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/(3+5*x)^(3/2)/(1-2*x)^(1/2)-40/363*(1-2*x)^(1/2)/(3+5*x)^(3/2)-160/3993*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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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} \[ -\frac {160 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}-\frac {40 \sqrt {1-2 x}}{363 (5 x+3)^{3/2}}+\frac {2}{11 (5 x+3)^{3/2} \sqrt {1-2 x}} \]

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 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)^{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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.80, size = 43, normalized size = 0.64 \[ -\frac {2 \, {\left (800 \, x^{2} + 520 \, x - 97\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

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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giac [B]  time = 1.16, size = 147, normalized size = 2.19 \[ -\frac {1}{63888} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {84 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6655 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {21 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{3993 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

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) + 84*(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*sqrt(10)*(5*x + 3)^(3/2
)*(21*(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 = 27, normalized size = 0.40 \[ \frac {\frac {1600}{3993} x^{2}+\frac {1040}{3993} x -\frac {194}{3993}}{\left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 64, normalized size = 0.96 \[ \frac {320 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {16}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{33 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]

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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mupad [B]  time = 0.22, size = 51, normalized size = 0.76 \[ \frac {\sqrt {5\,x+3}\,\left (\frac {64\,x^2}{3993}+\frac {208\,x}{19965}-\frac {194}{99825}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{5}+\frac {9\,\sqrt {1-2\,x}}{25}+x^2\,\sqrt {1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((208*x)/19965 + (64*x^2)/3993 - 194/99825))/((6*x*(1 - 2*x)^(1/2))/5 + (9*(1 - 2*x)^(1/2))/2
5 + x^2*(1 - 2*x)^(1/2))

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sympy [A]  time = 4.97, size = 230, normalized size = 3.43 \[ \begin {cases} - \frac {1600 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {880 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {242 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\- \frac {1600 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {880 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {242 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} & \text {otherwise} \end {cases} \]

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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