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

Optimal. Leaf size=96 \[ \frac {2 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {8182 \sqrt {1-2 x}}{219615 \sqrt {5 x+3}}-\frac {3679 \sqrt {1-2 x}}{19965 (5 x+3)^{3/2}}+\frac {49}{121 \sqrt {1-2 x} (5 x+3)^{3/2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {94, 89, 78, 37} \begin {gather*} \frac {2 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {8182 \sqrt {1-2 x}}{219615 \sqrt {5 x+3}}-\frac {3679 \sqrt {1-2 x}}{19965 (5 x+3)^{3/2}}+\frac {49}{121 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(19965*(3 + 5*x)^(3/2)) + (2*(2 + 3*x)^3)/(33*(1
- 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (8182*Sqrt[1 - 2*x])/(219615*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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rubi steps

\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{11} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{121} \int \frac {-\frac {617}{2}+99 x}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{19965 (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {4091 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{19965}\\ &=\frac {49}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{19965 (3+5 x)^{3/2}}+\frac {2 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {8182 \sqrt {1-2 x}}{219615 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 37, normalized size = 0.39 \begin {gather*} \frac {2 \left (19573 x^3+62232 x^2+52044 x+13040\right )}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(13040 + 52044*x + 62232*x^2 + 19573*x^3))/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.11, size = 69, normalized size = 0.72 \begin {gather*} -\frac {2 (5 x+3)^{3/2} \left (\frac {(1-2 x)^3}{(5 x+3)^3}+\frac {63 (1-2 x)^2}{(5 x+3)^2}-\frac {441 (1-2 x)}{5 x+3}-343\right )}{43923 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 5*x)^(3/2)*(-343 + (1 - 2*x)^3/(3 + 5*x)^3 + (63*(1 - 2*x)^2)/(3 + 5*x)^2 - (441*(1 - 2*x))/(3 + 5*x)
))/(43923*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.65, size = 53, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left (19573 \, x^{3} + 62232 \, x^{2} + 52044 \, x + 13040\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((2+3*x)^3/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

2/43923*(19573*x^3 + 62232*x^2 + 52044*x + 13040)*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 = 1.35, size = 160, normalized size = 1.67 \begin {gather*} -\frac {1}{17569200} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2508 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {98 \, {\left (17 \, \sqrt {5} {\left (5 \, x + 3\right )} + 99 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1098075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {627 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1098075 \, {\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((2+3*x)^3/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/17569200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 2508*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3)) + 98/1098075*(17*sqrt(5)*(5*x + 3) + 99*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1)^2 + 1/1098075*sqrt(10)*(5*x + 3)^(3/2)*(627*(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 = 0.33 \begin {gather*} \frac {\frac {39146}{43923} x^{3}+\frac {41488}{14641} x^{2}+\frac {34696}{14641} x +\frac {26080}{43923}}{\left (-2 x +1\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/43923*(19573*x^3+62232*x^2+52044*x+13040)/(5*x+3)^(3/2)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.47, size = 76, normalized size = 0.79 \begin {gather*} -\frac {19573 \, x}{219615 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {27 \, x^{2}}{10 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {19573}{4392300 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {95567 \, x}{36300 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {22039}{36300 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-19573/219615*x/sqrt(-10*x^2 - x + 3) + 27/10*x^2/(-10*x^2 - x + 3)^(3/2) - 19573/4392300/sqrt(-10*x^2 - x + 3
) + 95567/36300*x/(-10*x^2 - x + 3)^(3/2) + 22039/36300/(-10*x^2 - x + 3)^(3/2)

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mupad [B]  time = 0.31, size = 69, normalized size = 0.72 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {19573\,x^3}{1098075}+\frac {20744\,x^2}{366025}+\frac {17348\,x}{366025}+\frac {2608}{219615}\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((3*x + 2)^3/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)

[Out]

((5*x + 3)^(1/2)*((17348*x)/366025 + (20744*x^2)/366025 + (19573*x^3)/1098075 + 2608/219615))/((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 [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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