3.994 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=111 \[ 2 a^2 A \sqrt{x}+\frac{2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{11} B c^2 x^{11/2} \]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*(2*a*b*B + A*(b^2 + 2*a*c))
*x^(5/2))/5 + (2*(b^2*B + 2*A*b*c + 2*a*B*c)*x^(7/2))/7 + (2*c*(2*b*B + A*c)*x^(
9/2))/9 + (2*B*c^2*x^(11/2))/11

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Rubi [A]  time = 0.143014, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ 2 a^2 A \sqrt{x}+\frac{2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{11} B c^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^2)/Sqrt[x],x]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*(2*a*b*B + A*(b^2 + 2*a*c))
*x^(5/2))/5 + (2*(b^2*B + 2*A*b*c + 2*a*B*c)*x^(7/2))/7 + (2*c*(2*b*B + A*c)*x^(
9/2))/9 + (2*B*c^2*x^(11/2))/11

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Rubi in Sympy [A]  time = 19.6714, size = 122, normalized size = 1.1 \[ 2 A a^{2} \sqrt{x} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} + \frac{2 a x^{\frac{3}{2}} \left (2 A b + B a\right )}{3} + \frac{2 c x^{\frac{9}{2}} \left (A c + 2 B b\right )}{9} + x^{\frac{7}{2}} \left (\frac{4 A b c}{7} + \frac{4 B a c}{7} + \frac{2 B b^{2}}{7}\right ) + x^{\frac{5}{2}} \left (\frac{4 A a c}{5} + \frac{2 A b^{2}}{5} + \frac{4 B a b}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*sqrt(x) + 2*B*c**2*x**(11/2)/11 + 2*a*x**(3/2)*(2*A*b + B*a)/3 + 2*c*x*
*(9/2)*(A*c + 2*B*b)/9 + x**(7/2)*(4*A*b*c/7 + 4*B*a*c/7 + 2*B*b**2/7) + x**(5/2
)*(4*A*a*c/5 + 2*A*b**2/5 + 4*B*a*b/5)

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Mathematica [A]  time = 0.0875982, size = 93, normalized size = 0.84 \[ \frac{2 \sqrt{x} \left (3465 a^2 A+495 x^3 \left (2 a B c+2 A b c+b^2 B\right )+693 x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+1155 a x (a B+2 A b)+385 c x^4 (A c+2 b B)+315 B c^2 x^5\right )}{3465} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3465*a^2*A + 1155*a*(2*A*b + a*B)*x + 693*(2*a*b*B + A*(b^2 + 2*a*c)
)*x^2 + 495*(b^2*B + 2*A*b*c + 2*a*B*c)*x^3 + 385*c*(2*b*B + A*c)*x^4 + 315*B*c^
2*x^5))/3465

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Maple [A]  time = 0.01, size = 102, normalized size = 0.9 \[{\frac{630\,B{c}^{2}{x}^{5}+770\,A{c}^{2}{x}^{4}+1540\,B{x}^{4}bc+1980\,A{x}^{3}bc+1980\,aBc{x}^{3}+990\,B{b}^{2}{x}^{3}+2772\,aAc{x}^{2}+1386\,A{b}^{2}{x}^{2}+2772\,B{x}^{2}ab+4620\,aAbx+2310\,{a}^{2}Bx+6930\,A{a}^{2}}{3465}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*x^(1/2)*(315*B*c^2*x^5+385*A*c^2*x^4+770*B*b*c*x^4+990*A*b*c*x^3+990*B*a*
c*x^3+495*B*b^2*x^3+1386*A*a*c*x^2+693*A*b^2*x^2+1386*B*a*b*x^2+2310*A*a*b*x+115
5*B*a^2*x+3465*A*a^2)

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Maxima [A]  time = 0.721878, size = 126, normalized size = 1.14 \[ \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*c^2*x^(11/2) + 2/9*(2*B*b*c + A*c^2)*x^(9/2) + 2/7*(B*b^2 + 2*(B*a + A*b)
*c)*x^(7/2) + 2*A*a^2*sqrt(x) + 2/5*(2*B*a*b + A*b^2 + 2*A*a*c)*x^(5/2) + 2/3*(B
*a^2 + 2*A*a*b)*x^(3/2)

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Fricas [A]  time = 0.295014, size = 126, normalized size = 1.14 \[ \frac{2}{3465} \,{\left (315 \, B c^{2} x^{5} + 385 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 3465 \, A a^{2} + 693 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 1155 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*B*c^2*x^5 + 385*(2*B*b*c + A*c^2)*x^4 + 495*(B*b^2 + 2*(B*a + A*b)*c
)*x^3 + 3465*A*a^2 + 693*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 1155*(B*a^2 + 2*A*a*b
)*x)*sqrt(x)

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Sympy [A]  time = 12.513, size = 160, normalized size = 1.44 \[ 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{4 A a c x^{\frac{5}{2}}}{5} + \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{4 A b c x^{\frac{7}{2}}}{7} + \frac{2 A c^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{4 B a c x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} + \frac{4 B b c x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(3/2)/3 + 4*A*a*c*x**(5/2)/5 + 2*A*b**2*x**(5/2)/5
+ 4*A*b*c*x**(7/2)/7 + 2*A*c**2*x**(9/2)/9 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(
5/2)/5 + 4*B*a*c*x**(7/2)/7 + 2*B*b**2*x**(7/2)/7 + 4*B*b*c*x**(9/2)/9 + 2*B*c**
2*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.270639, size = 139, normalized size = 1.25 \[ \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B b c x^{\frac{9}{2}} + \frac{2}{9} \, A c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, B a c x^{\frac{7}{2}} + \frac{4}{7} \, A b c x^{\frac{7}{2}} + \frac{4}{5} \, B a b x^{\frac{5}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a c x^{\frac{5}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} + 2 \, A a^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*c^2*x^(11/2) + 4/9*B*b*c*x^(9/2) + 2/9*A*c^2*x^(9/2) + 2/7*B*b^2*x^(7/2)
+ 4/7*B*a*c*x^(7/2) + 4/7*A*b*c*x^(7/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2)
+ 4/5*A*a*c*x^(5/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2) + 2*A*a^2*sqrt(x)