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

Optimal. Leaf size=84 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^2}{33 (1-2 x)^{3/2}}-\frac {(95621-33462 x) \sqrt {5 x+3}}{14520 \sqrt {1-2 x}}+\frac {1593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40 \sqrt {10}} \]

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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, 143, 54, 216} \begin {gather*} \frac {7 \sqrt {5 x+3} (3 x+2)^2}{33 (1-2 x)^{3/2}}-\frac {(95621-33462 x) \sqrt {5 x+3}}{14520 \sqrt {1-2 x}}+\frac {1593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((95621 - 33462*x)*Sqrt[3 + 5*x])/(14520*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^2*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2))
+ (1593*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40*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 143

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

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)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x) \left (155+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593}{80} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{40 \sqrt {5}}\\ &=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 73, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {5 x+3} \left (39204 x^2-261664 x+83301\right )}{14520 (1-2 x)^{3/2}}-\frac {1593 \sqrt {1-2 x} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{40 \sqrt {20 x-10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/14520*(Sqrt[3 + 5*x]*(83301 - 261664*x + 39204*x^2))/(1 - 2*x)^(3/2) - (1593*Sqrt[1 - 2*x]*ArcSinh[Sqrt[5/1
1]*Sqrt[-1 + 2*x]])/(40*Sqrt[-10 + 20*x])

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IntegrateAlgebraic [A]  time = 0.14, size = 109, normalized size = 1.30 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {578211 (1-2 x)^2}{(5 x+3)^2}-\frac {153860 (1-2 x)}{5 x+3}+13720\right )}{14520 (1-2 x)^{3/2} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )}-\frac {1593 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{40 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 5*x)^(3/2)*(13720 - (578211*(1 - 2*x)^2)/(3 + 5*x)^2 - (153860*(1 - 2*x))/(3 + 5*x)))/(14520*(1 - 2*x)^(
3/2)*(2 + (5*(1 - 2*x))/(3 + 5*x))) - (1593*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(40*Sqrt[10])

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fricas [A]  time = 1.37, size = 91, normalized size = 1.08 \begin {gather*} -\frac {578259 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, 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 (39204 \, x^{2} - 261664 \, x + 83301\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{290400 \, {\left (4 \, x^{2} - 4 \, x + 1\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)^(1/2),x, algorithm="fricas")

[Out]

-1/290400*(578259*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*
x^2 + x - 3)) + 20*(39204*x^2 - 261664*x + 83301)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.23, size = 71, normalized size = 0.85 \begin {gather*} \frac {1593}{400} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9801 \, \sqrt {5} {\left (5 \, x + 3\right )} - 385886 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 6360321 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1815000 \, {\left (2 \, x - 1\right )}^{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)^(1/2),x, algorithm="giac")

[Out]

1593/400*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1815000*(4*(9801*sqrt(5)*(5*x + 3) - 385886*sqrt(5))
*(5*x + 3) + 6360321*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.02, size = 120, normalized size = 1.43 \begin {gather*} \frac {\left (2313036 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-784080 \sqrt {-10 x^{2}-x +3}\, x^{2}-2313036 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5233280 \sqrt {-10 x^{2}-x +3}\, x +578259 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1666020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{290400 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \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)^(1/2),x)

[Out]

1/290400*(2313036*10^(1/2)*x^2*arcsin(20/11*x+1/11)-2313036*10^(1/2)*x*arcsin(20/11*x+1/11)-784080*(-10*x^2-x+
3)^(1/2)*x^2+578259*10^(1/2)*arcsin(20/11*x+1/11)+5233280*(-10*x^2-x+3)^(1/2)*x-1666020*(-10*x^2-x+3)^(1/2))*(
5*x+3)^(1/2)*(-2*x+1)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.30, size = 76, normalized size = 0.90 \begin {gather*} \frac {1593}{800} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{40} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{132 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {11123 \, \sqrt {-10 \, x^{2} - x + 3}}{1452 \, {\left (2 \, x - 1\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)^(1/2),x, algorithm="maxima")

[Out]

1593/800*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 27/40*sqrt(-10*x^2 - x + 3) + 343/132*sqrt(-10*x^2 - x + 3)/
(4*x^2 - 4*x + 1) + 11123/1452*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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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 (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d 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)^(1/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, dx \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)**(1/2),x)

[Out]

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

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