3.1268 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.142823, 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 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{448 c^4 d^5}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{64 c^4 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{11/2}}{704 c^4 d^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.9841, size = 116, normalized size = 0.96 \[ \frac{\left (- 4 a c + b^{2}\right )^{3}}{64 c^{4} d \sqrt{b d + 2 c d x}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{64 c^{4} d^{3}} - \frac{3 \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}}{448 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{11}{2}}}{704 c^{4} d^{7}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.152367, size = 158, normalized size = 1.31 \[ \frac{b^2 c^2 \left (77 a^2+11 a c x^2+18 c^2 x^4\right )+b c^3 x \left (77 a^2+66 a c x^2+21 c^2 x^4\right )+c^3 \left (-77 a^3+77 a^2 c x^2+33 a c^2 x^4+7 c^3 x^6\right )-b^4 c \left (22 a+c x^2\right )+b^3 c^2 x \left (c x^2-22 a\right )+2 b^6+2 b^5 c x}{77 c^4 d \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^6 + 2*b^5*c*x + b^3*c^2*x*(-22*a + c*x^2) - b^4*c*(22*a + c*x^2) + b^2*c^2*
(77*a^2 + 11*a*c*x^2 + 18*c^2*x^4) + b*c^3*x*(77*a^2 + 66*a*c*x^2 + 21*c^2*x^4)
+ c^3*(-77*a^3 + 77*a^2*c*x^2 + 33*a*c^2*x^4 + 7*c^3*x^6))/(77*c^4*d*Sqrt[d*(b +
2*c*x)])

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Maple [A]  time = 0.01, size = 173, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -7\,{c}^{6}{x}^{6}-21\,b{c}^{5}{x}^{5}-33\,a{c}^{5}{x}^{4}-18\,{b}^{2}{c}^{4}{x}^{4}-66\,ab{c}^{4}{x}^{3}-{b}^{3}{c}^{3}{x}^{3}-77\,{a}^{2}{c}^{4}{x}^{2}-11\,a{b}^{2}{c}^{3}{x}^{2}+{b}^{4}{c}^{2}{x}^{2}-77\,{a}^{2}b{c}^{3}x+22\,a{b}^{3}{c}^{2}x-2\,{b}^{5}cx+77\,{a}^{3}{c}^{3}-77\,{a}^{2}{b}^{2}{c}^{2}+22\,a{b}^{4}c-2\,{b}^{6} \right ) }{77\,{c}^{4}} \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-1/77*(2*c*x+b)*(-7*c^6*x^6-21*b*c^5*x^5-33*a*c^5*x^4-18*b^2*c^4*x^4-66*a*b*c^4*
x^3-b^3*c^3*x^3-77*a^2*c^4*x^2-11*a*b^2*c^3*x^2+b^4*c^2*x^2-77*a^2*b*c^3*x+22*a*
b^3*c^2*x-2*b^5*c*x+77*a^3*c^3-77*a^2*b^2*c^2+22*a*b^4*c-2*b^6)/c^4/(2*c*d*x+b*d
)^(3/2)

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Maxima [A]  time = 0.684501, size = 184, normalized size = 1.52 \[ \frac{\frac{77 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{\sqrt{2 \, c d x + b d} c^{3}} - \frac{33 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 77 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} d^{4} - 7 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{c^{3} d^{6}}}{4928 \, c d} \]

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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.209797, size = 221, normalized size = 1.83 \[ \frac{7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 2 \, b^{6} - 22 \, a b^{4} c + 77 \, a^{2} b^{2} c^{2} - 77 \, a^{3} c^{3} + 3 \,{\left (6 \, b^{2} c^{4} + 11 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} + 66 \, a b c^{4}\right )} x^{3} -{\left (b^{4} c^{2} - 11 \, a b^{2} c^{3} - 77 \, a^{2} c^{4}\right )} x^{2} +{\left (2 \, b^{5} c - 22 \, a b^{3} c^{2} + 77 \, a^{2} b c^{3}\right )} x}{77 \, \sqrt{2 \, c d x + b d} c^{4} d} \]

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)^(3/2),x, algorithm="fricas")

[Out]

1/77*(7*c^6*x^6 + 21*b*c^5*x^5 + 2*b^6 - 22*a*b^4*c + 77*a^2*b^2*c^2 - 77*a^3*c^
3 + 3*(6*b^2*c^4 + 11*a*c^5)*x^4 + (b^3*c^3 + 66*a*b*c^4)*x^3 - (b^4*c^2 - 11*a*
b^2*c^3 - 77*a^2*c^4)*x^2 + (2*b^5*c - 22*a*b^3*c^2 + 77*a^2*b*c^3)*x)/(sqrt(2*c
*d*x + b*d)*c^4*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{3}}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**3/(d*(b + 2*c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.245675, size = 252, normalized size = 2.08 \[ \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{64 \, \sqrt{2 \, c d x + b d} c^{4} d} + \frac{77 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c^{40} d^{74} - 616 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} c^{41} d^{74} + 1232 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a^{2} c^{42} d^{74} - 33 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b^{2} c^{40} d^{72} + 132 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} a c^{41} d^{72} + 7 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}} c^{40} d^{70}}{4928 \, c^{44} d^{77}} \]

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)^(3/2),x, algorithm="giac")

[Out]

1/64*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/(sqrt(2*c*d*x + b*d)*c^4*d
) + 1/4928*(77*(2*c*d*x + b*d)^(3/2)*b^4*c^40*d^74 - 616*(2*c*d*x + b*d)^(3/2)*a
*b^2*c^41*d^74 + 1232*(2*c*d*x + b*d)^(3/2)*a^2*c^42*d^74 - 33*(2*c*d*x + b*d)^(
7/2)*b^2*c^40*d^72 + 132*(2*c*d*x + b*d)^(7/2)*a*c^41*d^72 + 7*(2*c*d*x + b*d)^(
11/2)*c^40*d^70)/(c^44*d^77)