3.2767 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{429479 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3189375}+\frac{2}{27} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{362 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2835}+\frac{14318 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{70875}-\frac{429479 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{637875}-\frac{4457606 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375} \]

[Out]

(-429479*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/637875 + (14318*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
))/70875 + (362*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2835 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*
x)^(3/2))/27 - (4457606*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375 - (429479*Sqrt[1
1/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375

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Rubi [A]  time = 0.0658865, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{27} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{362 \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2835}+\frac{14318 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{70875}-\frac{429479 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{637875}-\frac{429479 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375}-\frac{4457606 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-429479*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/637875 + (14318*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
))/70875 + (362*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2835 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*
x)^(3/2))/27 - (4457606*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375 - (429479*Sqrt[1
1/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3189375

Rule 101

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

Rule 154

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] && IntegersQ[2*m, 2
*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx &=\frac{2}{27} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2}{27} \int \frac{\left (-51-\frac{181 x}{2}\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\\ &=\frac{362 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{2}{27} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{4 \int \frac{\left (-\frac{10167}{4}-\frac{21477 x}{4}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx}{2835}\\ &=\frac{14318 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{70875}+\frac{362 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{2}{27} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{8 \int \frac{\left (-\frac{91323}{4}-\frac{1288437 x}{8}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{212625}\\ &=-\frac{429479 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{637875}+\frac{14318 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{70875}+\frac{362 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{2}{27} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{8 \int \frac{\frac{18881943}{16}+\frac{6686409 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1913625}\\ &=-\frac{429479 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{637875}+\frac{14318 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{70875}+\frac{362 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{2}{27} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{4724269 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6378750}+\frac{4457606 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3189375}\\ &=-\frac{429479 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{637875}+\frac{14318 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{70875}+\frac{362 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2}}{2835}+\frac{2}{27} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{4457606 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375}-\frac{429479 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3189375}\\ \end{align*}

Mathematica [A]  time = 0.148035, size = 102, normalized size = 0.53 \[ \frac{5257595 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (945000 x^3-1192500 x^2+232110 x+343207\right )+8915212 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{9568125 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(343207 + 232110*x - 1192500*x^2 + 945000*x^3) + 8915212*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 5257595*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(956
8125*Sqrt[2])

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Maple [C]  time = 0.012, size = 155, normalized size = 0.8 \begin{align*} -{\frac{1}{574087500\,{x}^{3}[phone]\,{x}^{2}-133953750\,x-114817500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -850500000\,{x}^{6}+5257595\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +8915212\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-549367200\,{x}^{3}-402719730\,{x}^{2}[phone]\,x+61777260 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/19136250*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-850500000*x^6+5257595*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1
/2)*(1-2*x)^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))+8915212*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*
(1-2*x)^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))[phone]*x^5+812376000*x^4-549367200*x^3-402719
730*x^2+113853270*x+61777260)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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