3.1450 \(\int \frac{(a+b x)^5}{(a c+b c x)^{9/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{3/2}}{3 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(3/2))/(3*b*c^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0041991, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ \frac{2 (a c+b c x)^{3/2}}{3 b c^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^(9/2),x]

[Out]

(2*(a*c + b*c*x)^(3/2))/(3*b*c^6)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^{9/2}} \, dx &=\frac{\int \sqrt{a c+b c x} \, dx}{c^5}\\ &=\frac{2 (a c+b c x)^{3/2}}{3 b c^6}\\ \end{align*}

Mathematica [A]  time = 0.0134821, size = 26, normalized size = 1.18 \[ \frac{2 (a+b x) \sqrt{c (a+b x)}}{3 b c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^(9/2),x]

[Out]

(2*(a + b*x)*Sqrt[c*(a + b*x)])/(3*b*c^5)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 23, normalized size = 1.1 \begin{align*}{\frac{2\, \left ( bx+a \right ) ^{6}}{3\,b} \left ( bcx+ac \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^(9/2),x)

[Out]

2/3*(b*x+a)^6/b/(b*c*x+a*c)^(9/2)

________________________________________________________________________________________

Maxima [A]  time = 0.966638, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (b c x + a c\right )}^{\frac{3}{2}}}{3 \, b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(9/2),x, algorithm="maxima")

[Out]

2/3*(b*c*x + a*c)^(3/2)/(b*c^6)

________________________________________________________________________________________

Fricas [A]  time = 1.96451, size = 55, normalized size = 2.5 \begin{align*} \frac{2 \, \sqrt{b c x + a c}{\left (b x + a\right )}}{3 \, b c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(9/2),x, algorithm="fricas")

[Out]

2/3*sqrt(b*c*x + a*c)*(b*x + a)/(b*c^5)

________________________________________________________________________________________

Sympy [A]  time = 7.32454, size = 53, normalized size = 2.41 \begin{align*} \begin{cases} \frac{2 a \sqrt{a c + b c x}}{3 b c^{5}} + \frac{2 x \sqrt{a c + b c x}}{3 c^{5}} & \text{for}\: b \neq 0 \\\frac{a^{5} x}{\left (a c\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**(9/2),x)

[Out]

Piecewise((2*a*sqrt(a*c + b*c*x)/(3*b*c**5) + 2*x*sqrt(a*c + b*c*x)/(3*c**5), Ne(b, 0)), (a**5*x/(a*c)**(9/2),
True))

________________________________________________________________________________________

Giac [B]  time = 1.07177, size = 73, normalized size = 3.32 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{b c x + a c} a - \frac{3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}}{c}\right )}}{3 \, b c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(9/2),x, algorithm="giac")

[Out]

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