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

Optimal. Leaf size=128 \[ -\frac {3}{50} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {3 \sqrt {1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac {917953 \sqrt {1-2 x} (5 x+3)^{3/2}}{128000}-\frac {30292449 \sqrt {1-2 x} \sqrt {5 x+3}}{512000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512000 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \begin {gather*} -\frac {3}{50} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {3 \sqrt {1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac {917953 \sqrt {1-2 x} (5 x+3)^{3/2}}{128000}-\frac {30292449 \sqrt {1-2 x} \sqrt {5 x+3}}{512000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30292449*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512000 - (917953*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/128000 - (3*Sqrt[1 - 2
*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/50 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7889 + 3900*x))/16000 + (333216939*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512000*Sqrt[10])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 100

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

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 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{50} \int \frac {\left (-311-\frac {975 x}{2}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {917953 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{32000}\\ &=-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {30292449 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{256000}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1024000}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{512000 \sqrt {5}}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 88, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (6912000 x^4+26870400 x^3+46785120 x^2+51453140 x+49229901\right )+333216939 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{5120000 \sqrt {2 x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5120000*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(49229901 + 51453140*x + 46785120*x^2 + 26870400*x^
3 + 6912000*x^4) + 333216939*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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IntegrateAlgebraic [A]  time = 0.24, size = 141, normalized size = 1.10 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {1721161875 (1-2 x)^4}{(5 x+3)^4}+\frac {3212835500 (1-2 x)^3}{(5 x+3)^3}+\frac {2349918720 (1-2 x)^2}{(5 x+3)^2}+\frac {828154320 (1-2 x)}{5 x+3}+131554256\right )}{512000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{512000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*Sqrt[1 - 2*x]*(131554256 + (1721161875*(1 - 2*x)^4)/(3 + 5*x)^4 + (3212835500*(1 - 2*x)^3)/(3 + 5*x)^3 +
 (2349918720*(1 - 2*x)^2)/(3 + 5*x)^2 + (828154320*(1 - 2*x))/(3 + 5*x)))/(512000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2
*x))/(3 + 5*x))^5) - (333216939*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(512000*Sqrt[10])

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fricas [A]  time = 0.85, size = 77, normalized size = 0.60 \begin {gather*} -\frac {1}{512000} \, {\left (6912000 \, x^{4} + 26870400 \, x^{3} + 46785120 \, x^{2} + 51453140 \, x + 49229901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {333216939}{10240000} \, \sqrt {10} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/512000*(6912000*x^4 + 26870400*x^3 + 46785120*x^2 + 51453140*x + 49229901)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3
33216939/10240000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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giac [A]  time = 1.30, size = 72, normalized size = 0.56 \begin {gather*} -\frac {1}{25600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (24 \, {\left (36 \, {\left (80 \, x + 167\right )} {\left (5 \, x + 3\right )} + 27809\right )} {\left (5 \, x + 3\right )} + 4589765\right )} {\left (5 \, x + 3\right )} + 151462245\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1666084695 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25600000*sqrt(5)*(2*(4*(24*(36*(80*x + 167)*(5*x + 3) + 27809)*(5*x + 3) + 4589765)*(5*x + 3) + 151462245)*
sqrt(5*x + 3)*sqrt(-10*x + 5) - 1666084695*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))

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maple [A]  time = 0.01, size = 121, normalized size = 0.95 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-537408000 \sqrt {-10 x^{2}-x +3}\, x^{3}-935702400 \sqrt {-10 x^{2}-x +3}\, x^{2}-1029062800 \sqrt {-10 x^{2}-x +3}\, x +333216939 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-984598020 \sqrt {-10 x^{2}-x +3}\right )}{10240000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10240000*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(-138240000*(-10*x^2-x+3)^(1/2)*x^4-537408000*(-10*x^2-x+3)^(1/2)*x^3-
935702400*(-10*x^2-x+3)^(1/2)*x^2+333216939*10^(1/2)*arcsin(20/11*x+1/11)-1029062800*(-10*x^2-x+3)^(1/2)*x-984
598020*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.28, size = 92, normalized size = 0.72 \begin {gather*} -\frac {27}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {8397}{160} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {292407}{3200} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2572657}{25600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {333216939}{10240000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {49229901}{512000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/2*sqrt(-10*x^2 - x + 3)*x^4 - 8397/160*sqrt(-10*x^2 - x + 3)*x^3 - 292407/3200*sqrt(-10*x^2 - x + 3)*x^2 -
 2572657/25600*sqrt(-10*x^2 - x + 3)*x - 333216939/10240000*sqrt(10)*arcsin(-20/11*x - 1/11) - 49229901/512000
*sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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