3.443 \(\int x^{5/2} (a+b x)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2} \]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(13/2))/13

________________________________________________________________________________________

Rubi [A]  time = 0.0106187, antiderivative size = 51, 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} \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(13/2))/13

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 x^{5/2} (a+b x)^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{7/2}+3 a b^2 x^{9/2}+b^3 x^{11/2}\right ) \, dx\\ &=\frac{2}{7} a^3 x^{7/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0112686, size = 39, normalized size = 0.76 \[ \frac{2 x^{7/2} \left (1001 a^2 b x+429 a^3+819 a b^2 x^2+231 b^3 x^3\right )}{3003} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x)^3,x]

[Out]

(2*x^(7/2)*(429*a^3 + 1001*a^2*b*x + 819*a*b^2*x^2 + 231*b^3*x^3))/3003

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 36, normalized size = 0.7 \begin{align*}{\frac{462\,{b}^{3}{x}^{3}+1638\,a{b}^{2}{x}^{2}+2002\,{a}^{2}bx+858\,{a}^{3}}{3003}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x+a)^3,x)

[Out]

2/3003*x^(7/2)*(231*b^3*x^3+819*a*b^2*x^2+1001*a^2*b*x+429*a^3)

________________________________________________________________________________________

Maxima [A]  time = 1.07651, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x+a)^3,x, algorithm="maxima")

[Out]

2/13*b^3*x^(13/2) + 6/11*a*b^2*x^(11/2) + 2/3*a^2*b*x^(9/2) + 2/7*a^3*x^(7/2)

________________________________________________________________________________________

Fricas [A]  time = 1.53442, size = 104, normalized size = 2.04 \begin{align*} \frac{2}{3003} \,{\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x+a)^3,x, algorithm="fricas")

[Out]

2/3003*(231*b^3*x^6 + 819*a*b^2*x^5 + 1001*a^2*b*x^4 + 429*a^3*x^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 5.68782, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 2*a**2*b*x**(9/2)/3 + 6*a*b**2*x**(11/2)/11 + 2*b**3*x**(13/2)/13

________________________________________________________________________________________

Giac [A]  time = 1.28215, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x+a)^3,x, algorithm="giac")

[Out]

2/13*b^3*x^(13/2) + 6/11*a*b^2*x^(11/2) + 2/3*a^2*b*x^(9/2) + 2/7*a^3*x^(7/2)