3.2566 \(\int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac {2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {20 \sqrt {1-2 x}}{121 \sqrt {5 x+3}} \]

[Out]

2/11/(1-2*x)^(1/2)/(3+5*x)^(1/2)-20/121*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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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} \[ \frac {2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {20 \sqrt {1-2 x}}{121 \sqrt {5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.60 \[ \frac {2 (20 x+1)}{121 \sqrt {1-2 x} \sqrt {5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + 20*x))/(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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fricas [A]  time = 0.80, size = 31, normalized size = 0.69 \[ -\frac {2 \, {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{121 \, {\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/121*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)

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giac [B]  time = 1.15, size = 87, normalized size = 1.93 \[ -\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{242 \, \sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{605 \, {\left (2 \, x - 1\right )}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{121 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/242*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4/605*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1) + 2/121*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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maple [A]  time = 0.00, size = 22, normalized size = 0.49 \[ \frac {\frac {40 x}{121}+\frac {2}{121}}{\sqrt {-2 x +1}\, \sqrt {5 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.62, size = 30, normalized size = 0.67 \[ \frac {40 \, x}{121 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2}{121 \, \sqrt {-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

40/121*x/sqrt(-10*x^2 - x + 3) + 2/121/sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 2.40, size = 34, normalized size = 0.76 \[ \frac {\sqrt {5\,x+3}\,\left (\frac {8\,x}{121}+\frac {2}{605}\right )}{x\,\sqrt {1-2\,x}+\frac {3\,\sqrt {1-2\,x}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((8*x)/121 + 2/605))/(x*(1 - 2*x)^(1/2) + (3*(1 - 2*x)^(1/2))/5)

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sympy [A]  time = 1.79, size = 116, normalized size = 2.58 \[ \begin {cases} \frac {40 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{605 - 1210 x} - \frac {22 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{605 - 1210 x} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\\frac {40 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{605 - 1210 x} - \frac {22 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{605 - 1210 x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((40*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(605 - 1210*x) - 22*sqrt(10)*sqrt(-1 + 11/(10*(x
+ 3/5)))/(605 - 1210*x), 11/(10*Abs(x + 3/5)) > 1), (40*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(605
- 1210*x) - 22*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(605 - 1210*x), True))

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