3.265 \(\int x^{5/2} (a+b x^2) \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{7} a x^{7/2}+\frac {2}{11} b x^{11/2} \]

[Out]

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

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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} \[ \frac {2}{7} a x^{7/2}+\frac {2}{11} b x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2),x]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {2}{7} a x^{7/2}+\frac {2}{11} b x^{11/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2),x]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11

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fricas [A]  time = 0.68, size = 18, normalized size = 0.86 \[ \frac {2}{77} \, {\left (7 \, b x^{5} + 11 \, a x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a),x, algorithm="fricas")

[Out]

2/77*(7*b*x^5 + 11*a*x^3)*sqrt(x)

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giac [A]  time = 0.60, size = 13, normalized size = 0.62 \[ \frac {2}{11} \, b x^{\frac {11}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a),x, algorithm="giac")

[Out]

2/11*b*x^(11/2) + 2/7*a*x^(7/2)

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {2 \left (7 b \,x^{2}+11 a \right ) x^{\frac {7}{2}}}{77} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x^2+a),x)

[Out]

2/77*x^(7/2)*(7*b*x^2+11*a)

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maxima [A]  time = 1.29, size = 13, normalized size = 0.62 \[ \frac {2}{11} \, b x^{\frac {11}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a),x, algorithm="maxima")

[Out]

2/11*b*x^(11/2) + 2/7*a*x^(7/2)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.71 \[ \frac {2\,x^{7/2}\,\left (7\,b\,x^2+11\,a\right )}{77} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + b*x^2),x)

[Out]

(2*x^(7/2)*(11*a + 7*b*x^2))/77

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sympy [A]  time = 2.41, size = 19, normalized size = 0.90 \[ \frac {2 a x^{\frac {7}{2}}}{7} + \frac {2 b x^{\frac {11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**2+a),x)

[Out]

2*a*x**(7/2)/7 + 2*b*x**(11/2)/11

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