right )} \sqrt{2 \, c d x + b d}}{21 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^(9/2),x, algorithm="fricas")

[Out]

-1/21*(7*c^2*x^2 + 7*b*c*x + b^2 + 3*a*c)*sqrt(2*c*d*x + b*d)/(16*c^6*d^5*x^4 + 32*b*c^5*d^5*x^3 + 24*b^2*c^4*
d^5*x^2 + 8*b^3*c^3*d^5*x + b^4*c^2*d^5)

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Sympy [A]  time = 10.6182, size = 360, normalized size = 6.55 \begin{align*} \begin{cases} - \frac{3 a c \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac{b^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac{7 b c x \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac{7 c^{2} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} & \text{for}\: c \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\left (b d\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**(9/2),x)

[Out]

Piecewise((-3*a*c*sqrt(b*d + 2*c*d*x)/(21*b**4*c**2*d**5 + 168*b**3*c**3*d**5*x + 504*b**2*c**4*d**5*x**2 + 67
2*b*c**5*d**5*x**3 + 336*c**6*d**5*x**4) - b**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**2*d**5 + 168*b**3*c**3*d**5*x
+ 504*b**2*c**4*d**5*x**2 + 672*b*c**5*d**5*x**3 + 336*c**6*d**5*x**4) - 7*b*c*x*sqrt(b*d + 2*c*d*x)/(21*b**4*
c**2*d**5 + 168*b**3*c**3*d**5*x + 504*b**2*c**4*d**5*x**2 + 672*b*c**5*d**5*x**3 + 336*c**6*d**5*x**4) - 7*c*
*2*x**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**2*d**5 + 168*b**3*c**3*d**5*x + 504*b**2*c**4*d**5*x**2 + 672*b*c**5*d
**5*x**3 + 336*c**6*d**5*x**4), Ne(c, 0)), ((a*x + b*x**2/2)/(b*d)**(9/2), True))

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Giac [A]  time = 1.15317, size = 65, normalized size = 1.18 \begin{align*} \frac{3 \, b^{2} d^{2} - 12 \, a c d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{2}}{84 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^(9/2),x, algorithm="giac")

[Out]

1/84*(3*b^2*d^2 - 12*a*c*d^2 - 7*(2*c*d*x + b*d)^2)/((2*c*d*x + b*d)^(7/2)*c^2*d^3)