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

Optimal. Leaf size=47 \[ 2 a^2 b x^{3/2}+2 a^3 \sqrt{x}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2} \]

[Out]

2*a^3*Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*b^2*x^(5/2))/5 + (2*b^3*x^(7/2))/7

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Rubi [A]  time = 0.0114421, antiderivative size = 47, normalized size of antiderivative = 1., 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} \[ 2 a^2 b x^{3/2}+2 a^3 \sqrt{x}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*b^2*x^(5/2))/5 + (2*b^3*x^(7/2))/7

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

Mathematica [A]  time = 0.0102448, size = 39, normalized size = 0.83 \[ \frac{2}{35} \sqrt{x} \left (35 a^2 b x+35 a^3+21 a b^2 x^2+5 b^3 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(35*a^3 + 35*a^2*b*x + 21*a*b^2*x^2 + 5*b^3*x^3))/35

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Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \begin{align*}{\frac{10\,{b}^{3}{x}^{3}+42\,a{b}^{2}{x}^{2}+70\,{a}^{2}bx+70\,{a}^{3}}{35}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*x^(1/2)*(5*b^3*x^3+21*a*b^2*x^2+35*a^2*b*x+35*a^3)

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Maxima [A]  time = 1.02586, size = 47, normalized size = 1. \begin{align*} \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 2 \, a^{3} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44814, size = 85, normalized size = 1.81 \begin{align*} \frac{2}{35} \,{\left (5 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 35 \, a^{2} b x + 35 \, a^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*b^3*x^3 + 21*a*b^2*x^2 + 35*a^2*b*x + 35*a^3)*sqrt(x)

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Sympy [A]  time = 0.644315, size = 46, normalized size = 0.98 \begin{align*} 2 a^{3} \sqrt{x} + 2 a^{2} b x^{\frac{3}{2}} + \frac{6 a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19342, size = 47, normalized size = 1. \begin{align*} \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 2 \, a^{3} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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