\(\int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx\) [1451]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2 \sqrt {a c+b c x}}{b c^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, 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} \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2 \sqrt {a c+b c x}}{b c^6} \]

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^(11/2),x]

[Out]

(2*Sqrt[a*c + b*c*x])/(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 \frac {1}{\sqrt {a c+b c x}} \, dx}{c^5} \\ & = \frac {2 \sqrt {a c+b c x}}{b c^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2 (a+b x)}{b c^5 \sqrt {c (a+b x)}} \]

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^(11/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {c \left (b x +a \right )}}{b \,c^{6}}\)	\(18\)
derivativedivides	\(\frac {2 \sqrt {b c x +a c}}{b \,c^{6}}\)	\(19\)
default	\(\frac {2 \sqrt {b c x +a c}}{b \,c^{6}}\)	\(19\)
trager	\(\frac {2 \sqrt {b c x +a c}}{b \,c^{6}}\)	\(19\)
gosper	\(\frac {2 \left (b x +a \right )^{6}}{b \left (b c x +a c \right )^{\frac {11}{2}}}\)	\(23\)
risch	\(\frac {2 b x +2 a}{c^{5} b \sqrt {c \left (b x +a \right )}}\)	\(23\)

[In]

int((b*x+a)^5/(b*c*x+a*c)^(11/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2 \, \sqrt {b c x + a c}}{b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(11/2),x, algorithm="fricas")

[Out]

2*sqrt(b*c*x + a*c)/(b*c^6)

Sympy [A] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\begin {cases} \frac {2 \sqrt {a c + b c x}}{b c^{6}} & \text {for}\: b \neq 0 \\\frac {a^{5} x}{\left (a c\right )^{\frac {11}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**(11/2),x)

[Out]

Piecewise((2*sqrt(a*c + b*c*x)/(b*c**6), Ne(b, 0)), (a**5*x/(a*c)**(11/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2 \, \sqrt {b c x + a c}}{b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(11/2),x, algorithm="maxima")

[Out]

2*sqrt(b*c*x + a*c)/(b*c^6)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2 \, \sqrt {b c x + a c}}{b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(11/2),x, algorithm="giac")

[Out]

2*sqrt(b*c*x + a*c)/(b*c^6)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{11/2}} \, dx=\frac {2\,\sqrt {c\,\left (a+b\,x\right )}}{b\,c^6} \]

[In]

int((a + b*x)^5/(a*c + b*c*x)^(11/2),x)

[Out]

(2*(c*(a + b*x))^(1/2))/(b*c^6)