3.27.13 \(\int \frac {1}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\) [2613]

Optimal. Leaf size=45 \[ \frac {2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {20 \sqrt {3+5 x}}{363 \sqrt {1-2 x}} \]

[Out]

2/33*(3+5*x)^(1/2)/(1-2*x)^(3/2)+20/363*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \begin {gather*} \frac {20 \sqrt {5 x+3}}{363 \sqrt {1-2 x}}+\frac {2 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (20*Sqrt[3 + 5*x])/(363*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]

Rule 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {10}{33} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {20 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 27, normalized size = 0.60 \begin {gather*} -\frac {2 \sqrt {3+5 x} (-21+20 x)}{363 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 5*x]*(-21 + 20*x))/(363*(1 - 2*x)^(3/2))

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Maple [A]
time = 0.08, size = 34, normalized size = 0.76

method	result	size
gosper	\(-\frac {2 \sqrt {3+5 x}\, \left (-21+20 x \right )}{363 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(22\)
default	\(\frac {2 \sqrt {3+5 x}}{33 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {20 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/33*(3+5*x)^(1/2)/(1-2*x)^(3/2)+20/363*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Maxima [A]
time = 0.52, size = 48, normalized size = 1.07 \begin {gather*} \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {20 \, \sqrt {-10 \, x^{2} - x + 3}}{363 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/33*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) - 20/363*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]
time = 0.49, size = 33, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (20 \, x - 21\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(20*x - 21)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 1.16, size = 177, normalized size = 3.93 \begin {gather*} \begin {cases} \frac {100 \sqrt {10} \left (x + \frac {3}{5}\right )}{3630 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}} - \frac {165 \sqrt {10}}{3630 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \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 {100 \sqrt {10} i \left (x + \frac {3}{5}\right )}{3630 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}} + \frac {165 \sqrt {10} i}{3630 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right ) - 3993 \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)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Piecewise((100*sqrt(10)*(x + 3/5)/(3630*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(-1 + 11/(10*(x + 3/
5)))) - 165*sqrt(10)/(3630*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(-1 + 11/(10*(x + 3/5)))), 1/Abs(
x + 3/5) > 10/11), (-100*sqrt(10)*I*(x + 3/5)/(3630*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(1 - 11/(
10*(x + 3/5)))) + 165*sqrt(10)*I/(3630*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(1 - 11/(10*(x + 3/5))
)), True))

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Giac [A]
time = 1.32, size = 39, normalized size = 0.87 \begin {gather*} -\frac {2 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1815 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1815*(4*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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Mupad [B]
time = 0.19, size = 34, normalized size = 0.76 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {20\,x}{363}-\frac {7}{121}\right )}{x\,\sqrt {1-2\,x}-\frac {\sqrt {1-2\,x}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((20*x)/363 - 7/121))/(x*(1 - 2*x)^(1/2) - (1 - 2*x)^(1/2)/2)

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