3.2899 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=219 \[ \frac{6770629 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{157500}+\frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{1397}{210} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{24358}{875} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{6770629 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{31500}+\frac{112543103 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{78750} \]

[Out]

(6770629*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/31500 + (24358*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)
)/875 + (1397*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/210 + (5*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^
(3/2))/3 + ((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (112543103*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/78750 + (6770629*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/157500

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Rubi [A]  time = 0.0721299, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{1397}{210} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{24358}{875} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{6770629 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{31500}+\frac{6770629 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{157500}+\frac{112543103 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{78750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6770629*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/31500 + (24358*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)
)/875 + (1397*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/210 + (5*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^
(3/2))/3 + ((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (112543103*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/78750 + (6770629*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/157500

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{(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^{5/2} \sqrt{3+5 x} \left (\frac{93}{2}+75 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{1}{45} \int \frac{\left (-\frac{13095}{2}-\frac{20955 x}{2}\right ) (2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{1397}{210} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{\int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (\frac{2776275}{4}+1096110 x\right )}{\sqrt{1-2 x}} \, dx}{1575}\\ &=\frac{24358}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{1397}{210} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{\int \frac{\left (-\frac{99001845}{2}-\frac{304678305 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{39375}\\ &=\frac{6770629 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{31500}+\frac{24358}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{1397}{210} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{\int \frac{\frac{12824947395}{8}+\frac{5064439635 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{354375}\\ &=\frac{6770629 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{31500}+\frac{24358}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{1397}{210} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{74476919 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{315000}-\frac{112543103 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{78750}\\ &=\frac{6770629 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{31500}+\frac{24358}{875} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{1397}{210} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{5}{3} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{112543103 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{78750}+\frac{6770629 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{157500}\\ \end{align*}

Mathematica [A]  time = 0.223393, size = 120, normalized size = 0.55 \[ \frac{226741655 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^4+2002500 x^3+4128030 x^2+6609296 x-12044593\right )-450172412 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{945000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-12044593 + 6609296*x + 4128030*x^2 + 2002500*x^3 + 472500*x^4) - 450172412*
Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 226741655*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2])/(945000*Sqrt[1 - 2*x])

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Maple [C]  time = 0.018, size = 155, normalized size = 0.7 \begin{align*} -{\frac{1}{28350000\,{x}^{3}[phone]\,{x}^{2}-6615000\,x-5670000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -212625000\,{x}^{6}[phone]\,\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 ) -450172412\,\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 ) -1170450000\,{x}^{5}-3084088500\,{x}^{4}-5687610300\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+2168026740 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/945000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-212625000*x^6+226741655*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))-450172412*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))-1170450000*x^5-3084088500*x^4-5687610300*x^3+9
09722730*x^2+5675744730*x+2168026740)/(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 (3 \, x + 2\right )}^{\frac{7}{2}}}{{\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)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(3*x + 2)^(7/2)/(-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 (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(4*x^2 - 4*x +
1), 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)*(3+5*x)**(3/2)/(1-2*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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\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)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

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