3.1452 \(\int \frac {(a+b x)^5}{(a c+b c x)^{13/2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac {2}{b c^6 \sqrt {a c+b c x}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ -\frac {2}{b c^6 \sqrt {a c+b c x}} \]

Antiderivative was successfully verified.

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.20 \[ -\frac {2 (a+b x)}{b c^5 (c (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.41, size = 29, normalized size = 1.45 \[ -\frac {2 \, \sqrt {b c x + a c}}{b^{2} c^{7} x + a b c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.81, size = 18, normalized size = 0.90 \[ -\frac {2}{\sqrt {b c x + a c} b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.00, size = 23, normalized size = 1.15 \[ -\frac {2 \left (b x +a \right )^{6}}{\left (b c x +a c \right )^{\frac {13}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 18, normalized size = 0.90 \[ -\frac {2}{\sqrt {b c x + a c} b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.03, size = 17, normalized size = 0.85 \[ -\frac {2}{b\,c^6\,\sqrt {c\,\left (a+b\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 39.43, size = 48, normalized size = 2.40 \[ \begin {cases} - \frac {2 \sqrt {a c + b c x}}{a b c^{7} + b^{2} c^{7} x} & \text {for}\: a \neq 0 \\- \frac {2}{b^{\frac {3}{2}} c^{\frac {13}{2}} \sqrt {x}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[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(a, 0)), (-2/(b**(3/2)*c**(13/2)*sqrt(x)), True))

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