\(\int \frac {(a+b x)^3}{\sqrt {x}} \, dx\) [446]

Optimal result

Integrand size = 13, antiderivative size = 47 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}+2 a^2 b x^{3/2}+\frac {6}{5} a b^2 x^{5/2}+\frac {2}{7} b^3 x^{7/2} \]

[Out]

2*a^2*b*x^(3/2)+6/5*a*b^2*x^(5/2)+2/7*b^3*x^(7/2)+2*a^3*x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}+2 a^2 b x^{3/2}+\frac {6}{5} a b^2 x^{5/2}+\frac {2}{7} b^3 x^{7/2} \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=\frac {2}{35} \sqrt {x} \left (35 a^3+35 a^2 b x+21 a b^2 x^2+5 b^3 x^3\right ) \]

[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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74

method	result	size
trager	\(\left (\frac {2}{7} b^{3} x^{3}+\frac {6}{5} a \,b^{2} x^{2}+2 a^{2} b x +2 a^{3}\right ) \sqrt {x}\)	\(35\)
gosper	\(\frac {2 \sqrt {x}\, \left (5 b^{3} x^{3}+21 a \,b^{2} x^{2}+35 a^{2} b x +35 a^{3}\right )}{35}\)	\(36\)
derivativedivides	\(2 a^{2} b \,x^{\frac {3}{2}}+\frac {6 a \,b^{2} x^{\frac {5}{2}}}{5}+\frac {2 b^{3} x^{\frac {7}{2}}}{7}+2 a^{3} \sqrt {x}\)	\(36\)
default	\(2 a^{2} b \,x^{\frac {3}{2}}+\frac {6 a \,b^{2} x^{\frac {5}{2}}}{5}+\frac {2 b^{3} x^{\frac {7}{2}}}{7}+2 a^{3} \sqrt {x}\)	\(36\)
risch	\(\frac {2 \sqrt {x}\, \left (5 b^{3} x^{3}+21 a \,b^{2} x^{2}+35 a^{2} b x +35 a^{3}\right )}{35}\)	\(36\)

[In]

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

[Out]

(2/7*b^3*x^3+6/5*a*b^2*x^2+2*a^2*b*x+2*a^3)*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=\frac {2}{35} \, {\left (5 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 35 \, a^{2} b x + 35 \, a^{3}\right )} \sqrt {x} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=2 a^{3} \sqrt {x} + 2 a^{2} b x^{\frac {3}{2}} + \frac {6 a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 b^{3} x^{\frac {7}{2}}}{7} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=\frac {2}{7} \, b^{3} x^{\frac {7}{2}} + \frac {6}{5} \, a b^{2} x^{\frac {5}{2}} + 2 \, a^{2} b x^{\frac {3}{2}} + 2 \, a^{3} \sqrt {x} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=\frac {2}{7} \, b^{3} x^{\frac {7}{2}} + \frac {6}{5} \, a b^{2} x^{\frac {5}{2}} + 2 \, a^{2} b x^{\frac {3}{2}} + 2 \, a^{3} \sqrt {x} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^3}{\sqrt {x}} \, dx=2\,a^3\,\sqrt {x}+\frac {2\,b^3\,x^{7/2}}{7}+2\,a^2\,b\,x^{3/2}+\frac {6\,a\,b^2\,x^{5/2}}{5} \]

[In]

int((a + b*x)^3/x^(1/2),x)

[Out]

2*a^3*x^(1/2) + (2*b^3*x^(7/2))/7 + 2*a^2*b*x^(3/2) + (6*a*b^2*x^(5/2))/5