3.1267 \(\int \frac{\left (a+b x+c x^2\right )^3}{\sqrt{b d+2 c d x}} \, dx\)
Optimal. Leaf size=121 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}{64 c^4 d}+\frac{(b d+2 c d x)^{13/2}}{832 c^4 d^7} \]
[Out]
-((b^2 - 4*a*c)^3*Sqrt[b*d + 2*c*d*x])/(64*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*
c*d*x)^(5/2))/(320*c^4*d^3) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2))/(192*c^4*d^5
) + (b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7)
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Rubi [A] time = 0.148915, 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
\[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac{\left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}{64 c^4 d}+\frac{(b d+2 c d x)^{13/2}}{832 c^4 d^7} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3/Sqrt[b*d + 2*c*d*x],x]
[Out]
-((b^2 - 4*a*c)^3*Sqrt[b*d + 2*c*d*x])/(64*c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*
c*d*x)^(5/2))/(320*c^4*d^3) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2))/(192*c^4*d^5
) + (b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7)
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Rubi in Sympy [A] time = 36.2086, size = 116, normalized size = 0.96 \[ - \frac{\left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x}}{64 c^{4} d} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{320 c^{4} d^{3}} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{9}{2}}}{192 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{13}{2}}}{832 c^{4} d^{7}} \]
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)**(1/2),x)
[Out]
-(-4*a*c + b**2)**3*sqrt(b*d + 2*c*d*x)/(64*c**4*d) + 3*(-4*a*c + b**2)**2*(b*d
+ 2*c*d*x)**(5/2)/(320*c**4*d**3) - (-4*a*c + b**2)*(b*d + 2*c*d*x)**(9/2)/(192*
c**4*d**5) + (b*d + 2*c*d*x)**(13/2)/(832*c**4*d**7)
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Mathematica [A] time = 0.187481, size = 159, normalized size = 1.31 \[ \frac{\left (b^2 c^2 \left (-117 a^2+39 a c x^2+40 c^2 x^4\right )+b c^3 x \left (117 a^2+130 a c x^2+45 c^2 x^4\right )+c^3 \left (195 a^3+117 a^2 c x^2+65 a c^2 x^4+15 c^3 x^6\right )+b^4 c \left (26 a-3 c x^2\right )+b^3 c^2 x \left (5 c x^2-26 a\right )-2 b^6+2 b^5 c x\right ) \sqrt{d (b+2 c x)}}{195 c^4 d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3/Sqrt[b*d + 2*c*d*x],x]
[Out]
(Sqrt[d*(b + 2*c*x)]*(-2*b^6 + 2*b^5*c*x + b^4*c*(26*a - 3*c*x^2) + b^3*c^2*x*(-
26*a + 5*c*x^2) + b^2*c^2*(-117*a^2 + 39*a*c*x^2 + 40*c^2*x^4) + b*c^3*x*(117*a^
2 + 130*a*c*x^2 + 45*c^2*x^4) + c^3*(195*a^3 + 117*a^2*c*x^2 + 65*a*c^2*x^4 + 15
*c^3*x^6)))/(195*c^4*d)
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Maple [A] time = 0.01, size = 174, normalized size = 1.4 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 15\,{c}^{6}{x}^{6}+45\,b{c}^{5}{x}^{5}+65\,a{c}^{5}{x}^{4}+40\,{b}^{2}{c}^{4}{x}^{4}+130\,ab{c}^{4}{x}^{3}+5\,{b}^{3}{c}^{3}{x}^{3}+117\,{a}^{2}{c}^{4}{x}^{2}+39\,a{b}^{2}{c}^{3}{x}^{2}-3\,{b}^{4}{c}^{2}{x}^{2}+117\,{a}^{2}b{c}^{3}x-26\,a{b}^{3}{c}^{2}x+2\,{b}^{5}cx+195\,{a}^{3}{c}^{3}-117\,{a}^{2}{b}^{2}{c}^{2}+26\,a{b}^{4}c-2\,{b}^{6} \right ) }{195\,{c}^{4}}{\frac{1}{\sqrt{2\,cdx+bd}}}} \]
