3.1257 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^{9/2}} \, dx\)

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 = 0.0719155, 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 \[ \frac{b^2-4 a c}{28 c^2 d (b d+2 c d x)^{7/2}}-\frac{1}{12 c^2 d^3 (b d+2 c d x)^{3/2}} \]

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))

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Rubi in Sympy [A]  time = 14.4274, size = 51, normalized size = 0.93 \[ \frac{- a c + \frac{b^{2}}{4}}{7 c^{2} d \left (b d + 2 c d x\right )^{\frac{7}{2}}} - \frac{1}{12 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**(9/2),x)

[Out]

(-a*c + b**2/4)/(7*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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Mathematica [A]  time = 0.0641054, 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 = 0.005, size = 44, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( 7\,{c}^{2}{x}^{2}+7\,bxc+3\,ac+{b}^{2} \right ) }{21\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{9}{2}}}} \]

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 = 0.692957, size = 62, normalized size = 1.13 \[ \frac{3 \,{\left (b^{2} - 4 \, a c\right )} d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{2}}{84 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{3}} \]

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 [A]  time = 0.210369, size = 111, normalized size = 2.02 \[ -\frac{7 \, c^{2} x^{2} + 7 \, b c x + b^{2} + 3 \, a c}{21 \,{\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )} \sqrt{2 \, c d x + b d}} \]

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)/((8*c^5*d^4*x^3 + 12*b*c^4*d^4*x^2 + 6
*b^2*c^3*d^4*x + b^3*c^2*d^4)*sqrt(2*c*d*x + b*d))

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Sympy [A]  time = 18.9647, size = 360, normalized size = 6.55 \[ \begin{cases} - \frac{3 a c \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac{b^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac{7 b c x \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} - \frac{7 c^{2} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{2} d^{5} + 168 b^{3} c^{3} d^{5} x + 504 b^{2} c^{4} d^{5} x^{2} + 672 b c^{5} d^{5} x^{3} + 336 c^{6} d^{5} x^{4}} & \text{for}\: c \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\left (b d\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \]

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 + 672*b*c**5*d**5*x**3 + 336*c**6*d**5*x**4) - b**2*sq
rt(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/XCAS [A]  time = 0.225699, size = 65, normalized size = 1.18 \[ \frac{3 \, b^{2} d^{2} - 12 \, a c d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{2}}{84 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{3}} \]

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)