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

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

[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 = 0.0115788, 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, Rules used = {270} \[ \frac{6}{5} a^2 b x^{5/2}+2 a^3 \sqrt{x}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{13} b^3 x^{13/2} \]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^3}{\sqrt{x}} \, dx &=\int \left (\frac{a^3}{\sqrt{x}}+3 a^2 b x^{3/2}+3 a b^2 x^{7/2}+b^3 x^{11/2}\right ) \, dx\\ &=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}\\ \end{align*}

Mathematica [A]  time = 0.009848, size = 41, normalized size = 0.84 \[ \frac{2}{195} \sqrt{x} \left (117 a^2 b x^2+195 a^3+65 a b^2 x^4+15 b^3 x^6\right ) \]

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 = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{30\,{b}^{3}{x}^{6}+130\,a{b}^{2}{x}^{4}+234\,{a}^{2}b{x}^{2}+390\,{a}^{3}}{195}\sqrt{x}} \end{align*}

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 = 2.33593, size = 47, normalized size = 0.96 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^(1/2),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 = 1.25466, size = 93, normalized size = 1.9 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^(1/2),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 = 2.2027, size = 48, normalized size = 0.98 \begin{align*} 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} \end{align*}

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 [A]  time = 2.1557, size = 47, normalized size = 0.96 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^(1/2),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)