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

Optimal. Leaf size=51 \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{15} b^3 x^{15/2} \]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(15/2))/15

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Rubi [A]  time = 0.0127798, antiderivative size = 51, 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} \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{15} b^3 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2)^3,x]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(15/2))/15

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 \sqrt{x} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 \sqrt{x}+3 a^2 b x^{5/2}+3 a b^2 x^{9/2}+b^3 x^{13/2}\right ) \, dx\\ &=\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{15} b^3 x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0097537, size = 41, normalized size = 0.8 \[ \frac{2 x^{3/2} \left (495 a^2 b x^2+385 a^3+315 a b^2 x^4+77 b^3 x^6\right )}{1155} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x^2)^3,x]

[Out]

(2*x^(3/2)*(385*a^3 + 495*a^2*b*x^2 + 315*a*b^2*x^4 + 77*b^3*x^6))/1155

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*}{\frac{154\,{b}^{3}{x}^{6}+630\,a{b}^{2}{x}^{4}+990\,{a}^{2}b{x}^{2}+770\,{a}^{3}}{1155}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*x^(3/2)*(77*b^3*x^6+315*a*b^2*x^4+495*a^2*b*x^2+385*a^3)

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Maxima [A]  time = 2.30333, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{15} \, b^{3} x^{\frac{15}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \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]

2/15*b^3*x^(15/2) + 6/11*a*b^2*x^(11/2) + 6/7*a^2*b*x^(7/2) + 2/3*a^3*x^(3/2)

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Fricas [A]  time = 1.2759, size = 99, normalized size = 1.94 \begin{align*} \frac{2}{1155} \,{\left (77 \, b^{3} x^{7} + 315 \, a b^{2} x^{5} + 495 \, a^{2} b x^{3} + 385 \, a^{3} x\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]

2/1155*(77*b^3*x^7 + 315*a*b^2*x^5 + 495*a^2*b*x^3 + 385*a^3*x)*sqrt(x)

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Sympy [A]  time = 2.11113, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{3}{2}}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{3} x^{\frac{15}{2}}}{15} \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*x**(3/2)/3 + 6*a**2*b*x**(7/2)/7 + 6*a*b**2*x**(11/2)/11 + 2*b**3*x**(15/2)/15

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Giac [A]  time = 2.9259, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{15} \, b^{3} x^{\frac{15}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \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]

2/15*b^3*x^(15/2) + 6/11*a*b^2*x^(11/2) + 6/7*a^2*b*x^(7/2) + 2/3*a^3*x^(3/2)