3.2918 \(\int \frac{(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{5057 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1250}+\frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}+\frac{312}{275} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{14517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{2750}+\frac{168123 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]

[Out]

(14517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2750 + (312*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/275
+ (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (168123*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/1250 + (5057*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1250

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Rubi [A]  time = 0.049833, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 154, 158, 113, 119} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}+\frac{312}{275} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{14517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{2750}+\frac{5057 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250}+\frac{168123 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(7/2)/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]

[Out]

(14517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2750 + (312*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/275
+ (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (168123*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/1250 + (5057*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1250

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 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)^{7/2}}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{1}{11} \int \frac{(2+3 x)^{3/2} \left (\frac{381}{2}+312 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{312}{275} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{1}{275} \int \frac{\left (-13425-\frac{43551 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{14517 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2750}+\frac{312}{275} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{\int \frac{\frac{1915857}{4}+\frac{1513107 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4125}\\ &=\frac{14517 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2750}+\frac{312}{275} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{15171 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2500}-\frac{504369 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{13750}\\ &=\frac{14517 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{2750}+\frac{312}{275} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{7 (2+3 x)^{5/2} \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{168123 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250}+\frac{5057 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1250}\\ \end{align*}

Mathematica [A]  time = 0.112899, size = 110, normalized size = 0.69 \[ \frac{169365 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-10 \sqrt{3 x+2} \sqrt{5 x+3} \left (2970 x^2+11154 x-27757\right )-336246 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{27500 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(7/2)/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]

[Out]

(-10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-27757 + 11154*x + 2970*x^2) - 336246*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]], -33/2] + 169365*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(27500*S
qrt[1 - 2*x])

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Maple [C]  time = 0.018, size = 145, normalized size = 0.9 \begin{align*} -{\frac{1}{825000\,{x}^{3}+632500\,{x}^{2}-192500\,x-165000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 169365\,\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 ) -336246\,\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 ) -445500\,{x}^{4}-2237400\,{x}^{3}+1866090\,{x}^{2}+4604590\,x+1665420 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(7/2)/(1-2*x)^(3/2)/(3+5*x)^(1/2),x)

[Out]

-1/27500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(169365*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))-336246*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-445500*x^4-2237400*x^3+1866090*x^2+4604590*x+1665420)/(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{7}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(7/2)/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(7/2)/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(20*x^3 - 8*x^2 - 7*x + 3), 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)**(7/2)/(1-2*x)**(3/2)/(3+5*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 (3 \, x + 2\right )}^{\frac{7}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(7/2)/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(7/2)/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)), x)