3.24.59 \(\int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}-\frac {21}{550} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (3660 x+8987)}{88000}+\frac {143283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 153, 147, 54, 216} \begin {gather*} -\frac {2 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}-\frac {21}{550} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (3660 x+8987)}{88000}+\frac {143283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*Sqrt[3 + 5*x]) - (21*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/550 - (21*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x]*(8987 + 3660*x))/88000 + (143283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000*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 153

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

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)^4}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-42-\frac {63 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {1}{825} \int \frac {(2+3 x) \left (\frac {6111}{2}+\frac {19215 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (8987+3660 x)}{88000}+\frac {143283 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16000}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (8987+3660 x)}{88000}+\frac {143283 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8000 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (8987+3660 x)}{88000}+\frac {143283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 87, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \left (237600 x^3+849420 x^2+1477575 x+632101\right )+1576113 \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{880000 \sqrt {2 x-1} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]

[Out]

-1/880000*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*(632101 + 1477575*x + 849420*x^2 + 237600*x^3) + 1576113*Sqrt[30 +
50*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]]))/(Sqrt[-1 + 2*x]*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.17, size = 125, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {1-2 x} \left (\frac {3200 (1-2 x)^3}{(5 x+3)^3}+\frac {39322185 (1-2 x)^2}{(5 x+3)^2}+\frac {41662320 (1-2 x)}{5 x+3}[phone]\right )}{88000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}-\frac {143283 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{8000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]

[Out]

-1/88000*(Sqrt[1 - 2*x]*(12903548 + (3200*(1 - 2*x)^3)/(3 + 5*x)^3 + (39322185*(1 - 2*x)^2)/(3 + 5*x)^2 + (416
62320*(1 - 2*x))/(3 + 5*x)))/(Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^3) - (143283*ArcTan[(Sqrt[5/2]*Sqrt[
1 - 2*x])/Sqrt[3 + 5*x]])/(8000*Sqrt[10])

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fricas [A]  time = 1.43, size = 86, normalized size = 0.76 \begin {gather*} -\frac {1576113 \, \sqrt {10} {\left (5 \, x + 3\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 (237600 \, x^{3} + 849420 \, x^{2} + 1477575 \, x + 632101\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1760000 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1760000*(1576113*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) + 20*(237600*x^3 + 849420*x^2 + 1477575*x + 632101)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.32, size = 124, normalized size = 1.10 \begin {gather*} -\frac {27}{200000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 71 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2407 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {143283}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{68750 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{34375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-27/200000*(4*(8*sqrt(5)*(5*x + 3) + 71*sqrt(5))*(5*x + 3) + 2407*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 143
283/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/68750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
/sqrt(5*x + 3) + 2/34375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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maple [A]  time = 0.01, size = 116, normalized size = 1.03 \begin {gather*} \frac {\left (-4752000 \sqrt {-10 x^{2}-x +3}\, x^{3}-16988400 \sqrt {-10 x^{2}-x +3}\, x^{2}+7880565 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-29551500 \sqrt {-10 x^{2}-x +3}\, x +4728339 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-12642020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{1760000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^4/(5*x+3)^(3/2)/(-2*x+1)^(1/2),x)

[Out]

1/1760000*(-4752000*(-10*x^2-x+3)^(1/2)*x^3+7880565*10^(1/2)*x*arcsin(20/11*x+1/11)-16988400*(-10*x^2-x+3)^(1/
2)*x^2+4728339*10^(1/2)*arcsin(20/11*x+1/11)-29551500*(-10*x^2-x+3)^(1/2)*x-12642020*(-10*x^2-x+3)^(1/2))*(-2*
x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [A]  time = 1.37, size = 82, normalized size = 0.73 \begin {gather*} -\frac {27}{50} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {143283}{160000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {3213}{2000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {95769}{40000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{6875 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-27/50*sqrt(-10*x^2 - x + 3)*x^2 + 143283/160000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 3213/2000*sqrt(-10*x
^2 - x + 3)*x - 95769/40000*sqrt(-10*x^2 - x + 3) - 2/6875*sqrt(-10*x^2 - x + 3)/(5*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 )}^4}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^4/((1 - 2*x)^(1/2)*(5*x + 3)^(3/2)),x)

[Out]

int((3*x + 2)^4/((1 - 2*x)^(1/2)*(5*x + 3)^(3/2)), 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)**4/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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