3.2469 \(\int \frac{(2+3 x)^2 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3}{40} \sqrt{1-2 x} (3 x+2) (5 x+3)^{5/2}-\frac{251}{800} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{14529 \sqrt{1-2 x} (5 x+3)^{3/2}}{6400}-\frac{479457 \sqrt{1-2 x} \sqrt{5 x+3}}{25600}+\frac{5274027 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

[Out]

(-479457*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 - (14529*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/6400 - (251*Sqrt[1 - 2*x]*
(3 + 5*x)^(5/2))/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(5/2))/40 + (5274027*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(25600*Sqrt[10])

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Rubi [A]  time = 0.0294783, antiderivative size = 121, normalized size of antiderivative = 1., 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 = {90, 80, 50, 54, 216} \[ -\frac{3}{40} \sqrt{1-2 x} (3 x+2) (5 x+3)^{5/2}-\frac{251}{800} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{14529 \sqrt{1-2 x} (5 x+3)^{3/2}}{6400}-\frac{479457 \sqrt{1-2 x} \sqrt{5 x+3}}{25600}+\frac{5274027 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-479457*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 - (14529*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/6400 - (251*Sqrt[1 - 2*x]*
(3 + 5*x)^(5/2))/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(5/2))/40 + (5274027*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(25600*Sqrt[10])

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 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 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 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 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx &=-\frac{3}{40} \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}-\frac{1}{40} \int \frac{\left (-244-\frac{753 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{251}{800} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{3}{40} \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac{14529 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{1600}\\ &=-\frac{14529 \sqrt{1-2 x} (3+5 x)^{3/2}}{6400}-\frac{251}{800} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{3}{40} \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac{479457 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{12800}\\ &=-\frac{479457 \sqrt{1-2 x} \sqrt{3+5 x}}{25600}-\frac{14529 \sqrt{1-2 x} (3+5 x)^{3/2}}{6400}-\frac{251}{800} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{3}{40} \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac{5274027 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{51200}\\ &=-\frac{479457 \sqrt{1-2 x} \sqrt{3+5 x}}{25600}-\frac{14529 \sqrt{1-2 x} (3+5 x)^{3/2}}{6400}-\frac{251}{800} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{3}{40} \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac{5274027 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25600 \sqrt{5}}\\ &=-\frac{479457 \sqrt{1-2 x} \sqrt{3+5 x}}{25600}-\frac{14529 \sqrt{1-2 x} (3+5 x)^{3/2}}{6400}-\frac{251}{800} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{3}{40} \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}+\frac{5274027 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{25600 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0387491, size = 65, normalized size = 0.54 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (144000 x^3+469600 x^2+698580 x+760653\right )-5274027 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{256000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(760653 + 698580*x + 469600*x^2 + 144000*x^3) - 5274027*Sqrt[10]*ArcSin[Sqrt[
5/11]*Sqrt[1 - 2*x]])/256000

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Maple [A]  time = 0.01, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{512000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -2880000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-9392000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+5274027\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -13971600\,x\sqrt{-10\,{x}^{2}-x+3}-15213060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-2880000*x^3*(-10*x^2-x+3)^(1/2)-9392000*x^2*(-10*x^2-x+3)^(1/2)+5274027
*10^(1/2)*arcsin(20/11*x+1/11)-13971600*x*(-10*x^2-x+3)^(1/2)-15213060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2
)

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Maxima [A]  time = 1.95388, size = 101, normalized size = 0.83 \begin{align*} -\frac{45}{8} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{587}{32} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{34929}{1280} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{5274027}{512000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{760653}{25600} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/8*sqrt(-10*x^2 - x + 3)*x^3 - 587/32*sqrt(-10*x^2 - x + 3)*x^2 - 34929/1280*sqrt(-10*x^2 - x + 3)*x - 5274
027/512000*sqrt(10)*arcsin(-20/11*x - 1/11) - 760653/25600*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.77015, size = 257, normalized size = 2.12 \begin{align*} -\frac{1}{25600} \,{\left (144000 \, x^{3} + 469600 \, x^{2} + 698580 \, x + 760653\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{5274027}{512000} \, \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25600*(144000*x^3 + 469600*x^2 + 698580*x + 760653)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 5274027/512000*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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Sympy [A]  time = 124.798, size = 398, normalized size = 3.29 \begin{align*} \frac{2 \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{12 \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} + \frac{18 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (\frac{2 \sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{7 \sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{3872} + \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*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 + 12*sqrt(5)*Piecew
ise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sqrt(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 + 18*sqrt(5)*Piecewise((14641*sqrt(2)*(2*sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993
+ 7*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 + sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 1
28*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(2
2)*sqrt(5*x + 3)/11)/128)/32, (x >= -3/5) & (x < 1/2)))/125

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Giac [A]  time = 1.29295, size = 85, normalized size = 0.7 \begin{align*} -\frac{1}{256000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (180 \, x + 371\right )}{\left (5 \, x + 3\right )} + 14529\right )}{\left (5 \, x + 3\right )} + 479457\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 5274027 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256000*sqrt(5)*(2*(4*(8*(180*x + 371)*(5*x + 3) + 14529)*(5*x + 3) + 479457)*sqrt(5*x + 3)*sqrt(-10*x + 5)
- 5274027*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))