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

Optimal. Leaf size=51 \[ \frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \]

[Out]

2/3*a^3*x^(3/2)+6/5*a^2*b*x^(5/2)+6/7*a*b^2*x^(7/2)+2/9*b^3*x^(9/2)

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Rubi [A]  time = 0.01, antiderivative size = 51, 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 = {43} \[ \frac {6}{5} a^2 b x^{5/2}+\frac {2}{3} a^3 x^{3/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x)^3,x]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(5/2))/5 + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(9/2))/9

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \sqrt {x} (a+b x)^3 \, dx &=\int \left (a^3 \sqrt {x}+3 a^2 b x^{3/2}+3 a b^2 x^{5/2}+b^3 x^{7/2}\right ) \, dx\\ &=\frac {2}{3} a^3 x^{3/2}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{7} a b^2 x^{7/2}+\frac {2}{9} b^3 x^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 0.76 \[ \frac {2}{315} x^{3/2} \left (105 a^3+189 a^2 b x+135 a b^2 x^2+35 b^3 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x)^3,x]

[Out]

(2*x^(3/2)*(105*a^3 + 189*a^2*b*x + 135*a*b^2*x^2 + 35*b^3*x^3))/315

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fricas [A]  time = 0.43, size = 38, normalized size = 0.75 \[ \frac {2}{315} \, {\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^3*x^4 + 135*a*b^2*x^3 + 189*a^2*b*x^2 + 105*a^3*x)*sqrt(x)

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giac [A]  time = 1.08, size = 35, normalized size = 0.69 \[ \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 6/7*a*b^2*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2/3*a^3*x^(3/2)

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maple [A]  time = 0.00, size = 36, normalized size = 0.71 \[ \frac {2 \left (35 b^{3} x^{3}+135 a \,b^{2} x^{2}+189 a^{2} b x +105 a^{3}\right ) x^{\frac {3}{2}}}{315} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*x^(3/2)*(35*b^3*x^3+135*a*b^2*x^2+189*a^2*b*x+105*a^3)

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maxima [A]  time = 1.26, size = 35, normalized size = 0.69 \[ \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 6/7*a*b^2*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2/3*a^3*x^(3/2)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.69 \[ \frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,b^3\,x^{9/2}}{9}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {6\,a\,b^2\,x^{7/2}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x)^3,x)

[Out]

(2*a^3*x^(3/2))/3 + (2*b^3*x^(9/2))/9 + (6*a^2*b*x^(5/2))/5 + (6*a*b^2*x^(7/2))/7

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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