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

Optimal. Leaf size=21 \[ \frac {2}{7} a x^{7/2}+\frac {2}{9} b x^{9/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {2}{7} a x^{7/2}+\frac {2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 0.81 \[ \frac {2}{63} x^{7/2} (9 a+7 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x),x]

[Out]

(2*x^(7/2)*(9*a + 7*b*x))/63

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 18, normalized size = 0.86 \[ \frac {2}{63} \, {\left (7 \, b x^{4} + 9 \, a x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(7*b*x^4 + 9*a*x^3)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.83, size = 13, normalized size = 0.62 \[ \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b*x^(9/2) + 2/7*a*x^(7/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.67 \[ \frac {2 \left (7 b x +9 a \right ) x^{\frac {7}{2}}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*x^(7/2)*(7*b*x+9*a)

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 13, normalized size = 0.62 \[ \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b*x^(9/2) + 2/7*a*x^(7/2)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 13, normalized size = 0.62 \[ \frac {2\,x^{7/2}\,\left (9\,a+7\,b\,x\right )}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + b*x),x)

[Out]

(2*x^(7/2)*(9*a + 7*b*x))/63

________________________________________________________________________________________

sympy [A]  time = 1.59, size = 19, normalized size = 0.90 \[ \frac {2 a x^{\frac {7}{2}}}{7} + \frac {2 b x^{\frac {9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(7/2)/7 + 2*b*x**(9/2)/9

________________________________________________________________________________________