3.2257 \(\int (a+b \sqrt {x})^4 x^m \, dx\)

Optimal. Leaf size=87 \[ \frac {a^4 x^{m+1}}{m+1}+\frac {8 a^3 b x^{m+\frac {3}{2}}}{2 m+3}+\frac {6 a^2 b^2 x^{m+2}}{m+2}+\frac {8 a b^3 x^{m+\frac {5}{2}}}{2 m+5}+\frac {b^4 x^{m+3}}{m+3} \]

[Out]

a^4*x^(1+m)/(1+m)+8*a^3*b*x^(3/2+m)/(3+2*m)+6*a^2*b^2*x^(2+m)/(2+m)+8*a*b^3*x^(5/2+m)/(5+2*m)+b^4*x^(3+m)/(3+m
)

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Rubi [A]  time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \frac {6 a^2 b^2 x^{m+2}}{m+2}+\frac {8 a^3 b x^{m+\frac {3}{2}}}{2 m+3}+\frac {a^4 x^{m+1}}{m+1}+\frac {8 a b^3 x^{m+\frac {5}{2}}}{2 m+5}+\frac {b^4 x^{m+3}}{m+3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^4*x^m,x]

[Out]

(a^4*x^(1 + m))/(1 + m) + (8*a^3*b*x^(3/2 + m))/(3 + 2*m) + (6*a^2*b^2*x^(2 + m))/(2 + m) + (8*a*b^3*x^(5/2 +
m))/(5 + 2*m) + (b^4*x^(3 + m))/(3 + m)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \left (a+b \sqrt {x}\right )^4 x^m \, dx &=\int \left (a^4 x^m+4 a^3 b x^{\frac {1}{2}+m}+6 a^2 b^2 x^{1+m}+4 a b^3 x^{\frac {3}{2}+m}+b^4 x^{2+m}\right ) \, dx\\ &=\frac {a^4 x^{1+m}}{1+m}+\frac {8 a^3 b x^{\frac {3}{2}+m}}{3+2 m}+\frac {6 a^2 b^2 x^{2+m}}{2+m}+\frac {8 a b^3 x^{\frac {5}{2}+m}}{5+2 m}+\frac {b^4 x^{3+m}}{3+m}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 78, normalized size = 0.90 \[ x^{m+1} \left (\frac {a^4}{m+1}+\frac {8 a^3 b \sqrt {x}}{2 m+3}+\frac {6 a^2 b^2 x}{m+2}+\frac {8 a b^3 x^{3/2}}{2 m+5}+\frac {b^4 x^2}{m+3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^4*x^m,x]

[Out]

x^(1 + m)*(a^4/(1 + m) + (8*a^3*b*Sqrt[x])/(3 + 2*m) + (6*a^2*b^2*x)/(2 + m) + (8*a*b^3*x^(3/2))/(5 + 2*m) + (
b^4*x^2)/(3 + m))

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fricas [B]  time = 1.37, size = 260, normalized size = 2.99 \[ \frac {{\left ({\left (4 \, b^{4} m^{4} + 28 \, b^{4} m^{3} + 71 \, b^{4} m^{2} + 77 \, b^{4} m + 30 \, b^{4}\right )} x^{3} + 6 \, {\left (4 \, a^{2} b^{2} m^{4} + 32 \, a^{2} b^{2} m^{3} + 91 \, a^{2} b^{2} m^{2} + 108 \, a^{2} b^{2} m + 45 \, a^{2} b^{2}\right )} x^{2} + {\left (4 \, a^{4} m^{4} + 36 \, a^{4} m^{3} + 119 \, a^{4} m^{2} + 171 \, a^{4} m + 90 \, a^{4}\right )} x + 8 \, {\left ({\left (2 \, a b^{3} m^{4} + 15 \, a b^{3} m^{3} + 40 \, a b^{3} m^{2} + 45 \, a b^{3} m + 18 \, a b^{3}\right )} x^{2} + {\left (2 \, a^{3} b m^{4} + 17 \, a^{3} b m^{3} + 52 \, a^{3} b m^{2} + 67 \, a^{3} b m + 30 \, a^{3} b\right )} x\right )} \sqrt {x}\right )} x^{m}}{4 \, m^{5} + 40 \, m^{4} + 155 \, m^{3} + 290 \, m^{2} + 261 \, m + 90} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2))^4,x, algorithm="fricas")

[Out]

((4*b^4*m^4 + 28*b^4*m^3 + 71*b^4*m^2 + 77*b^4*m + 30*b^4)*x^3 + 6*(4*a^2*b^2*m^4 + 32*a^2*b^2*m^3 + 91*a^2*b^
2*m^2 + 108*a^2*b^2*m + 45*a^2*b^2)*x^2 + (4*a^4*m^4 + 36*a^4*m^3 + 119*a^4*m^2 + 171*a^4*m + 90*a^4)*x + 8*((
2*a*b^3*m^4 + 15*a*b^3*m^3 + 40*a*b^3*m^2 + 45*a*b^3*m + 18*a*b^3)*x^2 + (2*a^3*b*m^4 + 17*a^3*b*m^3 + 52*a^3*
b*m^2 + 67*a^3*b*m + 30*a^3*b)*x)*sqrt(x))*x^m/(4*m^5 + 40*m^4 + 155*m^3 + 290*m^2 + 261*m + 90)

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giac [A]  time = 0.23, size = 106, normalized size = 1.22 \[ \frac {b^{4} x^{3} \sqrt {x}^{2 \, m}}{m + 3} + \frac {8 \, a b^{3} x^{\frac {5}{2}} \sqrt {x}^{2 \, m}}{2 \, m + 5} + \frac {6 \, a^{2} b^{2} x^{2} \sqrt {x}^{2 \, m}}{m + 2} + \frac {8 \, a^{3} b x^{\frac {3}{2}} \sqrt {x}^{2 \, m}}{2 \, m + 3} + \frac {a^{4} x \sqrt {x}^{2 \, m}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2))^4,x, algorithm="giac")

