3.2981 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{160297 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{30250 \sqrt{33}}+\frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{665 (3 x+2)^{5/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3284 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 \sqrt{5 x+3}}-\frac{153319 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{66550}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}} \]

[Out]

(3284*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*Sqrt[3 + 5*x]) - (665*(2 + 3*x)^(5/2))/(363*Sqrt[1 - 2*x]*Sqrt[3 +
 5*x]) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (153319*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
 5*x])/66550 - (5327983*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33]) - (160297*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33])

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Rubi [A]  time = 0.0676765, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{665 (3 x+2)^{5/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3284 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 \sqrt{5 x+3}}-\frac{153319 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{66550}-\frac{160297 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

(3284*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*Sqrt[3 + 5*x]) - (665*(2 + 3*x)^(5/2))/(363*Sqrt[1 - 2*x]*Sqrt[3 +
 5*x]) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (153319*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
 5*x])/66550 - (5327983*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33]) - (160297*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(30250*Sqrt[33])

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 150

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

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{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x)^{5/2} \left (\frac{359}{2}+306 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{363} \int \frac{\left (-\frac{23357}{2}-\frac{40023 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{2 \int \frac{\left (-\frac{425475}{2}-\frac{1379871 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{19965}\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{153319 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{66550}+\frac{2 \int \frac{\frac{60716097}{8}+\frac{47951847 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{299475}\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{153319 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{66550}+\frac{160297 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{60500}+\frac{5327983 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{332750}\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{153319 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{66550}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}-\frac{160297 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.257475, size = 102, normalized size = 0.55 \[ \frac{-5366165 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{5 \sqrt{6 x+4} \left (1078110 x^3-11321446 x^2-3117099 x+2438391\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}+10655966 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{998250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

((-5*Sqrt[4 + 6*x]*(2438391 - 3117099*x - 11321446*x^2 + 1078110*x^3))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 10655
966*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5366165*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2])/(998250*Sqrt[2])

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Maple [C]  time = 0.023, size = 238, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( 3993000\,x-1996500 \right ) \left ( 30\,{x}^{3}+23\,{x}^{2}-7\,x-6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 10732330\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-21311932\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5366165\,\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 ) +10655966\,\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 ) -32343300\,{x}^{4}+318081180\,{x}^{3}+319941890\,{x}^{2}-10809750\,x-48767820 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x)

[Out]

1/1996500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10732330*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66
^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-21311932*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^
(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-5366165*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*E
llipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))+10655966*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elli
pticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-32343300*x^4+318081180*x^3+319941890*x^2-10809750*x-48767820)/(2*x
-1)/(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 (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

integral(-(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(200*x^5 - 60*x^
4 - 138*x^3 + 47*x^2 + 24*x - 9), 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((2+3*x)**(9/2)/(1-2*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)), x)