3.2112 \(\int (a+b \sqrt {x}) x^3 \, dx\)

Optimal. Leaf size=19 \[ \frac {a x^4}{4}+\frac {2}{9} b x^{9/2} \]

[Out]

1/4*a*x^4+2/9*b*x^(9/2)

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Rubi [A]  time = 0.00, antiderivative size = 19, 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} \[ \frac {a x^4}{4}+\frac {2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])*x^3,x]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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 \left (a+b \sqrt {x}\right ) x^3 \, dx &=\int \left (a x^3+b x^{7/2}\right ) \, dx\\ &=\frac {a x^4}{4}+\frac {2}{9} b x^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {a x^4}{4}+\frac {2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x^3,x]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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fricas [A]  time = 1.24, size = 13, normalized size = 0.68 \[ \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2)),x, algorithm="fricas")

[Out]

2/9*b*x^(9/2) + 1/4*a*x^4

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giac [A]  time = 0.15, size = 13, normalized size = 0.68 \[ \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2)),x, algorithm="giac")

[Out]

2/9*b*x^(9/2) + 1/4*a*x^4

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ \frac {2 b \,x^{\frac {9}{2}}}{9}+\frac {a \,x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4+2/9*b*x^(9/2)

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maxima [B]  time = 0.92, size = 132, normalized size = 6.95 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{9}}{9 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{8} a}{4 \, b^{8}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{7} a^{2}}{b^{8}} - \frac {35 \, {\left (b \sqrt {x} + a\right )}^{6} a^{3}}{3 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{5} a^{4}}{b^{8}} - \frac {21 \, {\left (b \sqrt {x} + a\right )}^{4} a^{5}}{2 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{3} a^{6}}{3 \, b^{8}} - \frac {{\left (b \sqrt {x} + a\right )}^{2} a^{7}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2)),x, algorithm="maxima")

[Out]

2/9*(b*sqrt(x) + a)^9/b^8 - 7/4*(b*sqrt(x) + a)^8*a/b^8 + 6*(b*sqrt(x) + a)^7*a^2/b^8 - 35/3*(b*sqrt(x) + a)^6
*a^3/b^8 + 14*(b*sqrt(x) + a)^5*a^4/b^8 - 21/2*(b*sqrt(x) + a)^4*a^5/b^8 + 14/3*(b*sqrt(x) + a)^3*a^6/b^8 - (b
*sqrt(x) + a)^2*a^7/b^8

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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \[ \frac {a\,x^4}{4}+\frac {2\,b\,x^{9/2}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^(1/2)),x)

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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sympy [A]  time = 1.41, size = 15, normalized size = 0.79 \[ \frac {a x^{4}}{4} + \frac {2 b x^{\frac {9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 + 2*b*x**(9/2)/9

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