3.1445 \(\int (a+b x)^5 \sqrt{a c+b c x} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{13/2}}{13 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(13/2))/(13*b*c^6)

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Rubi [A]  time = 0.0045631, 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} \[ \frac{2 (a c+b c x)^{13/2}}{13 b c^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*Sqrt[a*c + b*c*x],x]

[Out]

(2*(a*c + b*c*x)^(13/2))/(13*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 (a+b x)^5 \sqrt{a c+b c x} \, dx &=\frac{\int (a c+b c x)^{11/2} \, dx}{c^5}\\ &=\frac{2 (a c+b c x)^{13/2}}{13 b c^6}\\ \end{align*}

Mathematica [A]  time = 0.0114057, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6 \sqrt{c (a+b x)}}{13 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5*Sqrt[a*c + b*c*x],x]

[Out]

(2*(a + b*x)^6*Sqrt[c*(a + b*x)])/(13*b)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{2\, \left ( bx+a \right ) ^{6}}{13\,b}\sqrt{bcx+ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(b*c*x+a*c)^(1/2),x)

[Out]

2/13*(b*x+a)^6*(b*c*x+a*c)^(1/2)/b

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Maxima [A]  time = 0.958328, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (b c x + a c\right )}^{\frac{13}{2}}}{13 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/13*(b*c*x + a*c)^(13/2)/(b*c^6)

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Fricas [B]  time = 1.93621, size = 161, normalized size = 7.32 \begin{align*} \frac{2 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \sqrt{b c x + a c}}{13 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^(1/2),x, algorithm="fricas")

[Out]

2/13*(b^6*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*sqrt(b*c*x +
a*c)/b

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Sympy [A]  time = 0.877335, size = 66, normalized size = 3. \begin{align*} \begin{cases} \frac{2 b^{\frac{11}{2}} \sqrt{c} \left (\frac{a}{b} + x\right )^{\frac{13}{2}}}{13} & \text{for}\: \left |{\frac{a}{b} + x}\right | < 1 \\b^{\frac{11}{2}} \sqrt{c}{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & \frac{15}{2} \\\frac{13}{2} & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} + b^{\frac{11}{2}} \sqrt{c}{G_{2, 2}^{0, 2}\left (\begin{matrix} \frac{15}{2}, 1 & \\ & \frac{13}{2}, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} & \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)**(1/2),x)

[Out]

Piecewise((2*b**(11/2)*sqrt(c)*(a/b + x)**(13/2)/13, Abs(a/b + x) < 1), (b**(11/2)*sqrt(c)*meijerg(((1,), (15/
2,)), ((13/2,), (0,)), a/b + x) + b**(11/2)*sqrt(c)*meijerg(((15/2, 1), ()), ((), (13/2, 0)), a/b + x), True))

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Giac [B]  time = 1.09542, size = 505, normalized size = 22.95 \begin{align*} \frac{2 \,{\left (3003 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{5} - \frac{3003 \,{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} a^{4}}{c} + \frac{858 \,{\left (35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} c^{2} - 42 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a c + 15 \,{\left (b c x + a c\right )}^{\frac{7}{2}}\right )} a^{3}}{c^{2}} - \frac{286 \,{\left (105 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} c^{3} - 189 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} c^{2} + 135 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a c - 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}}\right )} a^{2}}{c^{3}} + \frac{13 \,{\left (1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} c^{4} - 2772 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{3} c^{3} + 2970 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{2} c^{2} - 1540 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a c + 315 \,{\left (b c x + a c\right )}^{\frac{11}{2}}\right )} a}{c^{4}} - \frac{3003 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{5} c^{5} - 9009 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{4} c^{4} + 12870 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{3} c^{3} - 10010 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a^{2} c^{2} + 4095 \,{\left (b c x + a c\right )}^{\frac{11}{2}} a c - 693 \,{\left (b c x + a c\right )}^{\frac{13}{2}}}{c^{5}}\right )}}{9009 \, b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^(1/2),x, algorithm="giac")

[Out]

2/9009*(3003*(b*c*x + a*c)^(3/2)*a^5 - 3003*(5*(b*c*x + a*c)^(3/2)*a*c - 3*(b*c*x + a*c)^(5/2))*a^4/c + 858*(3
5*(b*c*x + a*c)^(3/2)*a^2*c^2 - 42*(b*c*x + a*c)^(5/2)*a*c + 15*(b*c*x + a*c)^(7/2))*a^3/c^2 - 286*(105*(b*c*x
+ a*c)^(3/2)*a^3*c^3 - 189*(b*c*x + a*c)^(5/2)*a^2*c^2 + 135*(b*c*x + a*c)^(7/2)*a*c - 35*(b*c*x + a*c)^(9/2)
)*a^2/c^3 + 13*(1155*(b*c*x + a*c)^(3/2)*a^4*c^4 - 2772*(b*c*x + a*c)^(5/2)*a^3*c^3 + 2970*(b*c*x + a*c)^(7/2)
*a^2*c^2 - 1540*(b*c*x + a*c)^(9/2)*a*c + 315*(b*c*x + a*c)^(11/2))*a/c^4 - (3003*(b*c*x + a*c)^(3/2)*a^5*c^5
- 9009*(b*c*x + a*c)^(5/2)*a^4*c^4 + 12870*(b*c*x + a*c)^(7/2)*a^3*c^3 - 10010*(b*c*x + a*c)^(9/2)*a^2*c^2 + 4
095*(b*c*x + a*c)^(11/2)*a*c - 693*(b*c*x + a*c)^(13/2))/c^5)/(b*c)