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

Optimal. Leaf size=47 \[ 2 a^2 b x^{3/2}-\frac{2 a^3}{\sqrt{x}}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{11} b^3 x^{11/2} \]

[Out]

(-2*a^3)/Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(11/2))/11

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Rubi [A]  time = 0.0116159, antiderivative size = 47, 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} \[ 2 a^2 b x^{3/2}-\frac{2 a^3}{\sqrt{x}}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{11} b^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(11/2))/11

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}{x^{3/2}} \, dx &=\int \left (\frac{a^3}{x^{3/2}}+3 a^2 b \sqrt{x}+3 a b^2 x^{5/2}+b^3 x^{9/2}\right ) \, dx\\ &=-\frac{2 a^3}{\sqrt{x}}+2 a^2 b x^{3/2}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{11} b^3 x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0107062, size = 41, normalized size = 0.87 \[ \frac{2 \left (77 a^2 b x^2-77 a^3+33 a b^2 x^4+7 b^3 x^6\right )}{77 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^(3/2),x]

[Out]

(2*(-77*a^3 + 77*a^2*b*x^2 + 33*a*b^2*x^4 + 7*b^3*x^6))/(77*Sqrt[x])

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Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \begin{align*} -{\frac{-14\,{b}^{3}{x}^{6}-66\,a{b}^{2}{x}^{4}-154\,{a}^{2}b{x}^{2}+154\,{a}^{3}}{77}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/77*(-7*b^3*x^6-33*a*b^2*x^4-77*a^2*b*x^2+77*a^3)/x^(1/2)

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Maxima [A]  time = 1.94537, size = 47, normalized size = 1. \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} - \frac{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^(3/2),x, algorithm="maxima")

[Out]

2/11*b^3*x^(11/2) + 6/7*a*b^2*x^(7/2) + 2*a^2*b*x^(3/2) - 2*a^3/sqrt(x)

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Fricas [A]  time = 1.15043, size = 88, normalized size = 1.87 \begin{align*} \frac{2 \,{\left (7 \, b^{3} x^{6} + 33 \, a b^{2} x^{4} + 77 \, a^{2} b x^{2} - 77 \, a^{3}\right )}}{77 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/77*(7*b^3*x^6 + 33*a*b^2*x^4 + 77*a^2*b*x^2 - 77*a^3)/sqrt(x)

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Sympy [A]  time = 2.40461, size = 46, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} + 2 a^{2} b x^{\frac{3}{2}} + \frac{6 a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/sqrt(x) + 2*a**2*b*x**(3/2) + 6*a*b**2*x**(7/2)/7 + 2*b**3*x**(11/2)/11

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Giac [A]  time = 2.23314, size = 47, normalized size = 1. \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} - \frac{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^(3/2),x, algorithm="giac")

[Out]

2/11*b^3*x^(11/2) + 6/7*a*b^2*x^(7/2) + 2*a^2*b*x^(3/2) - 2*a^3/sqrt(x)