3.1273 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}-\frac{3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac{\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac{\sqrt{b d+2 c d x}}{64 c^4 d^7} \]

[Out]

(b^2 - 4*a*c)^3/(704*c^4*d*(b*d + 2*c*d*x)^(11/2)) - (3*(b^2 - 4*a*c)^2)/(448*c^
4*d^3*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(64*c^4*d^5*(b*d + 2*c*d*x)^(3/2))
+ Sqrt[b*d + 2*c*d*x]/(64*c^4*d^7)

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Rubi [A]  time = 0.145706, antiderivative size = 121, 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 \[ \frac{b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}-\frac{3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac{\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac{\sqrt{b d+2 c d x}}{64 c^4 d^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(b^2 - 4*a*c)^3/(704*c^4*d*(b*d + 2*c*d*x)^(11/2)) - (3*(b^2 - 4*a*c)^2)/(448*c^
4*d^3*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(64*c^4*d^5*(b*d + 2*c*d*x)^(3/2))
+ Sqrt[b*d + 2*c*d*x]/(64*c^4*d^7)

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Rubi in Sympy [A]  time = 36.2946, size = 116, normalized size = 0.96 \[ \frac{\left (- 4 a c + b^{2}\right )^{3}}{704 c^{4} d \left (b d + 2 c d x\right )^{\frac{11}{2}}} - \frac{3 \left (- 4 a c + b^{2}\right )^{2}}{448 c^{4} d^{3} \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{- 4 a c + b^{2}}{64 c^{4} d^{5} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{b d + 2 c d x}}{64 c^{4} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(13/2),x)

[Out]

(-4*a*c + b**2)**3/(704*c**4*d*(b*d + 2*c*d*x)**(11/2)) - 3*(-4*a*c + b**2)**2/(
448*c**4*d**3*(b*d + 2*c*d*x)**(7/2)) + (-4*a*c + b**2)/(64*c**4*d**5*(b*d + 2*c
*d*x)**(3/2)) + sqrt(b*d + 2*c*d*x)/(64*c**4*d**7)

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Mathematica [A]  time = 0.258718, size = 87, normalized size = 0.72 \[ \frac{(b+2 c x)^7 \left (\frac{7 \left (b^2-4 a c\right )^3}{(b+2 c x)^6}-\frac{33 \left (b^2-4 a c\right )^2}{(b+2 c x)^4}+\frac{77 \left (b^2-4 a c\right )}{(b+2 c x)^2}+77\right )}{4928 c^4 (d (b+2 c x))^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]

[Out]

((b + 2*c*x)^7*(77 + (7*(b^2 - 4*a*c)^3)/(b + 2*c*x)^6 - (33*(b^2 - 4*a*c)^2)/(b
+ 2*c*x)^4 + (77*(b^2 - 4*a*c))/(b + 2*c*x)^2))/(4928*c^4*(d*(b + 2*c*x))^(13/2
))

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Maple [A]  time = 0.01, size = 174, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -77\,{c}^{6}{x}^{6}-231\,b{c}^{5}{x}^{5}+77\,a{c}^{5}{x}^{4}-308\,{b}^{2}{c}^{4}{x}^{4}+154\,ab{c}^{4}{x}^{3}-231\,{b}^{3}{c}^{3}{x}^{3}+33\,{a}^{2}{c}^{4}{x}^{2}+99\,a{b}^{2}{c}^{3}{x}^{2}-99\,{b}^{4}{c}^{2}{x}^{2}+33\,{a}^{2}b{c}^{3}x+22\,a{b}^{3}{c}^{2}x-22\,{b}^{5}cx+7\,{a}^{3}{c}^{3}+3\,{a}^{2}{b}^{2}{c}^{2}+2\,a{b}^{4}c-2\,{b}^{6} \right ) }{77\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(13/2),x)

[Out]

-1/77*(2*c*x+b)*(-77*c^6*x^6-231*b*c^5*x^5+77*a*c^5*x^4-308*b^2*c^4*x^4+154*a*b*
c^4*x^3-231*b^3*c^3*x^3+33*a^2*c^4*x^2+99*a*b^2*c^3*x^2-99*b^4*c^2*x^2+33*a^2*b*
c^3*x+22*a*b^3*c^2*x-22*b^5*c*x+7*a^3*c^3+3*a^2*b^2*c^2+2*a*b^4*c-2*b^6)/c^4/(2*
c*d*x+b*d)^(13/2)

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Maxima [A]  time = 0.69023, size = 186, normalized size = 1.54 \[ \frac{\frac{77 \, \sqrt{2 \, c d x + b d}}{c^{3} d^{6}} + \frac{77 \,{\left (2 \, c d x + b d\right )}^{4}{\left (b^{2} - 4 \, a c\right )} - 33 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{2} d^{2} + 7 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{4}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}} c^{3} d^{4}}}{4928 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(13/2),x, algorithm="maxima")

