3.2969 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=220 \[ -\frac{7738 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{84035}+\frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{4437 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{99 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]

[Out]

(99*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (1432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*x)^(5/
2)) - (4437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2 + 3*x)^(3/2)) - (27618*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*
Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (9206*Sqrt[33]*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035 - (7738*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8
4035

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Rubi [A]  time = 0.0812855, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{4437 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{99 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(99*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (1432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 + 3*x)^(5/
2)) - (4437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2 + 3*x)^(3/2)) - (27618*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*
Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (9206*Sqrt[33]*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035 - (7738*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8
4035

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 152

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{1}{21} \int \frac{\left (-\frac{147}{2}-150 x\right ) \sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{1}{147} \int \frac{-4959-\frac{17025 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{2 \int \frac{-\frac{72609}{4}-32220 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{5145}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{4 \int \frac{-\frac{91683}{4}-\frac{199665 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{108045}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{8 \int \frac{\frac{362655}{8}+\frac{621405 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{756315}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{27618 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{84035}+\frac{42559 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{84035}\\ &=\frac{99 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{1432 \sqrt{1-2 x} \sqrt{3+5 x}}{1715 (2+3 x)^{5/2}}-\frac{4437 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{3/2}}-\frac{27618 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}\\ \end{align*}

Mathematica [A]  time = 0.180134, size = 110, normalized size = 0.5 \[ \frac{3 \sqrt{2} \left (51765 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-9206 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (1491372 x^4+1056186 x^3-718167 x^2-640441 x-88623\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}}{252105} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[3 + 5*x]*(-88623 - 640441*x - 718167*x^2 + 1056186*x^3 + 1491372*x^4))/((1 - 2*x)^(3/2)*(2 + 3*x)^(5
/2)) + 3*Sqrt[2]*(-9206*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 51765*EllipticF[ArcSin[Sqrt[2/11]
*Sqrt[3 + 5*x]], -33/2]))/252105

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Maple [C]  time = 0.025, size = 406, normalized size = 1.9 \begin{align*} -{\frac{1}{252105\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 2795310\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-497124\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2329425\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-414270\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-621180\,\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}+110472\,\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}-621180\,\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 ) +110472\,\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}-844554\,{x}^{3}-10713412\,{x}^{2}-4728876\,x-531738 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/252105*(1-2*x)^(1/2)*(2795310*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3+5*x)^(1/2)*(2+
3*x)^(1/2)*(1-2*x)^(1/2)-497124*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3+5*x)^(1/2)*(2+3
*x)^(1/2)*(1-2*x)^(1/2)+2329425*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(2+3
*x)^(1/2)*(1-2*x)^(1/2)-414270*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*
x)^(1/2)*(1-2*x)^(1/2)-621180*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)+110472*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)-621180*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))+110472*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+19510092*x^4-844554*x^3-10713412*x^2-4728876*x-531738)/(2+3*x)^(5/2)/(2*x-1)^2/(3+5*x)^(1/
2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{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((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*(-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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{648 \, x^{7} + 756 \, x^{6} - 378 \, x^{5} - 609 \, x^{4} + 56 \, x^{3} + 168 \, x^{2} - 16}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(648*x^7 + 756*x^6 - 378*x^5 - 609*x^
4 + 56*x^3 + 168*x^2 - 16), 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((3+5*x)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**(7/2),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{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{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((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="giac")

[Out]

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