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)^(1/2),x)
[Out]
1/195*(2*c*x+b)*(15*c^6*x^6+45*b*c^5*x^5+65*a*c^5*x^4+40*b^2*c^4*x^4+130*a*b*c^4
*x^3+5*b^3*c^3*x^3+117*a^2*c^4*x^2+39*a*b^2*c^3*x^2-3*b^4*c^2*x^2+117*a^2*b*c^3*
x-26*a*b^3*c^2*x+2*b^5*c*x+195*a^3*c^3-117*a^2*b^2*c^2+26*a*b^4*c-2*b^6)/c^4/(2*
c*d*x+b*d)^(1/2)
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Maxima [A] time = 0.703864, size = 1050, normalized size = 8.68 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/sqrt(2*c*d*x + b*d),x, algorithm="maxima")
[Out]
1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 48048*a^2*(10*(3*sqrt(2*c*d*x + b*d)*
b*d - (2*c*d*x + b*d)^(3/2))*b/(c*d) - (15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c
*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))/(c*d^2)) + 572*a*(84*(15*sqrt(2
*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*
b^2/(c^2*d^2) - 36*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^
2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*b/(c^2*d^3) + (3
15*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*
x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2
))/(c^2*d^4)) - 3432*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*
b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*b^3/(c^3*d^3)
+ 572*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378
*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b
*d)^(9/2))*b^2/(c^3*d^4) - 130*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1155*(2*c*d*x
+ b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^3 - 990*(2*c*d*x + b*d)^
(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))*b/(c^
3*d^5) + 5*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d*x + b*d)^(3/2)*b^5*d^
5 + 9009*(2*c*d*x + b*d)^(5/2)*b^4*d^4 - 8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3 + 50
05*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^(11/2)*b*d + 231*(2*c*d*
x + b*d)^(13/2))/(c^3*d^6))/(c*d)
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Fricas [A] time = 0.210222, size = 223, normalized size = 1.84 \[ \frac{{\left (15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} - 2 \, b^{6} + 26 \, a b^{4} c - 117 \, a^{2} b^{2} c^{2} + 195 \, a^{3} c^{3} + 5 \,{\left (8 \, b^{2} c^{4} + 13 \, a c^{5}\right )} x^{4} + 5 \,{\left (b^{3} c^{3} + 26 \, a b c^{4}\right )} x^{3} - 3 \,{\left (b^{4} c^{2} - 13 \, a b^{2} c^{3} - 39 \, a^{2} c^{4}\right )} x^{2} +{\left (2 \, b^{5} c - 26 \, a b^{3} c^{2} + 117 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{195 \, c^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/sqrt(2*c*d*x + b*d),x, algorithm="fricas")
[Out]
1/195*(15*c^6*x^6 + 45*b*c^5*x^5 - 2*b^6 + 26*a*b^4*c - 117*a^2*b^2*c^2 + 195*a^
