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

Optimal. Leaf size=67 \[ -\frac {400 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}+\frac {40}{363 \sqrt {5 x+3} \sqrt {1-2 x}}+\frac {2}{33 \sqrt {5 x+3} (1-2 x)^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {400 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}+\frac {40}{363 \sqrt {5 x+3} \sqrt {1-2 x}}+\frac {2}{33 \sqrt {5 x+3} (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 40/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (400*Sqrt[1 - 2*x])/(3993*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}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {20}{33} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {40}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {200}{363} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {40}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {400 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 32, normalized size = 0.48 \begin {gather*} \frac {-1600 x^2+720 x+282}{3993 (1-2 x)^{3/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(282 + 720*x - 1600*x^2)/(3993*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.09, size = 54, normalized size = 0.81 \begin {gather*} -\frac {2 (5 x+3)^{3/2} \left (\frac {75 (1-2 x)^2}{(5 x+3)^2}-\frac {60 (1-2 x)}{5 x+3}-4\right )}{3993 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 5*x)^(3/2)*(-4 + (75*(1 - 2*x)^2)/(3 + 5*x)^2 - (60*(1 - 2*x))/(3 + 5*x)))/(3993*(1 - 2*x)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 1.46, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (800 \, x^{2} - 360 \, x - 141\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(800*x^2 - 360*x - 141)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(20*x^3 - 8*x^2 - 7*x + 3)

________________________________________________________________________________________

giac [B]  time = 1.00, size = 100, normalized size = 1.49 \begin {gather*} -\frac {5 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{2662 \, \sqrt {5 \, x + 3}} - \frac {8 \, {\left (5 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{19965 \, {\left (2 \, x - 1\right )}^{2}} + \frac {10 \, \sqrt {10} \sqrt {5 \, x + 3}}{1331 \, {\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/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-5/2662*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/19965*(5*sqrt(5)*(5*x + 3) - 33*sqrt(5
))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 10/1331*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 27, normalized size = 0.40 \begin {gather*} -\frac {2 \left (800 x^{2}-360 x -141\right )}{3993 \sqrt {5 x +3}\, \left (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(800*x^2-360*x-141)/(5*x+3)^(1/2)/(-2*x+1)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 64, normalized size = 0.96 \begin {gather*} \frac {800 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {40}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{33 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

800/3993*x/sqrt(-10*x^2 - x + 3) + 40/3993/sqrt(-10*x^2 - x + 3) - 2/33/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*
x^2 - x + 3))

________________________________________________________________________________________

mupad [B]  time = 2.44, size = 52, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {5\,x+3}\,\left (-\frac {160\,x^2}{3993}+\frac {24\,x}{1331}+\frac {47}{6655}\right )}{\frac {x\,\sqrt {1-2\,x}}{10}-\frac {3\,\sqrt {1-2\,x}}{10}+x^2\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((5*x + 3)^(1/2)*((24*x)/1331 - (160*x^2)/3993 + 47/6655))/((x*(1 - 2*x)^(1/2))/10 - (3*(1 - 2*x)^(1/2))/10 +
x^2*(1 - 2*x)^(1/2))

________________________________________________________________________________________

sympy [A]  time = 4.72, size = 230, normalized size = 3.43 \begin {gather*} \begin {cases} - \frac {8000 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} + \frac {13200 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} - \frac {3630 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\- \frac {8000 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} + \frac {13200 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} - \frac {3630 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} & \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)**(3/2),x)

[Out]

Piecewise((-8000*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(-878460*x + 399300*(x + 3/5)**2 - 43923)
+ 13200*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(-878460*x + 399300*(x + 3/5)**2 - 43923) - 3630*sqrt(
10)*sqrt(-1 + 11/(10*(x + 3/5)))/(-878460*x + 399300*(x + 3/5)**2 - 43923), 11/(10*Abs(x + 3/5)) > 1), (-8000*
sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(-878460*x + 399300*(x + 3/5)**2 - 43923) + 13200*sqrt(10)
*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(-878460*x + 399300*(x + 3/5)**2 - 43923) - 3630*sqrt(10)*I*sqrt(1 -
11/(10*(x + 3/5)))/(-878460*x + 399300*(x + 3/5)**2 - 43923), True))

________________________________________________________________________________________