3.157 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^{9/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 A b^2}{3 x^{3/2}}-\frac{2 b (2 A c+b B)}{\sqrt{x}}+2 c \sqrt{x} (A c+2 b B)+\frac{2}{3} B c^2 x^{3/2} \]

[Out]

(-2*A*b^2)/(3*x^(3/2)) - (2*b*(b*B + 2*A*c))/Sqrt[x] + 2*c*(2*b*B + A*c)*Sqrt[x]
+ (2*B*c^2*x^(3/2))/3

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Rubi [A]  time = 0.0951831, antiderivative size = 59, 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 \[ -\frac{2 A b^2}{3 x^{3/2}}-\frac{2 b (2 A c+b B)}{\sqrt{x}}+2 c \sqrt{x} (A c+2 b B)+\frac{2}{3} B c^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/x^(9/2),x]

[Out]

(-2*A*b^2)/(3*x^(3/2)) - (2*b*(b*B + 2*A*c))/Sqrt[x] + 2*c*(2*b*B + A*c)*Sqrt[x]
+ (2*B*c^2*x^(3/2))/3

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Rubi in Sympy [A]  time = 10.1803, size = 60, normalized size = 1.02 \[ - \frac{2 A b^{2}}{3 x^{\frac{3}{2}}} + \frac{2 B c^{2} x^{\frac{3}{2}}}{3} - \frac{2 b \left (2 A c + B b\right )}{\sqrt{x}} + 2 c \sqrt{x} \left (A c + 2 B b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**(9/2),x)

[Out]

-2*A*b**2/(3*x**(3/2)) + 2*B*c**2*x**(3/2)/3 - 2*b*(2*A*c + B*b)/sqrt(x) + 2*c*s
qrt(x)*(A*c + 2*B*b)

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Mathematica [A]  time = 0.0317382, size = 52, normalized size = 0.88 \[ \frac{2 B x \left (-3 b^2+6 b c x+c^2 x^2\right )-2 A \left (b^2+6 b c x-3 c^2 x^2\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^(9/2),x]

[Out]

(-2*A*(b^2 + 6*b*c*x - 3*c^2*x^2) + 2*B*x*(-3*b^2 + 6*b*c*x + c^2*x^2))/(3*x^(3/
2))

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Maple [A]  time = 0.008, size = 51, normalized size = 0.9 \[ -{\frac{-2\,B{c}^{2}{x}^{3}-6\,A{c}^{2}{x}^{2}-12\,B{x}^{2}bc+12\,Abcx+6\,{b}^{2}Bx+2\,{b}^{2}A}{3}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x^(9/2),x)

[Out]

-2/3/x^(3/2)*(-B*c^2*x^3-3*A*c^2*x^2-6*B*b*c*x^2+6*A*b*c*x+3*B*b^2*x+A*b^2)

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Maxima [A]  time = 0.685758, size = 69, normalized size = 1.17 \[ \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 2 \,{\left (2 \, B b c + A c^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A b^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^(9/2),x, algorithm="maxima")

[Out]

2/3*B*c^2*x^(3/2) + 2*(2*B*b*c + A*c^2)*sqrt(x) - 2/3*(A*b^2 + 3*(B*b^2 + 2*A*b*
c)*x)/x^(3/2)

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Fricas [A]  time = 0.286477, size = 68, normalized size = 1.15 \[ \frac{2 \,{\left (B c^{2} x^{3} - A b^{2} + 3 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^(9/2),x, algorithm="fricas")

[Out]

2/3*(B*c^2*x^3 - A*b^2 + 3*(2*B*b*c + A*c^2)*x^2 - 3*(B*b^2 + 2*A*b*c)*x)/x^(3/2
)

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Sympy [A]  time = 11.3728, size = 73, normalized size = 1.24 \[ - \frac{2 A b^{2}}{3 x^{\frac{3}{2}}} - \frac{4 A b c}{\sqrt{x}} + 2 A c^{2} \sqrt{x} - \frac{2 B b^{2}}{\sqrt{x}} + 4 B b c \sqrt{x} + \frac{2 B c^{2} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/x**(9/2),x)

[Out]

-2*A*b**2/(3*x**(3/2)) - 4*A*b*c/sqrt(x) + 2*A*c**2*sqrt(x) - 2*B*b**2/sqrt(x) +
4*B*b*c*sqrt(x) + 2*B*c**2*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.267836, size = 69, normalized size = 1.17 \[ \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 4 \, B b c \sqrt{x} + 2 \, A c^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B b^{2} x + 6 \, A b c x + A b^{2}\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^(9/2),x, algorithm="giac")

[Out]

2/3*B*c^2*x^(3/2) + 4*B*b*c*sqrt(x) + 2*A*c^2*sqrt(x) - 2/3*(3*B*b^2*x + 6*A*b*c
*x + A*b^2)/x^(3/2)