3.24.73 \(\int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac {8 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}} \]

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Rubi [A]  time = 0.01, 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 = {45, 37} \begin {gather*} -\frac {8 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) - (8*Sqrt[1 - 2*x])/(363*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]

Rule 45

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}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {4}{33} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}-\frac {8 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.60 \begin {gather*} -\frac {2 \sqrt {1-2 x} (20 x+23)}{363 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(23 + 20*x))/(363*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.07, size = 38, normalized size = 0.84 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+6\right )}{363 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(6 + (5*(1 - 2*x))/(3 + 5*x)))/(363*Sqrt[3 + 5*x])

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fricas [A]  time = 1.25, size = 33, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (20 \, x + 23\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(20*x + 23)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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giac [B]  time = 0.87, size = 121, normalized size = 2.69 \begin {gather*} -\frac {1}{29040} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {36 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {9 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1815 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29040*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 36*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3)) + 1/1815*sqrt(10)*(5*x + 3)^(3/2)*(9*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4
)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.00, size = 22, normalized size = 0.49 \begin {gather*} -\frac {2 \left (20 x +23\right ) \sqrt {-2 x +1}}{363 \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(20*x+23)/(5*x+3)^(3/2)*(-2*x+1)^(1/2)

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maxima [A]  time = 1.17, size = 48, normalized size = 1.07 \begin {gather*} -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {8 \, \sqrt {-10 \, x^{2} - x + 3}}{363 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/33*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 8/363*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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mupad [B]  time = 0.18, size = 51, normalized size = 1.13 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {16\,x^2}{1815}+\frac {52\,x}{9075}-\frac {46}{9075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{5}+\frac {9\,\sqrt {1-2\,x}}{25}+x^2\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((52*x)/9075 + (16*x^2)/1815 - 46/9075))/((6*x*(1 - 2*x)^(1/2))/5 + (9*(1 - 2*x)^(1/2))/25 +
x^2*(1 - 2*x)^(1/2))

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sympy [A]  time = 2.27, size = 102, normalized size = 2.27 \begin {gather*} \begin {cases} - \frac {8 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1815} - \frac {2 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{825 \left (x + \frac {3}{5}\right )} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\- \frac {8 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1815} - \frac {2 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{825 \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/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Piecewise((-8*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/1815 - 2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(825*(x + 3
/5)), 11/(10*Abs(x + 3/5)) > 1), (-8*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/1815 - 2*sqrt(10)*I*sqrt(1 - 11/(1
0*(x + 3/5)))/(825*(x + 3/5)), True))

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