3.2682 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{38723 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{48}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{183 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}-\frac{2486 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{203179 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (2486*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/21875
+ (183*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/4375 + (48*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/17
5 - (203179*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/218750 - (38723*EllipticF[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(109375*Sqrt[33])

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Rubi [A]  time = 0.0664199, antiderivative size = 189, 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 = {97, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{48}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{183 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}-\frac{2486 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{38723 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{203179 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (2486*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/21875
+ (183*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/4375 + (48*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/17
5 - (203179*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/218750 - (38723*EllipticF[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(109375*Sqrt[33])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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{\sqrt{1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{\left (\frac{17}{2}-24 x\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}+\frac{48}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{2}{175} \int \frac{(2+3 x)^{3/2} \left (-79+\frac{183 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}+\frac{183 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}+\frac{48}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{2 \int \frac{\sqrt{2+3 x} \left (\frac{11225}{4}+3729 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4375}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{2486 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{21875}+\frac{183 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}+\frac{48}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{2 \int \frac{-97239-\frac{609537 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{65625}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{2486 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{21875}+\frac{183 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}+\frac{48}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{38723 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{218750}+\frac{203179 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{218750}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{2486 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{21875}+\frac{183 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}+\frac{48}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{203179 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750}-\frac{38723 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.291267, size = 107, normalized size = 0.57 \[ \frac{-87010 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (33750 x^3+63225 x^2+25955 x+32\right )}{\sqrt{5 x+3}}+203179 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{656250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(32 + 25955*x + 63225*x^2 + 33750*x^3))/Sqrt[3 + 5*x] + 203179*Sqrt[2]*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 87010*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2
])/656250

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Maple [C]  time = 0.033, size = 150, normalized size = 0.8 \begin{align*}{\frac{1}{19687500\,{x}^{3}[phone]\,{x}^{2}-4593750\,x-3937500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 87010\,\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 ) -203179\,\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 ) +6075000\,{x}^{5}[phone]\,{x}^{4}+4543650\,{x}^{3}-3009090\,{x}^{2}-1556340\,x-1920 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/656250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(87010*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-203179*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))+6075000*x^5+12393000*x^4+4543650*x^3-3009090*x^2-1556340*x-1920)/(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{-2 \, x + 1}}{{\left (5 \, x + 3\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)^(1/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(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}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(7/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2),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)/(25*x^2 + 30*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)**(7/2)*(1-2*x)**(1/2)/(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{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\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)^(1/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

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