3.1271 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{64 c^4 d^5}-\frac{\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}+\frac{\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac{(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]

[Out]

(b^2 - 4*a*c)^3/(448*c^4*d*(b*d + 2*c*d*x)^(7/2)) - (b^2 - 4*a*c)^2/(64*c^4*d^3*
(b*d + 2*c*d*x)^(3/2)) - (3*(b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x])/(64*c^4*d^5) + (b
*d + 2*c*d*x)^(5/2)/(320*c^4*d^7)

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Rubi [A]  time = 0.144876, 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 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{64 c^4 d^5}-\frac{\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}+\frac{\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac{(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(b^2 - 4*a*c)^3/(448*c^4*d*(b*d + 2*c*d*x)^(7/2)) - (b^2 - 4*a*c)^2/(64*c^4*d^3*
(b*d + 2*c*d*x)^(3/2)) - (3*(b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*x])/(64*c^4*d^5) + (b
*d + 2*c*d*x)^(5/2)/(320*c^4*d^7)

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Rubi in Sympy [A]  time = 35.9773, size = 116, normalized size = 0.96 \[ \frac{\left (- 4 a c + b^{2}\right )^{3}}{448 c^{4} d \left (b d + 2 c d x\right )^{\frac{7}{2}}} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{64 c^{4} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} - \frac{3 \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x}}{64 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{320 c^{4} d^{7}} \]

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)**(9/2),x)

[Out]

(-4*a*c + b**2)**3/(448*c**4*d*(b*d + 2*c*d*x)**(7/2)) - (-4*a*c + b**2)**2/(64*
c**4*d**3*(b*d + 2*c*d*x)**(3/2)) - 3*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)/(64*c*
*4*d**5) + (b*d + 2*c*d*x)**(5/2)/(320*c**4*d**7)

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Mathematica [A]  time = 0.272018, size = 90, normalized size = 0.74 \[ \frac{(b+2 c x)^5 \left (\frac{5 \left (b^2-4 a c\right )^3}{(b+2 c x)^4}-\frac{35 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}+420 a c-98 b^2+28 b c x+28 c^2 x^2\right )}{2240 c^4 (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(9/2),x]

[Out]

((b + 2*c*x)^5*(-98*b^2 + 420*a*c + 28*b*c*x + 28*c^2*x^2 + (5*(b^2 - 4*a*c)^3)/
(b + 2*c*x)^4 - (35*(b^2 - 4*a*c)^2)/(b + 2*c*x)^2))/(2240*c^4*(d*(b + 2*c*x))^(
9/2))

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Maple [A]  time = 0.011, size = 163, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -7\,{c}^{6}{x}^{6}-21\,b{c}^{5}{x}^{5}-105\,a{c}^{5}{x}^{4}-210\,ab{c}^{4}{x}^{3}+35\,{b}^{3}{c}^{3}{x}^{3}+35\,{a}^{2}{c}^{4}{x}^{2}-175\,a{b}^{2}{c}^{3}{x}^{2}+35\,{b}^{4}{c}^{2}{x}^{2}+35\,{a}^{2}b{c}^{3}x-70\,a{b}^{3}{c}^{2}x+14\,{b}^{5}cx+5\,{a}^{3}{c}^{3}+5\,{a}^{2}{b}^{2}{c}^{2}-10\,a{b}^{4}c+2\,{b}^{6} \right ) }{35\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(9/2),x)

[Out]

-1/35*(2*c*x+b)*(-7*c^6*x^6-21*b*c^5*x^5-105*a*c^5*x^4-210*a*b*c^4*x^3+35*b^3*c^
3*x^3+35*a^2*c^4*x^2-175*a*b^2*c^3*x^2+35*b^4*c^2*x^2+35*a^2*b*c^3*x-70*a*b^3*c^
2*x+14*b^5*c*x+5*a^3*c^3+5*a^2*b^2*c^2-10*a*b^4*c+2*b^6)/c^4/(2*c*d*x+b*d)^(9/2)

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Maxima [A]  time = 0.68689, size = 192, normalized size = 1.59 \[ -\frac{\frac{5 \,{\left (7 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{2} -{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{3} d^{2}} + \frac{7 \,{\left (15 \, \sqrt{2 \, c d x + b d}{\left (b^{2} - 4 \, a c\right )} d^{2} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )}}{c^{3} d^{6}}}{2240 \, c d} \]

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)^(9/2),x, algorithm="maxima")

[Out]

