3.1266 \(\int \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=121 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac{(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]

[Out]

-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(3/2))/(192*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d +
2*c*d*x)^(7/2))/(448*c^4*d^3) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(11/2))/(704*c
^4*d^5) + (b*d + 2*c*d*x)^(15/2)/(960*c^4*d^7)

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Rubi [A]  time = 0.175291, 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 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac{(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(3/2))/(192*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d +
2*c*d*x)^(7/2))/(448*c^4*d^3) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(11/2))/(704*c
^4*d^5) + (b*d + 2*c*d*x)^(15/2)/(960*c^4*d^7)

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Rubi in Sympy [A]  time = 36.1603, size = 117, normalized size = 0.97 \[ - \frac{\left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{192 c^{4} d} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{7}{2}}}{448 c^{4} d^{3}} - \frac{3 \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{11}{2}}}{704 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{15}{2}}}{960 c^{4} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**3,x)

[Out]

-(-4*a*c + b**2)**3*(b*d + 2*c*d*x)**(3/2)/(192*c**4*d) + 3*(-4*a*c + b**2)**2*(
b*d + 2*c*d*x)**(7/2)/(448*c**4*d**3) - 3*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(11/2
)/(704*c**4*d**5) + (b*d + 2*c*d*x)**(15/2)/(960*c**4*d**7)

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Mathematica [A]  time = 0.238242, size = 162, normalized size = 1.34 \[ \frac{\left (15 b^2 c^2 \left (-11 a^2+15 a c x^2+14 c^2 x^4\right )+3 b c^3 x \left (165 a^2+210 a c x^2+77 c^2 x^4\right )+c^3 \left (385 a^3+495 a^2 c x^2+315 a c^2 x^4+77 c^3 x^6\right )-15 b^4 c \left (c x^2-2 a\right )+5 b^3 c^2 x \left (7 c x^2-18 a\right )-2 b^6+6 b^5 c x\right ) (d (b+2 c x))^{3/2}}{1155 c^4 d} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(3/2)*(-2*b^6 + 6*b^5*c*x - 15*b^4*c*(-2*a + c*x^2) + 5*b^3*c^2
*x*(-18*a + 7*c*x^2) + 15*b^2*c^2*(-11*a^2 + 15*a*c*x^2 + 14*c^2*x^4) + 3*b*c^3*
x*(165*a^2 + 210*a*c*x^2 + 77*c^2*x^4) + c^3*(385*a^3 + 495*a^2*c*x^2 + 315*a*c^
2*x^4 + 77*c^3*x^6)))/(1155*c^4*d)

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Maple [A]  time = 0.01, size = 174, normalized size = 1.4 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 77\,{c}^{6}{x}^{6}+231\,b{c}^{5}{x}^{5}+315\,a{c}^{5}{x}^{4}+210\,{b}^{2}{c}^{4}{x}^{4}+630\,ab{c}^{4}{x}^{3}+35\,{b}^{3}{c}^{3}{x}^{3}+495\,{a}^{2}{c}^{4}{x}^{2}+225\,a{b}^{2}{c}^{3}{x}^{2}-15\,{b}^{4}{c}^{2}{x}^{2}+495\,{a}^{2}b{c}^{3}x-90\,a{b}^{3}{c}^{2}x+6\,{b}^{5}cx+385\,{a}^{3}{c}^{3}-165\,{a}^{2}{b}^{2}{c}^{2}+30\,a{b}^{4}c-2\,{b}^{6} \right ) }{1155\,{c}^{4}}\sqrt{2\,cdx+bd}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^3,x)

[Out]

1/1155*(2*c*x+b)*(77*c^6*x^6+231*b*c^5*x^5+315*a*c^5*x^4+210*b^2*c^4*x^4+630*a*b
*c^4*x^3+35*b^3*c^3*x^3+495*a^2*c^4*x^2+225*a*b^2*c^3*x^2-15*b^4*c^2*x^2+495*a^2
*b*c^3*x-90*a*b^3*c^2*x+6*b^5*c*x+385*a^3*c^3-165*a^2*b^2*c^2+30*a*b^4*c-2*b^6)*
(2*c*d*x+b*d)^(1/2)/c^4

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Maxima [A]  time = 0.691391, size = 171, normalized size = 1.41 \[ -\frac{315 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 495 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} d^{4} + 385 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} d^{6} - 77 \,{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}{73920 \, c^{4} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/73920*(315*(2*c*d*x + b*d)^(11/2)*(b^2 - 4*a*c)*d^2 - 495*(b^4 - 8*a*b^2*c +
16*a^2*c^2)*(2*c*d*x + b*d)^(7/2)*d^4 + 385*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 -
64*a^3*c^3)*(2*c*d*x + b*d)^(3/2)*d^6 - 77*(2*c*d*x + b*d)^(15/2))/(c^4*d^7)

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Fricas [A]  time = 0.208061, size = 277, normalized size = 2.29 \[ \frac{{\left (154 \, c^{7} x^{7} + 539 \, b c^{6} x^{6} - 2 \, b^{7} + 30 \, a b^{5} c - 165 \, a^{2} b^{3} c^{2} + 385 \, a^{3} b c^{3} + 21 \,{\left (31 \, b^{2} c^{5} + 30 \, a c^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{3} c^{4} + 45 \, a b c^{5}\right )} x^{4} + 5 \,{\left (b^{4} c^{3} + 216 \, a b^{2} c^{4} + 198 \, a^{2} c^{5}\right )} x^{3} - 3 \,{\left (b^{5} c^{2} - 15 \, a b^{3} c^{3} - 495 \, a^{2} b c^{4}\right )} x^{2} +{\left (2 \, b^{6} c - 30 \, a b^{4} c^{2} + 165 \, a^{2} b^{2} c^{3} + 770 \, a^{3} c^{4}\right )} x\right )} \sqrt{2 \, c d x + b d}}{1155 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1155*(154*c^7*x^7 + 539*b*c^6*x^6 - 2*b^7 + 30*a*b^5*c - 165*a^2*b^3*c^2 + 385
*a^3*b*c^3 + 21*(31*b^2*c^5 + 30*a*c^6)*x^5 + 35*(8*b^3*c^4 + 45*a*b*c^5)*x^4 +
5*(b^4*c^3 + 216*a*b^2*c^4 + 198*a^2*c^5)*x^3 - 3*(b^5*c^2 - 15*a*b^3*c^3 - 495*
a^2*b*c^4)*x^2 + (2*b^6*c - 30*a*b^4*c^2 + 165*a^2*b^2*c^3 + 770*a^3*c^4)*x)*sqr
t(2*c*d*x + b*d)/c^4

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Sympy [A]  time = 4.05579, size = 151, normalized size = 1.25 \[ \frac{\frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}{192 c^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{448 c^{3} d^{2}} + \frac{\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{11}{2}}}{704 c^{3} d^{4}} + \frac{\left (b d + 2 c d x\right )^{\frac{15}{2}}}{960 c^{3} d^{6}}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**3,x)

[Out]

((b*d + 2*c*d*x)**(3/2)*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)/
(192*c**3) + (b*d + 2*c*d*x)**(7/2)*(48*a**2*c**2 - 24*a*b**2*c + 3*b**4)/(448*c
**3*d**2) + (12*a*c - 3*b**2)*(b*d + 2*c*d*x)**(11/2)/(704*c**3*d**4) + (b*d + 2
*c*d*x)**(15/2)/(960*c**3*d**6))/(c*d)

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GIAC/XCAS [A]  time = 0.236171, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done