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

Optimal. Leaf size=99 \[ -\frac {1}{10} \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}-\frac {181}{400} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {6269 \sqrt {1-2 x} \sqrt {5 x+3}}{1600}+\frac {68959 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]

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Rubi [A]  time = 0.02, antiderivative size = 99, 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 = {90, 80, 50, 54, 216} \begin {gather*} -\frac {1}{10} \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}-\frac {181}{400} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {6269 \sqrt {1-2 x} \sqrt {5 x+3}}{1600}+\frac {68959 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600 - (181*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/400 - (Sqrt[1 - 2*x]*(2 + 3*x)*
(3 + 5*x)^(3/2))/10 + (68959*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*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 80

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 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)^2 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}-\frac {1}{30} \int \frac {\left (-174-\frac {543 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {181}{400} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac {6269}{800} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {6269 \sqrt {1-2 x} \sqrt {3+5 x}}{1600}-\frac {181}{400} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac {68959 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200}\\ &=-\frac {6269 \sqrt {1-2 x} \sqrt {3+5 x}}{1600}-\frac {181}{400} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac {68959 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}}\\ &=-\frac {6269 \sqrt {1-2 x} \sqrt {3+5 x}}{1600}-\frac {181}{400} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}+\frac {68959 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 78, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (2400 x^2+6660 x+9401\right )+68959 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{16000 \sqrt {2 x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16000*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(9401 + 6660*x + 2400*x^2) + 68959*Sqrt[10]*ArcSinh[S
qrt[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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IntegrateAlgebraic [A]  time = 0.42, size = 115, normalized size = 1.16 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (-96 \sqrt {5} (5 x+3)^{5/2}-756 \sqrt {5} (5 x+3)^{3/2}-6269 \sqrt {5} \sqrt {5 x+3}\right )}{8000}-\frac {68959 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{800 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-6269*Sqrt[5]*Sqrt[3 + 5*x] - 756*Sqrt[5]*(3 + 5*x)^(3/2) - 96*Sqrt[5]*(3 + 5*x)^(5/2
)))/8000 - (68959*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(800*Sqrt[10])

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fricas [A]  time = 1.13, size = 67, normalized size = 0.68 \begin {gather*} -\frac {1}{1600} \, {\left (2400 \, x^{2} + 6660 \, x + 9401\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {68959}{32000} \, \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)^2*(3+5*x)^(1/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1600*(2400*x^2 + 6660*x + 9401)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 68959/32000*sqrt(10)*arctan(1/20*sqrt(10)*(2
0*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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giac [A]  time = 1.29, size = 54, normalized size = 0.55 \begin {gather*} -\frac {1}{16000} \, \sqrt {5} {\left (2 \, {\left (12 \, {\left (40 \, x + 87\right )} {\left (5 \, x + 3\right )} + 6269\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 68959 \, \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)^2*(3+5*x)^(1/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/16000*sqrt(5)*(2*(12*(40*x + 87)*(5*x + 3) + 6269)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 68959*sqrt(2)*arcsin(1/1
1*sqrt(22)*sqrt(5*x + 3)))

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maple [A]  time = 0.01, size = 87, normalized size = 0.88 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-48000 \sqrt {-10 x^{2}-x +3}\, x^{2}-133200 \sqrt {-10 x^{2}-x +3}\, x +68959 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-188020 \sqrt {-10 x^{2}-x +3}\right )}{32000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32000*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(-48000*(-10*x^2-x+3)^(1/2)*x^2+68959*10^(1/2)*arcsin(20/11*x+1/11)-13320
0*(-10*x^2-x+3)^(1/2)*x-188020*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.16, size = 58, normalized size = 0.59 \begin {gather*} \frac {68959}{32000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3}{20} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {321}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {10121}{1600} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

68959/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/20*(-10*x^2 - x + 3)^(3/2) - 321/80*sqrt(-10*x^2 - x +
3)*x - 10121/1600*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 7.45, size = 534, normalized size = 5.39 \begin {gather*} \frac {68959\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{8000}-\frac {\frac {30559\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {248379\,{\left (\sqrt {1-2\,x}-1\right )}^3}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {70541\,{\left (\sqrt {1-2\,x}-1\right )}^5}{31250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {70541\,{\left (\sqrt {1-2\,x}-1\right )}^7}{12500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {248379\,{\left (\sqrt {1-2\,x}-1\right )}^9}{10000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {30559\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{4000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {7168\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {95104\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32256\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {23776\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {48\,{\left (\sqrt {1-2\,x}-1\right )}^4}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^8}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {64}{15625}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(68959*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/8000 - ((30559*((1 - 2
*x)^(1/2) - 1))/(390625*(3^(1/2) - (5*x + 3)^(1/2))) - (248379*((1 - 2*x)^(1/2) - 1)^3)/(156250*(3^(1/2) - (5*
x + 3)^(1/2))^3) - (70541*((1 - 2*x)^(1/2) - 1)^5)/(31250*(3^(1/2) - (5*x + 3)^(1/2))^5) + (70541*((1 - 2*x)^(
1/2) - 1)^7)/(12500*(3^(1/2) - (5*x + 3)^(1/2))^7) + (248379*((1 - 2*x)^(1/2) - 1)^9)/(10000*(3^(1/2) - (5*x +
3)^(1/2))^9) - (30559*((1 - 2*x)^(1/2) - 1)^11)/(4000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (7168*3^(1/2)*((1 - 2
*x)^(1/2) - 1)^2)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (95104*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1
/2) - (5*x + 3)^(1/2))^4) + (32256*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^6) + (23
776*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (448*3^(1/2)*((1 - 2*x)^(1/2) - 1)^
10)/(25*(3^(1/2) - (5*x + 3)^(1/2))^10))/((192*((1 - 2*x)^(1/2) - 1)^2)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^2) +
(48*((1 - 2*x)^(1/2) - 1)^4)/(125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (32*((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2)
- (5*x + 3)^(1/2))^6) + (12*((1 - 2*x)^(1/2) - 1)^8)/(5*(3^(1/2) - (5*x + 3)^(1/2))^8) + (12*((1 - 2*x)^(1/2)
- 1)^10)/(5*(3^(1/2) - (5*x + 3)^(1/2))^10) + ((1 - 2*x)^(1/2) - 1)^12/(3^(1/2) - (5*x + 3)^(1/2))^12 + 64/156
25)

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sympy [A]  time = 19.36, size = 292, normalized size = 2.95 \begin {gather*} \frac {2 \sqrt {5} \left (\begin {cases} \frac {11 \sqrt {2} \left (- \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{2}\right )}{4} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{125} + \frac {12 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{125} + \frac {18 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + asin(sqrt(22)*sqrt(5*x + 3)/11)/2)
/4, (x >= -3/5) & (x < 1/2)))/125 + 12*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt
(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5)
& (x < 1/2)))/125 + 18*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sq
rt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22
)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/125

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