3.280 \(\int x^{7/2} (a+b x^2)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{13} a^2 b x^{13/2}+\frac{2}{9} a^3 x^{9/2}+\frac{6}{17} a b^2 x^{17/2}+\frac{2}{21} b^3 x^{21/2} \]

[Out]

(2*a^3*x^(9/2))/9 + (6*a^2*b*x^(13/2))/13 + (6*a*b^2*x^(17/2))/17 + (2*b^3*x^(21/2))/21

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Rubi [A]  time = 0.0125644, 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} \[ \frac{6}{13} a^2 b x^{13/2}+\frac{2}{9} a^3 x^{9/2}+\frac{6}{17} a b^2 x^{17/2}+\frac{2}{21} b^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2)^3,x]

[Out]

(2*a^3*x^(9/2))/9 + (6*a^2*b*x^(13/2))/13 + (6*a*b^2*x^(17/2))/17 + (2*b^3*x^(21/2))/21

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 x^{7/2} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^{7/2}+3 a^2 b x^{11/2}+3 a b^2 x^{15/2}+b^3 x^{19/2}\right ) \, dx\\ &=\frac{2}{9} a^3 x^{9/2}+\frac{6}{13} a^2 b x^{13/2}+\frac{6}{17} a b^2 x^{17/2}+\frac{2}{21} b^3 x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0103996, size = 41, normalized size = 0.8 \[ \frac{2 x^{9/2} \left (3213 a^2 b x^2+1547 a^3+2457 a b^2 x^4+663 b^3 x^6\right )}{13923} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2)^3,x]

[Out]

(2*x^(9/2)*(1547*a^3 + 3213*a^2*b*x^2 + 2457*a*b^2*x^4 + 663*b^3*x^6))/13923

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*}{\frac{1326\,{b}^{3}{x}^{6}+4914\,a{b}^{2}{x}^{4}+6426\,{a}^{2}b{x}^{2}+3094\,{a}^{3}}{13923}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^3,x)

[Out]

2/13923*x^(9/2)*(663*b^3*x^6+2457*a*b^2*x^4+3213*a^2*b*x^2+1547*a^3)

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Maxima [A]  time = 1.99198, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{21} \, b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, a b^{2} x^{\frac{17}{2}} + \frac{6}{13} \, a^{2} b x^{\frac{13}{2}} + \frac{2}{9} \, a^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

2/21*b^3*x^(21/2) + 6/17*a*b^2*x^(17/2) + 6/13*a^2*b*x^(13/2) + 2/9*a^3*x^(9/2)

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Fricas [A]  time = 1.19382, size = 109, normalized size = 2.14 \begin{align*} \frac{2}{13923} \,{\left (663 \, b^{3} x^{10} + 2457 \, a b^{2} x^{8} + 3213 \, a^{2} b x^{6} + 1547 \, a^{3} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

2/13923*(663*b^3*x^10 + 2457*a*b^2*x^8 + 3213*a^2*b*x^6 + 1547*a^3*x^4)*sqrt(x)

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Sympy [A]  time = 21.0104, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{9}{2}}}{9} + \frac{6 a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{3} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**3,x)

[Out]

2*a**3*x**(9/2)/9 + 6*a**2*b*x**(13/2)/13 + 6*a*b**2*x**(17/2)/17 + 2*b**3*x**(21/2)/21

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Giac [A]  time = 1.47747, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{21} \, b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, a b^{2} x^{\frac{17}{2}} + \frac{6}{13} \, a^{2} b x^{\frac{13}{2}} + \frac{2}{9} \, a^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^3,x, algorithm="giac")

[Out]

2/21*b^3*x^(21/2) + 6/17*a*b^2*x^(17/2) + 6/13*a^2*b*x^(13/2) + 2/9*a^3*x^(9/2)