3*c^3 + 5*(8*b^2*c^4 + 13*a*c^5)*x^4 + 5*(b^3*c^3 + 26*a*b*c^4)*x^3 - 3*(b^4*c^2
- 13*a*b^2*c^3 - 39*a^2*c^4)*x^2 + (2*b^5*c - 26*a*b^3*c^2 + 117*a^2*b*c^3)*x)*
sqrt(2*c*d*x + b*d)/(c^4*d)
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Sympy [A] time = 125.92, size = 1363, normalized size = 11.26 \[ \text{result too large to display} \]
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)**(1/2),x)
[Out]
Piecewise((-(a**3*b/sqrt(b*d + 2*c*d*x) + a**3*(-b*d/sqrt(b*d + 2*c*d*x) - sqrt(
b*d + 2*c*d*x))/d + 3*a**2*b**2*(-b*d/sqrt(b*d + 2*c*d*x) - sqrt(b*d + 2*c*d*x))
/(2*c*d) + 9*a**2*b*(b**2*d**2/sqrt(b*d + 2*c*d*x) + 2*b*d*sqrt(b*d + 2*c*d*x) -
(b*d + 2*c*d*x)**(3/2)/3)/(4*c*d**2) + 3*a**2*(-b**3*d**3/sqrt(b*d + 2*c*d*x) -
3*b**2*d**2*sqrt(b*d + 2*c*d*x) + b*d*(b*d + 2*c*d*x)**(3/2) - (b*d + 2*c*d*x)*
*(5/2)/5)/(4*c*d**3) + 3*a*b**3*(b**2*d**2/sqrt(b*d + 2*c*d*x) + 2*b*d*sqrt(b*d
+ 2*c*d*x) - (b*d + 2*c*d*x)**(3/2)/3)/(4*c**2*d**2) + 3*a*b**2*(-b**3*d**3/sqrt
(b*d + 2*c*d*x) - 3*b**2*d**2*sqrt(b*d + 2*c*d*x) + b*d*(b*d + 2*c*d*x)**(3/2) -
(b*d + 2*c*d*x)**(5/2)/5)/(2*c**2*d**3) + 15*a*b*(b**4*d**4/sqrt(b*d + 2*c*d*x)
+ 4*b**3*d**3*sqrt(b*d + 2*c*d*x) - 2*b**2*d**2*(b*d + 2*c*d*x)**(3/2) + 4*b*d*
(b*d + 2*c*d*x)**(5/2)/5 - (b*d + 2*c*d*x)**(7/2)/7)/(16*c**2*d**4) + 3*a*(-b**5
*d**5/sqrt(b*d + 2*c*d*x) - 5*b**4*d**4*sqrt(b*d + 2*c*d*x) + 10*b**3*d**3*(b*d
+ 2*c*d*x)**(3/2)/3 - 2*b**2*d**2*(b*d + 2*c*d*x)**(5/2) + 5*b*d*(b*d + 2*c*d*x)
**(7/2)/7 - (b*d + 2*c*d*x)**(9/2)/9)/(16*c**2*d**5) + b**4*(-b**3*d**3/sqrt(b*d
+ 2*c*d*x) - 3*b**2*d**2*sqrt(b*d + 2*c*d*x) + b*d*(b*d + 2*c*d*x)**(3/2) - (b*
d + 2*c*d*x)**(5/2)/5)/(8*c**3*d**3) + 5*b**3*(b**4*d**4/sqrt(b*d + 2*c*d*x) + 4
*b**3*d**3*sqrt(b*d + 2*c*d*x) - 2*b**2*d**2*(b*d + 2*c*d*x)**(3/2) + 4*b*d*(b*d
+ 2*c*d*x)**(5/2)/5 - (b*d + 2*c*d*x)**(7/2)/7)/(16*c**3*d**4) + 9*b**2*(-b**5*
d**5/sqrt(b*d + 2*c*d*x) - 5*b**4*d**4*sqrt(b*d + 2*c*d*x) + 10*b**3*d**3*(b*d +
2*c*d*x)**(3/2)/3 - 2*b**2*d**2*(b*d + 2*c*d*x)**(5/2) + 5*b*d*(b*d + 2*c*d*x)*
*(7/2)/7 - (b*d + 2*c*d*x)**(9/2)/9)/(32*c**3*d**5) + 7*b*(b**6*d**6/sqrt(b*d +
2*c*d*x) + 6*b**5*d**5*sqrt(b*d + 2*c*d*x) - 5*b**4*d**4*(b*d + 2*c*d*x)**(3/2)
+ 4*b**3*d**3*(b*d + 2*c*d*x)**(5/2) - 15*b**2*d**2*(b*d + 2*c*d*x)**(7/2)/7 + 2
*b*d*(b*d + 2*c*d*x)**(9/2)/3 - (b*d + 2*c*d*x)**(11/2)/11)/(64*c**3*d**6) + (-b
**7*d**7/sqrt(b*d + 2*c*d*x) - 7*b**6*d**6*sqrt(b*d + 2*c*d*x) + 7*b**5*d**5*(b*
d + 2*c*d*x)**(3/2) - 7*b**4*d**4*(b*d + 2*c*d*x)**(5/2) + 5*b**3*d**3*(b*d + 2*
c*d*x)**(7/2) - 7*b**2*d**2*(b*d + 2*c*d*x)**(9/2)/3 + 7*b*d*(b*d + 2*c*d*x)**(1
1/2)/11 - (b*d + 2*c*d*x)**(13/2)/13)/(64*c**3*d**7))/c, Ne(c, 0)), (Piecewise((
a**3*x, Eq(b, 0)), ((a + b*x)**4/(4*b), True))/sqrt(b*d), True))
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GIAC/XCAS [A] time = 0.23447, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/sqrt(2*c*d*x + b*d),x, algorithm="giac")
[Out]
Done