3.2552 \(\int \frac {(2+3 x)^3}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=84 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^2}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (10380 x+25003)}{8800}-\frac {56421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]

[Out]

-56421/8000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/11*(2+3*x)^2*(3+5*x)^(1/2)/(1-2*x)^(1/2)+3/8800*(25
003+10380*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 84, 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 = {98, 147, 54, 216} \[ \frac {7 \sqrt {5 x+3} (3 x+2)^2}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (10380 x+25003)}{8800}-\frac {56421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25003 + 10380*x))/8800 - (5
6421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 98

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x) \left (159+\frac {519 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (25003+10380 x)}{8800}-\frac {56421 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600}\\ &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (25003+10380 x)}{8800}-\frac {56421 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}}\\ &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (25003+10380 x)}{8800}-\frac {56421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 78, normalized size = 0.93 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (11880 x^2+51678 x-97409\right )-620631 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{88000 \sqrt {-(1-2 x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-97409 + 51678*x + 11880*x^2) - 620631*Sqrt[10]*(-1 + 2*x)*ArcSinh[Sqrt[5/1
1]*Sqrt[-1 + 2*x]])/(88000*Sqrt[-(1 - 2*x)^2])

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fricas [A]  time = 0.83, size = 81, normalized size = 0.96 \[ \frac {620631 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (11880 \, x^{2} + 51678 \, x - 97409\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{176000 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/176000*(620631*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) + 20*(11880*x^2 + 51678*x - 97409)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 1.01, size = 71, normalized size = 0.85 \[ -\frac {56421}{8000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (594 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 63 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 620695 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{220000 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-56421/8000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/220000*(594*(4*sqrt(5)*(5*x + 3) + 63*sqrt(5))*(5
*x + 3) - 620695*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.02, size = 106, normalized size = 1.26 \[ -\frac {\left (-237600 \sqrt {-10 x^{2}-x +3}\, x^{2}+1241262 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1033560 \sqrt {-10 x^{2}-x +3}\, x -620631 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1948180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{176000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/176000*(1241262*10^(1/2)*x*arcsin(20/11*x+1/11)-237600*(-10*x^2-x+3)^(1/2)*x^2-620631*10^(1/2)*arcsin(20/11
*x+1/11)-1033560*(-10*x^2-x+3)^(1/2)*x+1948180*(-10*x^2-x+3)^(1/2))*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(2*x-1)/(-10*
x^2-x+3)^(1/2)

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maxima [A]  time = 1.18, size = 65, normalized size = 0.77 \[ -\frac {56421}{16000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {27}{40} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {2619}{800} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{44 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-56421/16000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 27/40*sqrt(-10*x^2 - x + 3)*x + 2619/800*sqrt(-10*x^2 -
x + 3) - 343/44*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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