3.284 \(\int \frac{\left (a+b x^2\right )^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=49 \[ 2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{13} b^3 x^{13/2} \]

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*b^2*x^(9/2))/3 + (2*b^3*x^(13/2))/13

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Rubi [A]  time = 0.0400382, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ 2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{13} b^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*b^2*x^(9/2))/3 + (2*b^3*x^(13/2))/13

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Rubi in Sympy [A]  time = 6.32968, size = 48, normalized size = 0.98 \[ 2 a^{3} \sqrt{x} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**(1/2),x)

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a*b**2*x**(9/2)/3 + 2*b**3*x**(13/2)/13

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Mathematica [A]  time = 0.0125801, size = 41, normalized size = 0.84 \[ \frac{2}{195} \sqrt{x} \left (195 a^3+117 a^2 b x^2+65 a b^2 x^4+15 b^3 x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(195*a^3 + 117*a^2*b*x^2 + 65*a*b^2*x^4 + 15*b^3*x^6))/195

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Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \[{\frac{30\,{b}^{3}{x}^{6}+130\,a{b}^{2}{x}^{4}+234\,{a}^{2}b{x}^{2}+390\,{a}^{3}}{195}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^(1/2),x)

[Out]

2/195*x^(1/2)*(15*b^3*x^6+65*a*b^2*x^4+117*a^2*b*x^2+195*a^3)

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Maxima [A]  time = 1.34807, size = 47, normalized size = 0.96 \[ \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + 2 \, a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*b^3*x^(13/2) + 2/3*a*b^2*x^(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)

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Fricas [A]  time = 0.209291, size = 50, normalized size = 1.02 \[ \frac{2}{195} \,{\left (15 \, b^{3} x^{6} + 65 \, a b^{2} x^{4} + 117 \, a^{2} b x^{2} + 195 \, a^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/195*(15*b^3*x^6 + 65*a*b^2*x^4 + 117*a^2*b*x^2 + 195*a^3)*sqrt(x)

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Sympy [A]  time = 6.44597, size = 48, normalized size = 0.98 \[ 2 a^{3} \sqrt{x} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**(1/2),x)

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a*b**2*x**(9/2)/3 + 2*b**3*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.217871, size = 47, normalized size = 0.96 \[ \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + 2 \, a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*b^3*x^(13/2) + 2/3*a*b^2*x^(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)