\(\int (a+b x)^5 \sqrt {a c+b c x} \, dx\) [1445]

Optimal result

Integrand size = 20, antiderivative size = 22 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2 (a c+b c x)^{13/2}}{13 b c^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2 (a c+b c x)^{13/2}}{13 b c^6} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2 (a+b x)^6 \sqrt {c (a+b x)}}{13 b} \]

[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)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {2 \left (b c x +a c \right )^{\frac {13}{2}}}{13 b \,c^{6}}\)	\(19\)
default	\(\frac {2 \left (b c x +a c \right )^{\frac {13}{2}}}{13 b \,c^{6}}\)	\(19\)
pseudoelliptic	\(\frac {2 \left (b x +a \right )^{6} \sqrt {c \left (b x +a \right )}}{13 b}\)	\(22\)
gosper	\(\frac {2 \left (b x +a \right )^{6} \sqrt {b c x +a c}}{13 b}\)	\(23\)
trager	\(\frac {2 \left (b^{6} x^{6}+6 a \,x^{5} b^{5}+15 a^{2} x^{4} b^{4}+20 a^{3} x^{3} b^{3}+15 a^{4} x^{2} b^{2}+6 a^{5} x b +a^{6}\right ) \sqrt {b c x +a c}}{13 b}\)	\(76\)
risch	\(\frac {2 c \left (b^{6} x^{6}+6 a \,x^{5} b^{5}+15 a^{2} x^{4} b^{4}+20 a^{3} x^{3} b^{3}+15 a^{4} x^{2} b^{2}+6 a^{5} x b +a^{6}\right ) \left (b x +a \right )}{13 b \sqrt {c \left (b x +a \right )}}\)	\(81\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (18) = 36\).

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.41 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \sqrt {b c x + a c}}{13 \, b} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (19) = 38\).

Time = 0.66 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.00 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\begin {cases} \frac {2 b^{\frac {11}{2}} \sqrt {c} \left (\frac {a}{b} + x\right )^{\frac {13}{2}}}{13} & \text {for}\: \left |{\frac {a}{b} + x}\right | < 1 \\b^{\frac {11}{2}} \sqrt {c} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {15}{2} \\\frac {13}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} + b^{\frac {11}{2}} \sqrt {c} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {15}{2}, 1 & \\ & \frac {13}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2 \, {\left (b c x + a c\right )}^{\frac {13}{2}}}{13 \, b c^{6}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (18) = 36\).

Time = 0.32 (sec) , antiderivative size = 495, normalized size of antiderivative = 22.50 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2 \, {\left (3003 \, \sqrt {b c x + a c} a^{6} - \frac {6006 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{5}}{c} + \frac {3003 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{4}}{c^{2}} - \frac {1716 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{3}}{c^{3}} + \frac {143 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a^{2}}{c^{4}} - \frac {26 \, {\left (693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}\right )} a}{c^{5}} + \frac {3003 \, \sqrt {b c x + a c} a^{6} c^{6} - 6006 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{5} c^{5} + 9009 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{4} c^{4} - 8580 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{3} c^{3} + 5005 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{2} c^{2} - 1638 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a c + 231 \, {\left (b c x + a c\right )}^{\frac {13}{2}}}{c^{6}}\right )}}{3003 \, b} \]

[In]

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

[Out]

2/3003*(3003*sqrt(b*c*x + a*c)*a^6 - 6006*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^5/c + 3003*(15*sqr
t(b*c*x + a*c)*a^2*c^2 - 10*(b*c*x + a*c)^(3/2)*a*c + 3*(b*c*x + a*c)^(5/2))*a^4/c^2 - 1716*(35*sqrt(b*c*x + a
*c)*a^3*c^3 - 35*(b*c*x + a*c)^(3/2)*a^2*c^2 + 21*(b*c*x + a*c)^(5/2)*a*c - 5*(b*c*x + a*c)^(7/2))*a^3/c^3 + 1
43*(315*sqrt(b*c*x + a*c)*a^4*c^4 - 420*(b*c*x + a*c)^(3/2)*a^3*c^3 + 378*(b*c*x + a*c)^(5/2)*a^2*c^2 - 180*(b
*c*x + a*c)^(7/2)*a*c + 35*(b*c*x + a*c)^(9/2))*a^2/c^4 - 26*(693*sqrt(b*c*x + a*c)*a^5*c^5 - 1155*(b*c*x + a*
c)^(3/2)*a^4*c^4 + 1386*(b*c*x + a*c)^(5/2)*a^3*c^3 - 990*(b*c*x + a*c)^(7/2)*a^2*c^2 + 385*(b*c*x + a*c)^(9/2
)*a*c - 63*(b*c*x + a*c)^(11/2))*a/c^5 + (3003*sqrt(b*c*x + a*c)*a^6*c^6 - 6006*(b*c*x + a*c)^(3/2)*a^5*c^5 +
9009*(b*c*x + a*c)^(5/2)*a^4*c^4 - 8580*(b*c*x + a*c)^(7/2)*a^3*c^3 + 5005*(b*c*x + a*c)^(9/2)*a^2*c^2 - 1638*
(b*c*x + a*c)^(11/2)*a*c + 231*(b*c*x + a*c)^(13/2))/c^6)/b

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int (a+b x)^5 \sqrt {a c+b c x} \, dx=\frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{13/2}}{13\,b\,c^6} \]

[In]

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

[Out]

(2*(c*(a + b*x))^(13/2))/(13*b*c^6)