[Out]

b^4*x^3*sqrt(x)^(2*m)/(m + 3) + 8*a*b^3*x^(5/2)*sqrt(x)^(2*m)/(2*m + 5) + 6*a^2*b^2*x^2*sqrt(x)^(2*m)/(m + 2)
+ 8*a^3*b*x^(3/2)*sqrt(x)^(2*m)/(2*m + 3) + a^4*x*sqrt(x)^(2*m)/(m + 1)

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \left (b \sqrt {x}+a \right )^{4} x^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^(1/2)+a)^4,x)

[Out]

int(x^m*(b*x^(1/2)+a)^4,x)

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maxima [A]  time = 0.90, size = 83, normalized size = 0.95 \[ \frac {b^{4} x^{m + 3}}{m + 3} + \frac {8 \, a b^{3} x^{m + \frac {5}{2}}}{2 \, m + 5} + \frac {6 \, a^{2} b^{2} x^{m + 2}}{m + 2} + \frac {8 \, a^{3} b x^{m + \frac {3}{2}}}{2 \, m + 3} + \frac {a^{4} x^{m + 1}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2))^4,x, algorithm="maxima")

[Out]

b^4*x^(m + 3)/(m + 3) + 8*a*b^3*x^(m + 5/2)/(2*m + 5) + 6*a^2*b^2*x^(m + 2)/(m + 2) + 8*a^3*b*x^(m + 3/2)/(2*m
+ 3) + a^4*x^(m + 1)/(m + 1)

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mupad [B]  time = 1.70, size = 292, normalized size = 3.36 \[ \frac {b^4\,x^m\,x^3\,\left (4\,m^4+28\,m^3+71\,m^2+77\,m+30\right )}{4\,m^5+40\,m^4+155\,m^3+290\,m^2+261\,m+90}+\frac {a^4\,x\,x^m\,\left (4\,m^4+36\,m^3+119\,m^2+171\,m+90\right )}{4\,m^5+40\,m^4+155\,m^3+290\,m^2+261\,m+90}+\frac {8\,a\,b^3\,x^m\,x^{5/2}\,\left (2\,m^4+15\,m^3+40\,m^2+45\,m+18\right )}{4\,m^5+40\,m^4+155\,m^3+290\,m^2+261\,m+90}+\frac {8\,a^3\,b\,x^m\,x^{3/2}\,\left (2\,m^4+17\,m^3+52\,m^2+67\,m+30\right )}{4\,m^5+40\,m^4+155\,m^3+290\,m^2+261\,m+90}+\frac {6\,a^2\,b^2\,x^m\,x^2\,\left (4\,m^4+32\,m^3+91\,m^2+108\,m+45\right )}{4\,m^5+40\,m^4+155\,m^3+290\,m^2+261\,m+90} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^(1/2))^4,x)

[Out]

(b^4*x^m*x^3*(77*m + 71*m^2 + 28*m^3 + 4*m^4 + 30))/(261*m + 290*m^2 + 155*m^3 + 40*m^4 + 4*m^5 + 90) + (a^4*x
*x^m*(171*m + 119*m^2 + 36*m^3 + 4*m^4 + 90))/(261*m + 290*m^2 + 155*m^3 + 40*m^4 + 4*m^5 + 90) + (8*a*b^3*x^m
*x^(5/2)*(45*m + 40*m^2 + 15*m^3 + 2*m^4 + 18))/(261*m + 290*m^2 + 155*m^3 + 40*m^4 + 4*m^5 + 90) + (8*a^3*b*x
^m*x^(3/2)*(67*m + 52*m^2 + 17*m^3 + 2*m^4 + 30))/(261*m + 290*m^2 + 155*m^3 + 40*m^4 + 4*m^5 + 90) + (6*a^2*b
^2*x^m*x^2*(108*m + 91*m^2 + 32*m^3 + 4*m^4 + 45))/(261*m + 290*m^2 + 155*m^3 + 40*m^4 + 4*m^5 + 90)

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sympy [A]  time = 16.33, size = 117, normalized size = 1.34 \[ a^{4} \left (\begin {cases} \frac {x^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + 8 a^{3} b \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{m}}{2 m + 3} & \text {for}\: m \neq - \frac {3}{2} \\\log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + 6 a^{2} b^{2} \left (\begin {cases} \frac {x^{2} x^{m}}{m + 2} & \text {for}\: m \neq -2 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + 4 a b^{3} \left (\begin {cases} \frac {2 x^{\frac {5}{2}} x^{m}}{2 m + 5} & \text {for}\: m \neq - \frac {5}{2} \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + b^{4} \left (\begin {cases} \frac {x^{3} x^{m}}{m + 3} & \text {for}\: m \neq -3 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a+b*x**(1/2))**4,x)

[Out]

a**4*Piecewise((x**(m + 1)/(m + 1), Ne(m, -1)), (log(x), True)) + 8*a**3*b*Piecewise((x**(3/2)*x**m/(2*m + 3),
Ne(m, -3/2)), (log(sqrt(x)), True)) + 6*a**2*b**2*Piecewise((x**2*x**m/(m + 2), Ne(m, -2)), (log(x), True)) +
4*a*b**3*Piecewise((2*x**(5/2)*x**m/(2*m + 5), Ne(m, -5/2)), (log(x), True)) + b**4*Piecewise((x**3*x**m/(m +
3), Ne(m, -3)), (log(x), True))

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