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

Optimal. Leaf size=63 \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{11} B c^2 x^{11/2} \]

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*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 = 0.0921423, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{11} B c^2 x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*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 = 10.2858, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} + \frac{2 b x^{\frac{7}{2}} \left (2 A c + B b\right )}{7} + \frac{2 c x^{\frac{9}{2}} \left (A c + 2 B b\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**2*x**(5/2)/5 + 2*B*c**2*x**(11/2)/11 + 2*b*x**(7/2)*(2*A*c + B*b)/7 + 2*c
*x**(9/2)*(A*c + 2*B*b)/9

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Mathematica [A]  time = 0.0317734, size = 51, normalized size = 0.81 \[ \frac{2 x^{5/2} \left (693 A b^2+385 c x^2 (A c+2 b B)+495 b x (2 A c+b B)+315 B c^2 x^3\right )}{3465} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 52, normalized size = 0.8 \[{\frac{630\,B{c}^{2}{x}^{3}+770\,A{c}^{2}{x}^{2}+1540\,B{x}^{2}bc+1980\,Abcx+990\,{b}^{2}Bx+1386\,{b}^{2}A}{3465}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*x^(5/2)*(315*B*c^2*x^3+385*A*c^2*x^2+770*B*b*c*x^2+990*A*b*c*x+495*B*b^2*
x+693*A*b^2)

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Maxima [A]  time = 0.683364, size = 69, normalized size = 1.1 \[ \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.287236, size = 76, normalized size = 1.21 \[ \frac{2}{3465} \,{\left (315 \, B c^{2} x^{5} + 693 \, A b^{2} x^{2} + 385 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \,{\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.40358, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{4 A b c x^{\frac{7}{2}}}{7} + \frac{2 A c^{2} x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} + \frac{4 B b c x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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*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 = 0.269695, size = 72, normalized size = 1.14 \[ \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B b c x^{\frac{9}{2}} + \frac{2}{9} \, A c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A b c x^{\frac{7}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^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*A*b*c*x^(7/2) + 2/5*A*b^2*x^(5/2)