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

Optimal. Leaf size=21 \[ \frac {2}{3} a x^{3/2}+\frac {2}{7} b x^{7/2} \]

[Out]

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

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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} \[ \frac {2}{3} a x^{3/2}+\frac {2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {2}{3} a x^{3/2}+\frac {2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x^2),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

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fricas [A]  time = 0.72, size = 16, normalized size = 0.76 \[ \frac {2}{21} \, {\left (3 \, b x^{3} + 7 \, a x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*x^(1/2),x, algorithm="fricas")

[Out]

2/21*(3*b*x^3 + 7*a*x)*sqrt(x)

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giac [A]  time = 0.61, size = 13, normalized size = 0.62 \[ \frac {2}{7} \, b x^{\frac {7}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*x^(1/2),x, algorithm="giac")

[Out]

2/7*b*x^(7/2) + 2/3*a*x^(3/2)

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {2 \left (3 b \,x^{2}+7 a \right ) x^{\frac {3}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*x^(3/2)*(3*b*x^2+7*a)

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maxima [A]  time = 1.31, size = 13, normalized size = 0.62 \[ \frac {2}{7} \, b x^{\frac {7}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*x^(1/2),x, algorithm="maxima")

[Out]

2/7*b*x^(7/2) + 2/3*a*x^(3/2)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.71 \[ \frac {2\,x^{3/2}\,\left (3\,b\,x^2+7\,a\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x^2),x)

[Out]

(2*x^(3/2)*(7*a + 3*b*x^2))/21

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sympy [A]  time = 1.30, size = 19, normalized size = 0.90 \[ \frac {2 a x^{\frac {3}{2}}}{3} + \frac {2 b x^{\frac {7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*x**(1/2),x)

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7

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