3.26.2 \(\int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\) [2502]

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

[Out]

-2/11*(1-2*x)^(1/2)/(3+5*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 {1-2 x}}{11 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

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Maple [A]
time = 0.08, size = 17, normalized size = 0.77

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.60, size = 21, normalized size = 0.95 \begin {gather*} -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{11 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]
time = 1.12, size = 16, normalized size = 0.73 \begin {gather*} -\frac {2 \, \sqrt {-2 \, x + 1}}{11 \, \sqrt {5 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*sqrt(-2*x + 1)/sqrt(5*x + 3)

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Sympy [C] Result contains complex when optimal does not.
time = 0.49, size = 53, normalized size = 2.41 \begin {gather*} \begin {cases} - \frac {2 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{55} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {2 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{55} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/55, 1/Abs(x + 3/5) > 10/11), (-2*sqrt(10)*I*sqrt(1 - 11/(1
0*(x + 3/5)))/55, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (16) = 32\).
time = 0.58, size = 61, normalized size = 2.77 \begin {gather*} -\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{110 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{55 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/110*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/55*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))

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Mupad [B]
time = 2.47, size = 34, normalized size = 1.55 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {4\,x}{55}-\frac {2}{55}\right )}{x\,\sqrt {1-2\,x}+\frac {3\,\sqrt {1-2\,x}}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((4*x)/55 - 2/55))/(x*(1 - 2*x)^(1/2) + (3*(1 - 2*x)^(1/2))/5)

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