-1/2240*(5*(7*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^2 - (b^6 - 12*a*b^4
*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^2)/((2*c*d*x + b*d)^(7/2)*c^3*d^2) + 7*(15*s
qrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^2 - (2*c*d*x + b*d)^(5/2))/(c^3*d^6))/(c*d)

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Fricas [A]  time = 0.208483, size = 265, normalized size = 2.19 \[ \frac{7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 105 \, a c^{5} x^{4} - 2 \, b^{6} + 10 \, a b^{4} c - 5 \, a^{2} b^{2} c^{2} - 5 \, a^{3} c^{3} - 35 \,{\left (b^{3} c^{3} - 6 \, a b c^{4}\right )} x^{3} - 35 \,{\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} - 7 \,{\left (2 \, b^{5} c - 10 \, a b^{3} c^{2} + 5 \, a^{2} b c^{3}\right )} x}{35 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(2*c*d*x + b*d)^(9/2),x, algorithm="fricas")

[Out]

1/35*(7*c^6*x^6 + 21*b*c^5*x^5 + 105*a*c^5*x^4 - 2*b^6 + 10*a*b^4*c - 5*a^2*b^2*
c^2 - 5*a^3*c^3 - 35*(b^3*c^3 - 6*a*b*c^4)*x^3 - 35*(b^4*c^2 - 5*a*b^2*c^3 + a^2
*c^4)*x^2 - 7*(2*b^5*c - 10*a*b^3*c^2 + 5*a^2*b*c^3)*x)/((8*c^7*d^4*x^3 + 12*b*c
^6*d^4*x^2 + 6*b^2*c^5*d^4*x + b^3*c^4*d^4)*sqrt(2*c*d*x + b*d))

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Sympy [A]  time = 28.484, size = 1394, normalized size = 11.52 \[ \text{result too large to display} \]

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)**(9/2),x)

[Out]

Piecewise((-5*a**3*c**3*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d
**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) -
5*a**2*b**2*c**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x +
840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 35*a**2
*b*c**3*x*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b*
*2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 35*a**2*c**4*x
**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**
6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 10*a*b**4*c*sqrt(b*d
+ 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2
+ 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 70*a*b**3*c**2*x*sqrt(b*d + 2*c*
d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*
b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 175*a*b**2*c**3*x**2*sqrt(b*d + 2*c*d*x
)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c
**7*d**5*x**3 + 560*c**8*d**5*x**4) + 210*a*b*c**4*x**3*sqrt(b*d + 2*c*d*x)/(35*
b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d*
*5*x**3 + 560*c**8*d**5*x**4) + 105*a*c**5*x**4*sqrt(b*d + 2*c*d*x)/(35*b**4*c**
4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3
+ 560*c**8*d**5*x**4) - 2*b**6*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3
*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x
**4) - 14*b**5*c*x*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x
+ 840*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 35*b*
*4*c**2*x**2*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840
*b**2*c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) - 35*b**3*c**
3*x**3*sqrt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*
c**6*d**5*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 21*b*c**5*x**5*sq
rt(b*d + 2*c*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5
*x**2 + 1120*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4) + 7*c**6*x**6*sqrt(b*d + 2*c
*d*x)/(35*b**4*c**4*d**5 + 280*b**3*c**5*d**5*x + 840*b**2*c**6*d**5*x**2 + 1120
*b*c**7*d**5*x**3 + 560*c**8*d**5*x**4), 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)**(9/2), True))

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GIAC/XCAS [A]  time = 0.244974, size = 251, normalized size = 2.07 \[ \frac{b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{2} b^{4} + 56 \,{\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 112 \,{\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{448 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{4} d^{3}} - \frac{15 \, \sqrt{2 \, c d x + b d} b^{2} c^{16} d^{30} - 60 \, \sqrt{2 \, c d x + b d} a c^{17} d^{30} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{16} d^{28}}{320 \, c^{20} d^{35}} \]

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)^(9/2),x, algorithm="giac")

[Out]

1/448*(b^6*d^2 - 12*a*b^4*c*d^2 + 48*a^2*b^2*c^2*d^2 - 64*a^3*c^3*d^2 - 7*(2*c*d
*x + b*d)^2*b^4 + 56*(2*c*d*x + b*d)^2*a*b^2*c - 112*(2*c*d*x + b*d)^2*a^2*c^2)/
((2*c*d*x + b*d)^(7/2)*c^4*d^3) - 1/320*(15*sqrt(2*c*d*x + b*d)*b^2*c^16*d^30 -
60*sqrt(2*c*d*x + b*d)*a*c^17*d^30 - (2*c*d*x + b*d)^(5/2)*c^16*d^28)/(c^20*d^35
)