[Out]

1/4928*(77*sqrt(2*c*d*x + b*d)/(c^3*d^6) + (77*(2*c*d*x + b*d)^4*(b^2 - 4*a*c) -
33*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^2*d^2 + 7*(b^6 - 12*a*b^4*c +
48*a^2*b^2*c^2 - 64*a^3*c^3)*d^4)/((2*c*d*x + b*d)^(11/2)*c^3*d^4))/(c*d)

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Fricas [A]  time = 0.210005, size = 321, normalized size = 2.65 \[ \frac{77 \, c^{6} x^{6} + 231 \, b c^{5} x^{5} + 2 \, b^{6} - 2 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 77 \,{\left (4 \, b^{2} c^{4} - a c^{5}\right )} x^{4} + 77 \,{\left (3 \, b^{3} c^{3} - 2 \, a b c^{4}\right )} x^{3} + 33 \,{\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - a^{2} c^{4}\right )} x^{2} + 11 \,{\left (2 \, b^{5} c - 2 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x}{77 \,{\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(13/2),x, algorithm="fricas")

[Out]

1/77*(77*c^6*x^6 + 231*b*c^5*x^5 + 2*b^6 - 2*a*b^4*c - 3*a^2*b^2*c^2 - 7*a^3*c^3
+ 77*(4*b^2*c^4 - a*c^5)*x^4 + 77*(3*b^3*c^3 - 2*a*b*c^4)*x^3 + 33*(3*b^4*c^2 -
3*a*b^2*c^3 - a^2*c^4)*x^2 + 11*(2*b^5*c - 2*a*b^3*c^2 - 3*a^2*b*c^3)*x)/((32*c
^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4
*c^5*d^6*x + b^5*c^4*d^6)*sqrt(2*c*d*x + b*d))

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Sympy [A]  time = 100.957, size = 1975, normalized size = 16.32 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(13/2),x)

[Out]

Piecewise((-7*a**3*c**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d
**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d
**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 3*a**2*b**2*c**2*sqr
t(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7
*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**
7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2*b*c**3*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c
**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**
7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x*
*6) - 33*a**2*c**4*x**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d
**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d
**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 2*a*b**4*c*sqrt(b*d
+ 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2
+ 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5
+ 4928*c**10*d**7*x**6) - 22*a*b**3*c**2*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d*
*7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3
+ 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) -
99*a*b**2*c**3*x**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*
x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*
x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 154*a*b*c**4*x**3*sqrt(b
*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x*
*2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x
**5 + 4928*c**10*d**7*x**6) - 77*a*c**5*x**4*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d
**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**
3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) +
2*b**6*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**
4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784
*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 22*b**5*c*x*sqrt(b*d + 2*c*d*x)/(77*
b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c*
*7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d
**7*x**6) + 99*b**4*c**2*x**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*
c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*
c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 231*b**3*c**3*
x**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c
**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*
c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 308*b**2*c**4*x**4*sqrt(b*d + 2*c*d*x)/
(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**
3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**
10*d**7*x**6) + 231*b*c**5*x**5*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**
5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**
2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) + 77*c**6*x**6
*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*
d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9
*d**7*x**5 + 4928*c**10*d**7*x**6), Ne(c, 0)), ((a**3*x + 3*a**2*b*x**2/2 + a*b*
*2*x**3 + b**3*x**4/4)/(b*d)**(13/2), True))

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GIAC/XCAS [A]  time = 0.25101, size = 238, normalized size = 1.97 \[ \frac{\sqrt{2 \, c d x + b d}}{64 \, c^{4} d^{7}} + \frac{7 \, b^{6} d^{4} - 84 \, a b^{4} c d^{4} + 336 \, a^{2} b^{2} c^{2} d^{4} - 448 \, a^{3} c^{3} d^{4} - 33 \,{\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 264 \,{\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 528 \,{\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 77 \,{\left (2 \, c d x + b d\right )}^{4} b^{2} - 308 \,{\left (2 \, c d x + b d\right )}^{4} a c}{4928 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}} c^{4} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(13/2),x, algorithm="giac")

[Out]

1/64*sqrt(2*c*d*x + b*d)/(c^4*d^7) + 1/4928*(7*b^6*d^4 - 84*a*b^4*c*d^4 + 336*a^
2*b^2*c^2*d^4 - 448*a^3*c^3*d^4 - 33*(2*c*d*x + b*d)^2*b^4*d^2 + 264*(2*c*d*x +
b*d)^2*a*b^2*c*d^2 - 528*(2*c*d*x + b*d)^2*a^2*c^2*d^2 + 77*(2*c*d*x + b*d)^4*b^
2 - 308*(2*c*d*x + b*d)^4*a*c)/((2*c*d*x + b*d)^(11/2)*c^4*d^5)