3.1258 \(\int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac{(b d+2 c d x)^{13/2}}{208 c^3 d^5} \]

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5/2))/(80*c^3*d) - ((b^2 - 4*a*c)*(b*d + 2*c*d
*x)^(9/2))/(72*c^3*d^3) + (b*d + 2*c*d*x)^(13/2)/(208*c^3*d^5)

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Rubi [A]  time = 0.147152, 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 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{72 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{80 c^3 d}+\frac{(b d+2 c d x)^{13/2}}{208 c^3 d^5} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5/2))/(80*c^3*d) - ((b^2 - 4*a*c)*(b*d + 2*c*d
*x)^(9/2))/(72*c^3*d^3) + (b*d + 2*c*d*x)^(13/2)/(208*c^3*d^5)

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Rubi in Sympy [A]  time = 28.8273, size = 82, normalized size = 0.93 \[ \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{80 c^{3} d} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{9}{2}}}{72 c^{3} d^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{13}{2}}}{208 c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(5/2)/(80*c**3*d) - (-4*a*c + b**2)*(b*d + 2
*c*d*x)**(9/2)/(72*c**3*d**3) + (b*d + 2*c*d*x)**(13/2)/(208*c**3*d**5)

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Mathematica [A]  time = 0.109338, size = 92, normalized size = 1.05 \[ \frac{\left (c^2 \left (117 a^2+130 a c x^2+45 c^2 x^4\right )+b^2 c \left (35 c x^2-26 a\right )+10 b c^2 x \left (13 a+9 c x^2\right )+2 b^4-10 b^3 c x\right ) (d (b+2 c x))^{5/2}}{585 c^3 d} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(5/2)*(2*b^4 - 10*b^3*c*x + 10*b*c^2*x*(13*a + 9*c*x^2) + b^2*c
*(-26*a + 35*c*x^2) + c^2*(117*a^2 + 130*a*c*x^2 + 45*c^2*x^4)))/(585*c^3*d)

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Maple [A]  time = 0.01, size = 96, normalized size = 1.1 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 45\,{c}^{4}{x}^{4}+90\,b{x}^{3}{c}^{3}+130\,a{c}^{3}{x}^{2}+35\,{b}^{2}{c}^{2}{x}^{2}+130\,ab{c}^{2}x-10\,{b}^{3}cx+117\,{a}^{2}{c}^{2}-26\,ac{b}^{2}+2\,{b}^{4} \right ) }{585\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/585*(2*c*x+b)*(45*c^4*x^4+90*b*c^3*x^3+130*a*c^3*x^2+35*b^2*c^2*x^2+130*a*b*c^
2*x-10*b^3*c*x+117*a^2*c^2-26*a*b^2*c+2*b^4)*(2*c*d*x+b*d)^(3/2)/c^3

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Maxima [A]  time = 0.686126, size = 109, normalized size = 1.24 \[ -\frac{130 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 117 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} d^{4} - 45 \,{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}{9360 \, c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9360*(130*(2*c*d*x + b*d)^(9/2)*(b^2 - 4*a*c)*d^2 - 117*(b^4 - 8*a*b^2*c + 16
*a^2*c^2)*(2*c*d*x + b*d)^(5/2)*d^4 - 45*(2*c*d*x + b*d)^(13/2))/(c^3*d^5)

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Fricas [A]  time = 0.208711, size = 221, normalized size = 2.51 \[ \frac{{\left (180 \, c^{6} d x^{6} + 540 \, b c^{5} d x^{5} + 5 \,{\left (109 \, b^{2} c^{4} + 104 \, a c^{5}\right )} d x^{4} + 10 \,{\left (19 \, b^{3} c^{3} + 104 \, a b c^{4}\right )} d x^{3} + 3 \,{\left (b^{4} c^{2} + 182 \, a b^{2} c^{3} + 156 \, a^{2} c^{4}\right )} d x^{2} - 2 \,{\left (b^{5} c - 13 \, a b^{3} c^{2} - 234 \, a^{2} b c^{3}\right )} d x +{\left (2 \, b^{6} - 26 \, a b^{4} c + 117 \, a^{2} b^{2} c^{2}\right )} d\right )} \sqrt{2 \, c d x + b d}}{585 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/585*(180*c^6*d*x^6 + 540*b*c^5*d*x^5 + 5*(109*b^2*c^4 + 104*a*c^5)*d*x^4 + 10*
(19*b^3*c^3 + 104*a*b*c^4)*d*x^3 + 3*(b^4*c^2 + 182*a*b^2*c^3 + 156*a^2*c^4)*d*x
^2 - 2*(b^5*c - 13*a*b^3*c^2 - 234*a^2*b*c^3)*d*x + (2*b^6 - 26*a*b^4*c + 117*a^
2*b^2*c^2)*d)*sqrt(2*c*d*x + b*d)/c^3

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Sympy [A]  time = 8.79901, size = 695, normalized size = 7.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*b*d*Piecewise((x*sqrt(b*d), Eq(c, 0)), (0, Eq(d, 0)), ((b*d + 2*c*d*x)**(3/
2)/(3*c*d), True)) + a**2*(-b*d*(b*d + 2*c*d*x)**(3/2)/3 + (b*d + 2*c*d*x)**(5/2
)/5)/(c*d) + a*b**2*(-b*d*(b*d + 2*c*d*x)**(3/2)/3 + (b*d + 2*c*d*x)**(5/2)/5)/(
c**2*d) + 3*a*b*(b**2*d**2*(b*d + 2*c*d*x)**(3/2)/3 - 2*b*d*(b*d + 2*c*d*x)**(5/
2)/5 + (b*d + 2*c*d*x)**(7/2)/7)/(2*c**2*d**2) + a*(-b**3*d**3*(b*d + 2*c*d*x)**
(3/2)/3 + 3*b**2*d**2*(b*d + 2*c*d*x)**(5/2)/5 - 3*b*d*(b*d + 2*c*d*x)**(7/2)/7
+ (b*d + 2*c*d*x)**(9/2)/9)/(2*c**2*d**3) + b**3*(b**2*d**2*(b*d + 2*c*d*x)**(3/
2)/3 - 2*b*d*(b*d + 2*c*d*x)**(5/2)/5 + (b*d + 2*c*d*x)**(7/2)/7)/(4*c**3*d**2)
+ b**2*(-b**3*d**3*(b*d + 2*c*d*x)**(3/2)/3 + 3*b**2*d**2*(b*d + 2*c*d*x)**(5/2)
/5 - 3*b*d*(b*d + 2*c*d*x)**(7/2)/7 + (b*d + 2*c*d*x)**(9/2)/9)/(2*c**3*d**3) +
5*b*(b**4*d**4*(b*d + 2*c*d*x)**(3/2)/3 - 4*b**3*d**3*(b*d + 2*c*d*x)**(5/2)/5 +
6*b**2*d**2*(b*d + 2*c*d*x)**(7/2)/7 - 4*b*d*(b*d + 2*c*d*x)**(9/2)/9 + (b*d +
2*c*d*x)**(11/2)/11)/(16*c**3*d**4) + (-b**5*d**5*(b*d + 2*c*d*x)**(3/2)/3 + b**
4*d**4*(b*d + 2*c*d*x)**(5/2) - 10*b**3*d**3*(b*d + 2*c*d*x)**(7/2)/7 + 10*b**2*
d**2*(b*d + 2*c*d*x)**(9/2)/9 - 5*b*d*(b*d + 2*c*d*x)**(11/2)/11 + (b*d + 2*c*d*
x)**(13/2)/13)/(16*c**3*d**5)

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GIAC/XCAS [A]  time = 0.240189, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done