3.2576 \(\int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{160 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x}}{363 (5 x+3)^{3/2}}+\frac{2}{11 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (40*Sqrt[1 - 2*x])/(363*(3 + 5*x)^(3/2)) - (160*Sqrt[1 - 2*x])/(3993*Sq
rt[3 + 5*x])

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Rubi [A]  time = 0.0102942, antiderivative size = 67, normalized size of antiderivative = 1., 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} \[ -\frac{160 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x}}{363 (5 x+3)^{3/2}}+\frac{2}{11 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (40*Sqrt[1 - 2*x])/(363*(3 + 5*x)^(3/2)) - (160*Sqrt[1 - 2*x])/(3993*Sq
rt[3 + 5*x])

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])

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]

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{20}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x}}{363 (3+5 x)^{3/2}}+\frac{80}{363} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x}}{363 (3+5 x)^{3/2}}-\frac{160 \sqrt{1-2 x}}{3993 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0072954, size = 32, normalized size = 0.48 \[ \frac{2 \left (800 x^2+520 x-97\right )}{3993 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-97 + 520*x + 800*x^2))/(3993*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.003, size = 27, normalized size = 0.4 \begin{align*}{\frac{1600\,{x}^{2}+1040\,x-194}{3993} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3993*(800*x^2+520*x-97)/(3+5*x)^(3/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.16982, size = 86, normalized size = 1.28 \begin{align*} \frac{320 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{16}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{33 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

320/3993*x/sqrt(-10*x^2 - x + 3) + 16/3993/sqrt(-10*x^2 - x + 3) - 2/33/(5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt(-1
0*x^2 - x + 3))

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Fricas [A]  time = 1.5157, size = 122, normalized size = 1.82 \begin{align*} -\frac{2 \,{\left (800 \, x^{2} + 520 \, x - 97\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(800*x^2 + 520*x - 97)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 22.9272, size = 230, normalized size = 3.43 \begin{align*} \begin{cases} - \frac{1600 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{880 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{242 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{1600 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{880 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{242 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1600*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(-219615*x + 199650*(x + 3/5)**2 - 131769)
+ 880*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(-219615*x + 199650*(x + 3/5)**2 - 131769) + 242*sqrt(1
0)*sqrt(-1 + 11/(10*(x + 3/5)))/(-219615*x + 199650*(x + 3/5)**2 - 131769), 11/(10*Abs(x + 3/5)) > 1), (-1600*
sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(-219615*x + 199650*(x + 3/5)**2 - 131769) + 880*sqrt(10)*
I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(-219615*x + 199650*(x + 3/5)**2 - 131769) + 242*sqrt(10)*I*sqrt(1 - 1
1/(10*(x + 3/5)))/(-219615*x + 199650*(x + 3/5)**2 - 131769), True))

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Giac [B]  time = 2.41198, size = 205, normalized size = 3.06 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{63888 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{7 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{5324 \, \sqrt{5 \, x + 3}} - \frac{8 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{6655 \,{\left (2 \, x - 1\right )}} + \frac{{\left (\frac{21 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{3993 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/63888*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 7/5324*sqrt(10)*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 8/6655*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/3993*(21*sqrt(10)
*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^3