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

Optimal. Leaf size=89 \[ -\frac {1600 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}-\frac {400 \sqrt {1-2 x}}{3993 (5 x+3)^{3/2}}+\frac {20}{121 (5 x+3)^{3/2} \sqrt {1-2 x}}+\frac {2}{33 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {1600 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}-\frac {400 \sqrt {1-2 x}}{3993 (5 x+3)^{3/2}}+\frac {20}{121 (5 x+3)^{3/2} \sqrt {1-2 x}}+\frac {2}{33 (5 x+3)^{3/2} (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 20/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (400*Sqrt[1 - 2*x])/(3993*(3
+ 5*x)^(3/2)) - (1600*Sqrt[1 - 2*x])/(43923*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)^{5/2}} \, dx &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {10}{11} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {20}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {200}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {20}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {400 \sqrt {1-2 x}}{3993 (3+5 x)^{3/2}}+\frac {800 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{3993}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {20}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {400 \sqrt {1-2 x}}{3993 (3+5 x)^{3/2}}-\frac {1600 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 0.42 \begin {gather*} \frac {-32000 x^3-4800 x^2+14280 x+722}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(722 + 14280*x - 4800*x^2 - 32000*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.10, size = 70, normalized size = 0.79 \begin {gather*} -\frac {2 (5 x+3)^{3/2} \left (\frac {125 (1-2 x)^3}{(5 x+3)^3}+\frac {450 (1-2 x)^2}{(5 x+3)^2}-\frac {180 (1-2 x)}{5 x+3}-8\right )}{43923 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 5*x)^(3/2)*(-8 + (125*(1 - 2*x)^3)/(3 + 5*x)^3 + (450*(1 - 2*x)^2)/(3 + 5*x)^2 - (180*(1 - 2*x))/(3 +
5*x)))/(43923*(1 - 2*x)^(3/2))

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fricas [A]  time = 0.87, size = 53, normalized size = 0.60 \begin {gather*} -\frac {2 \, {\left (16000 \, x^{3} + 2400 \, x^{2} - 7140 \, x - 361\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(16000*x^3 + 2400*x^2 - 7140*x - 361)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x +
9)

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giac [B]  time = 1.11, size = 160, normalized size = 1.80 \begin {gather*} -\frac {5}{702768} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {132 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {8 \, {\left (16 \, \sqrt {5} {\left (5 \, x + 3\right )} - 99 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615 \, {\left (2 \, x - 1\right )}^{2}} + \frac {5 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {33 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{43923 \, {\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/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-5/702768*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 132*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3)) - 8/219615*(16*sqrt(5)*(5*x + 3) - 99*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^
2 + 5/43923*sqrt(10)*(5*x + 3)^(3/2)*(33*(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 = 32, normalized size = 0.36 \begin {gather*} -\frac {2 \left (16000 x^{3}+2400 x^{2}-7140 x -361\right )}{43923 \left (5 x +3\right )^{\frac {3}{2}} \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)^(5/2),x)

[Out]

-2/43923*(16000*x^3+2400*x^2-7140*x-361)/(5*x+3)^(3/2)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.54, size = 59, normalized size = 0.66 \begin {gather*} \frac {3200 \, x}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {160}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {40 \, x}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3200/43923*x/sqrt(-10*x^2 - x + 3) + 160/43923/sqrt(-10*x^2 - x + 3) + 40/363*x/(-10*x^2 - x + 3)^(3/2) + 2/36
3/(-10*x^2 - x + 3)^(3/2)

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mupad [B]  time = 2.45, size = 69, normalized size = 0.78 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (-\frac {640\,x^3}{43923}-\frac {32\,x^2}{14641}+\frac {476\,x}{73205}+\frac {361}{1098075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\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)^(5/2)),x)

[Out]

((5*x + 3)^(1/2)*((476*x)/73205 - (32*x^2)/14641 - (640*x^3)/43923 + 361/1098075))/((6*x*(1 - 2*x)^(1/2))/25 +
(9*(1 - 2*x)^(1/2))/50 - (7*x^2*(1 - 2*x)^(1/2))/10 - x^3*(1 - 2*x)^(1/2))

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sympy [B]  time = 8.56, size = 391, normalized size = 4.39 \begin {gather*} \begin {cases} \frac {32000 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{3}}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} - \frac {52800 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} + \frac {14520 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} + \frac {2662 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {32000 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{3}}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} - \frac {52800 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} + \frac {14520 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} + \frac {2662 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 5314683 x - 4392300 \left (x + \frac {3}{5}\right )^{3} + 9663060 \left (x + \frac {3}{5}\right )^{2} - \frac {15944049}{5}} & \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)**(5/2),x)

[Out]

Piecewise((32000*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**3/(-5314683*x - 4392300*(x + 3/5)**3 + 96630
60*(x + 3/5)**2 - 15944049/5) - 52800*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(-5314683*x - 4392300
*(x + 3/5)**3 + 9663060*(x + 3/5)**2 - 15944049/5) + 14520*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(-5
314683*x - 4392300*(x + 3/5)**3 + 9663060*(x + 3/5)**2 - 15944049/5) + 2662*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5
)))/(-5314683*x - 4392300*(x + 3/5)**3 + 9663060*(x + 3/5)**2 - 15944049/5), 11/(10*Abs(x + 3/5)) > 1), (32000
*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**3/(-5314683*x - 4392300*(x + 3/5)**3 + 9663060*(x + 3/5)**2
- 15944049/5) - 52800*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(-5314683*x - 4392300*(x + 3/5)**3
+ 9663060*(x + 3/5)**2 - 15944049/5) + 14520*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(-5314683*x - 43
92300*(x + 3/5)**3 + 9663060*(x + 3/5)**2 - 15944049/5) + 2662*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(-531468
3*x - 4392300*(x + 3/5)**3 + 9663060*(x + 3/5)**2 - 15944049/5), True))

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