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

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

[Out]

2/9*a*x^(9/2)+2/13*b*x^(13/2)

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \frac {2}{9} a x^{9/2}+\frac {2}{13} b x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2),x]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ \frac {2}{9} a x^{9/2}+\frac {2}{13} b x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2),x]

[Out]

(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13

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fricas [A]  time = 0.72, size = 18, normalized size = 0.86 \[ \frac {2}{117} \, {\left (9 \, b x^{6} + 13 \, a x^{4}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/117*(9*b*x^6 + 13*a*x^4)*sqrt(x)

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giac [A]  time = 0.63, size = 13, normalized size = 0.62 \[ \frac {2}{13} \, b x^{\frac {13}{2}} + \frac {2}{9} \, a x^{\frac {9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*b*x^(13/2) + 2/9*a*x^(9/2)

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {2 \left (9 b \,x^{2}+13 a \right ) x^{\frac {9}{2}}}{117} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/117*x^(9/2)*(9*b*x^2+13*a)

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maxima [A]  time = 1.33, size = 13, normalized size = 0.62 \[ \frac {2}{13} \, b x^{\frac {13}{2}} + \frac {2}{9} \, a x^{\frac {9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*b*x^(13/2) + 2/9*a*x^(9/2)

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mupad [B]  time = 4.50, size = 15, normalized size = 0.71 \[ \frac {2\,x^{9/2}\,\left (9\,b\,x^2+13\,a\right )}{117} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + b*x^2),x)

[Out]

(2*x^(9/2)*(13*a + 9*b*x^2))/117

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sympy [A]  time = 5.45, size = 19, normalized size = 0.90 \[ \frac {2 a x^{\frac {9}{2}}}{9} + \frac {2 b x^{\frac {13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(9/2)/9 + 2*b*x**(13/2)/13

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