3.1256 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^{7/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{b^2-4 a c}{20 c^2 d (b d+2 c d x)^{5/2}}-\frac{1}{4 c^2 d^3 \sqrt{b d+2 c d x}} \]

[Out]

(b^2 - 4*a*c)/(20*c^2*d*(b*d + 2*c*d*x)^(5/2)) - 1/(4*c^2*d^3*Sqrt[b*d + 2*c*d*x
])

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Rubi [A]  time = 0.0725494, 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 \[ \frac{b^2-4 a c}{20 c^2 d (b d+2 c d x)^{5/2}}-\frac{1}{4 c^2 d^3 \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(7/2),x]

[Out]

(b^2 - 4*a*c)/(20*c^2*d*(b*d + 2*c*d*x)^(5/2)) - 1/(4*c^2*d^3*Sqrt[b*d + 2*c*d*x
])

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Rubi in Sympy [A]  time = 14.341, size = 51, normalized size = 0.93 \[ \frac{- a c + \frac{b^{2}}{4}}{5 c^{2} d \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{1}{4 c^{2} d^{3} \sqrt{b d + 2 c d x}} \]

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)**(7/2),x)

[Out]

(-a*c + b**2/4)/(5*c**2*d*(b*d + 2*c*d*x)**(5/2)) - 1/(4*c**2*d**3*sqrt(b*d + 2*
c*d*x))

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Mathematica [A]  time = 0.0495865, size = 44, normalized size = 0.8 \[ \frac{-c \left (a+5 c x^2\right )-b^2-5 b c x}{5 c^2 d (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 43, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( 5\,{c}^{2}{x}^{2}+5\,bxc+ac+{b}^{2} \right ) }{5\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(2*c*d*x+b*d)^(7/2),x)

[Out]

-1/5*(2*c*x+b)*(5*c^2*x^2+5*b*c*x+a*c+b^2)/c^2/(2*c*d*x+b*d)^(7/2)

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Maxima [A]  time = 0.692713, size = 61, normalized size = 1.11 \[ \frac{{\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \,{\left (2 \, c d x + b d\right )}^{2}}{20 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2),x, algorithm="maxima")

[Out]

1/20*((b^2 - 4*a*c)*d^2 - 5*(2*c*d*x + b*d)^2)/((2*c*d*x + b*d)^(5/2)*c^2*d^3)

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Fricas [A]  time = 0.210797, size = 90, normalized size = 1.64 \[ -\frac{5 \, c^{2} x^{2} + 5 \, b c x + b^{2} + a c}{5 \,{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2),x, algorithm="fricas")

[Out]

-1/5*(5*c^2*x^2 + 5*b*c*x + b^2 + a*c)/((4*c^4*d^3*x^2 + 4*b*c^3*d^3*x + b^2*c^2
*d^3)*sqrt(2*c*d*x + b*d))

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Sympy [A]  time = 8.70146, size = 298, normalized size = 5.42 \[ \begin{cases} - \frac{a c \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} - \frac{b^{2} \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} - \frac{5 b c x \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} - \frac{5 c^{2} x^{2} \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} & \text{for}\: c \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\left (b d\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**(7/2),x)

[Out]

Piecewise((-a*c*sqrt(b*d + 2*c*d*x)/(5*b**3*c**2*d**4 + 30*b**2*c**3*d**4*x + 60
*b*c**4*d**4*x**2 + 40*c**5*d**4*x**3) - b**2*sqrt(b*d + 2*c*d*x)/(5*b**3*c**2*d
**4 + 30*b**2*c**3*d**4*x + 60*b*c**4*d**4*x**2 + 40*c**5*d**4*x**3) - 5*b*c*x*s
qrt(b*d + 2*c*d*x)/(5*b**3*c**2*d**4 + 30*b**2*c**3*d**4*x + 60*b*c**4*d**4*x**2
+ 40*c**5*d**4*x**3) - 5*c**2*x**2*sqrt(b*d + 2*c*d*x)/(5*b**3*c**2*d**4 + 30*b
**2*c**3*d**4*x + 60*b*c**4*d**4*x**2 + 40*c**5*d**4*x**3), Ne(c, 0)), ((a*x + b
*x**2/2)/(b*d)**(7/2), True))

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GIAC/XCAS [A]  time = 0.229283, size = 63, normalized size = 1.15 \[ \frac{b^{2} d^{2} - 4 \, a c d^{2} - 5 \,{\left (2 \, c d x + b d\right )}^{2}}{20 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2),x, algorithm="giac")

[Out]

1/20*(b^2*d^2 - 4*a*c*d^2 - 5*(2*c*d*x + b*d)^2)/((2*c*d*x + b*d)^(5/2)*c^2*d^3)