3.1263 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{7/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5} \]

[Out]

-(b^2 - 4*a*c)^2/(80*c^3*d*(b*d + 2*c*d*x)^(5/2)) + (b^2 - 4*a*c)/(8*c^3*d^3*Sqr
t[b*d + 2*c*d*x]) + (b*d + 2*c*d*x)^(3/2)/(48*c^3*d^5)

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Rubi [A]  time = 0.114441, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2 - 4*a*c)^2/(80*c^3*d*(b*d + 2*c*d*x)^(5/2)) + (b^2 - 4*a*c)/(8*c^3*d^3*Sqr
t[b*d + 2*c*d*x]) + (b*d + 2*c*d*x)^(3/2)/(48*c^3*d^5)

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Rubi in Sympy [A]  time = 28.5275, size = 82, normalized size = 0.93 \[ - \frac{\left (- 4 a c + b^{2}\right )^{2}}{80 c^{3} d \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{- 4 a c + b^{2}}{8 c^{3} d^{3} \sqrt{b d + 2 c d x}} + \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{48 c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-4*a*c + b**2)**2/(80*c**3*d*(b*d + 2*c*d*x)**(5/2)) + (-4*a*c + b**2)/(8*c**3
*d**3*sqrt(b*d + 2*c*d*x)) + (b*d + 2*c*d*x)**(3/2)/(48*c**3*d**5)

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Mathematica [A]  time = 0.213819, size = 73, normalized size = 0.83 \[ \frac{(b+2 c x)^4 \left (-\frac{3 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{30 \left (b^2-4 a c\right )}{b+2 c x}+5 b+10 c x\right )}{240 c^3 (d (b+2 c x))^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b + 2*c*x)^4*(5*b + 10*c*x - (3*(b^2 - 4*a*c)^2)/(b + 2*c*x)^3 + (30*(b^2 - 4*
a*c))/(b + 2*c*x)))/(240*c^3*(d*(b + 2*c*x))^(7/2))

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Maple [A]  time = 0.01, size = 96, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -5\,{c}^{4}{x}^{4}-10\,b{x}^{3}{c}^{3}+30\,a{c}^{3}{x}^{2}-15\,{b}^{2}{c}^{2}{x}^{2}+30\,ab{c}^{2}x-10\,{b}^{3}cx+3\,{a}^{2}{c}^{2}+6\,ac{b}^{2}-2\,{b}^{4} \right ) }{15\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(2*c*x+b)*(-5*c^4*x^4-10*b*c^3*x^3+30*a*c^3*x^2-15*b^2*c^2*x^2+30*a*b*c^2*
x-10*b^3*c*x+3*a^2*c^2+6*a*b^2*c-2*b^4)/c^3/(2*c*d*x+b*d)^(7/2)

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Maxima [A]  time = 0.687442, size = 126, normalized size = 1.43 \[ \frac{\frac{5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{c^{2} d^{4}} + \frac{3 \,{\left (10 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} -{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{2}}}{240 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(5*(2*c*d*x + b*d)^(3/2)/(c^2*d^4) + 3*(10*(2*c*d*x + b*d)^2*(b^2 - 4*a*c)
- (b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2)/((2*c*d*x + b*d)^(5/2)*c^2*d^2))/(c*d)

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Fricas [A]  time = 0.211129, size = 162, normalized size = 1.84 \[ \frac{5 \, c^{4} x^{4} + 10 \, b c^{3} x^{3} + 2 \, b^{4} - 6 \, a b^{2} c - 3 \, a^{2} c^{2} + 15 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} x}{15 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(5*c^4*x^4 + 10*b*c^3*x^3 + 2*b^4 - 6*a*b^2*c - 3*a^2*c^2 + 15*(b^2*c^2 - 2
*a*c^3)*x^2 + 10*(b^3*c - 3*a*b*c^2)*x)/((4*c^5*d^3*x^2 + 4*b*c^4*d^3*x + b^2*c^
3*d^3)*sqrt(2*c*d*x + b*d))

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Sympy [A]  time = 10.9249, size = 688, normalized size = 7.82 \[ \begin{cases} - \frac{3 a^{2} c^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{6 a b^{2} c \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{30 a b c^{2} x \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{30 a c^{3} x^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{2 b^{4} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{10 b^{3} c x \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{15 b^{2} c^{2} x^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{10 b c^{3} x^{3} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{5 c^{4} x^{4} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\left (b d\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*a**2*c**2*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d*
*4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) - 6*a*b**2*c*sqrt(b*d + 2*c*d*
x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d*
*4*x**3) - 30*a*b*c**2*x*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d
**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) - 30*a*c**3*x**2*sqrt(b*d + 2
*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c*
*6*d**4*x**3) + 2*b**4*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**
4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 10*b**3*c*x*sqrt(b*d + 2*c*d*
x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d*
*4*x**3) + 15*b**2*c**2*x**2*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c*
*4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 10*b*c**3*x**3*sqrt(b*d
+ 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 12
0*c**6*d**4*x**3) + 5*c**4*x**4*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2
*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3), Ne(c, 0)), ((a**2*x +
a*b*x**2 + b**2*x**3/3)/(b*d)**(7/2), True))

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GIAC/XCAS [A]  time = 0.234635, size = 134, normalized size = 1.52 \[ \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{48 \, c^{3} d^{5}} - \frac{b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2} - 10 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} + 40 \,{\left (2 \, c d x + b d\right )}^{2} a c}{80 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*c*d*x + b*d)^(3/2)/(c^3*d^5) - 1/80*(b^4*d^2 - 8*a*b^2*c*d^2 + 16*a^2*c^
2*d^2 - 10*(2*c*d*x + b*d)^2*b^2 + 40*(2*c*d*x + b*d)^2*a*c)/((2*c*d*x + b*d)^(5
/2)*c^3*d^3)