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

Optimal. Leaf size=34 \[ \frac {2 (a+b x)^{7/2}}{7 b^2}-\frac {2 a (a+b x)^{5/2}}{5 b^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 34, 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} \begin {gather*} \frac {2 (a+b x)^{7/2}}{7 b^2}-\frac {2 a (a+b x)^{5/2}}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(3/2),x]

[Out]

(-2*a*(a + b*x)^(5/2))/(5*b^2) + (2*(a + b*x)^(7/2))/(7*b^2)

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.71 \begin {gather*} \frac {2 (a+b x)^{5/2} (5 b x-2 a)}{35 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2)*(-2*a + 5*b*x))/(35*b^2)

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IntegrateAlgebraic [A]  time = 0.01, size = 46, normalized size = 1.35 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (2 a^3-a^2 b x-8 a b^2 x^2-5 b^3 x^3\right )}{35 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x*(a + b*x)^(3/2),x]

[Out]

(-2*Sqrt[a + b*x]*(2*a^3 - a^2*b*x - 8*a*b^2*x^2 - 5*b^3*x^3))/(35*b^2)

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fricas [A]  time = 0.81, size = 41, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x + a}}{35 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*b^3*x^3 + 8*a*b^2*x^2 + a^2*b*x - 2*a^3)*sqrt(b*x + a)/b^2

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giac [B]  time = 1.24, size = 119, normalized size = 3.50 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{2}}{b} + \frac {14 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a}{b} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )}}{b}\right )}}{105 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*a^2/b + 14*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt
(b*x + a)*a^2)*a/b + 3*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a
^3)/b)/b

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maple [A]  time = 0.00, size = 21, normalized size = 0.62 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-5 b x +2 a \right )}{35 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(b*x+a)^(5/2)*(-5*b*x+2*a)/b^2

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maxima [A]  time = 1.35, size = 26, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(b*x + a)^(7/2)/b^2 - 2/5*(b*x + a)^(5/2)*a/b^2

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mupad [B]  time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {14\,a\,{\left (a+b\,x\right )}^{5/2}-10\,{\left (a+b\,x\right )}^{7/2}}{35\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^(3/2),x)

[Out]

-(14*a*(a + b*x)^(5/2) - 10*(a + b*x)^(7/2))/(35*b^2)

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sympy [A]  time = 0.74, size = 80, normalized size = 2.35 \begin {gather*} \begin {cases} - \frac {4 a^{3} \sqrt {a + b x}}{35 b^{2}} + \frac {2 a^{2} x \sqrt {a + b x}}{35 b} + \frac {16 a x^{2} \sqrt {a + b x}}{35} + \frac {2 b x^{3} \sqrt {a + b x}}{7} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*a**3*sqrt(a + b*x)/(35*b**2) + 2*a**2*x*sqrt(a + b*x)/(35*b) + 16*a*x**2*sqrt(a + b*x)/35 + 2*b*
x**3*sqrt(a + b*x)/7, Ne(b, 0)), (a**(3/2)*x**2/2, True))

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