\(\int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx\) [1452]

Optimal result

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

[Out]

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

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

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^(13/2),x]

[Out]

-2/(b*c^6*Sqrt[a*c + b*c*x])

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx=-\frac {2 (a+b x)}{b c^5 (c (a+b x))^{3/2}} \]

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^(13/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {2}{b \,c^{6} \sqrt {c \left (b x +a \right )}}\)	\(18\)
derivativedivides	\(-\frac {2}{b \,c^{6} \sqrt {b c x +a c}}\)	\(19\)
default	\(-\frac {2}{b \,c^{6} \sqrt {b c x +a c}}\)	\(19\)
gosper	\(-\frac {2 \left (b x +a \right )^{6}}{b \left (b c x +a c \right )^{\frac {13}{2}}}\)	\(23\)
trager	\(-\frac {2 \sqrt {b c x +a c}}{c^{7} b \left (b x +a \right )}\)	\(26\)

[In]

int((b*x+a)^5/(b*c*x+a*c)^(13/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx=-\frac {2 \, \sqrt {b c x + a c}}{b^{2} c^{7} x + a b c^{7}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(13/2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 2.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx=\begin {cases} - \frac {2 \sqrt {a c + b c x}}{a b c^{7} + b^{2} c^{7} x} & \text {for}\: b \neq 0 \\\frac {a^{5} x}{\left (a c\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**(13/2),x)

[Out]

Piecewise((-2*sqrt(a*c + b*c*x)/(a*b*c**7 + b**2*c**7*x), Ne(b, 0)), (a**5*x/(a*c)**(13/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx=-\frac {2}{\sqrt {b c x + a c} b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(13/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*c*x + a*c)*b*c^6)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx=-\frac {2}{\sqrt {b c x + a c} b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(13/2),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 \[ \int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx=-\frac {2}{b\,c^6\,\sqrt {c\,\left (a+b\,x\right )}} \]

[In]

int((a + b*x)^5/(a*c + b*c*x)^(13/2),x)

[Out]

-2/(b*c^6*(c*(a + b*x))^(1/2))