3.26.55 \(\int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\) [2555]

Optimal. Leaf size=22 \[ \frac {2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}} \]

[Out]

2/11*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} \frac {2 \sqrt {5 x+3}}{11 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]

[Out]

(2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]

[Out]

(2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x])

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Maple [A]
time = 0.07, size = 17, normalized size = 0.77

method	result	size
gosper	\(\frac {2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}\)	\(17\)
default	\(\frac {2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/11*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Maxima [A]
time = 0.52, size = 21, normalized size = 0.95 \begin {gather*} -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{11 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

-2/11*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]
time = 0.56, size = 23, normalized size = 1.05 \begin {gather*} -\frac {2 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-2/11*sqrt(5*x + 3)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 48, normalized size = 2.18 \begin {gather*} \begin {cases} \frac {\sqrt {10}}{11 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {\sqrt {10} i}{11 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Piecewise((sqrt(10)/(11*sqrt(-1 + 11/(10*(x + 3/5)))), 1/Abs(x + 3/5) > 10/11), (-sqrt(10)*I/(11*sqrt(1 - 11/(
10*(x + 3/5)))), True))

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Giac [A]
time = 0.64, size = 26, normalized size = 1.18 \begin {gather*} -\frac {2 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{55 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-2/55*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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Mupad [B]
time = 2.49, size = 16, normalized size = 0.73 \begin {gather*} \frac {2\,\sqrt {5\,x+3}}{11\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(3/2)*(5*x + 3)^(1/2)),x)

[Out]

(2*(5*x + 3)^(1/2))/(11*(1 - 2*x)^(1/2))

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