3.2859 \(\int \frac{(2+3 x)^{7/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{178879 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{43750 \sqrt{33}}-\frac{3}{35} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{333}{875} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{15553 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{8750}-\frac{270248 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{21875} \]

[Out]

(-15553*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/8750 - (333*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/87
5 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/35 - (270248*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/21875 - (178879*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(43750*Sqrt[33])

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Rubi [A]  time = 0.056494, antiderivative size = 158, 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 = {102, 154, 158, 113, 119} \[ -\frac{3}{35} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{333}{875} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{15553 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{8750}-\frac{178879 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{43750 \sqrt{33}}-\frac{270248 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{21875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15553*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/8750 - (333*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/87
5 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/35 - (270248*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/21875 - (178879*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(43750*Sqrt[33])

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 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)^{7/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\frac{3}{35} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{1}{35} \int \frac{\left (-\frac{409}{2}-333 x\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{333}{875} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{3}{35} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{1}{875} \int \frac{\sqrt{2+3 x} \left (\frac{28775}{2}+\frac{46659 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{15553 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{8750}-\frac{333}{875} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{3}{35} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{\int \frac{-\frac{2053113}{4}-810744 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{13125}\\ &=-\frac{15553 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{8750}-\frac{333}{875} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{3}{35} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{178879 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{87500}+\frac{270248 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{21875}\\ &=-\frac{15553 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{8750}-\frac{333}{875} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{3}{35} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{270248 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{21875}-\frac{178879 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{43750 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.228817, size = 97, normalized size = 0.61 \[ \frac{-544355 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (6750 x^2+18990 x+25213\right )+1080992 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{131250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(25213 + 18990*x + 6750*x^2) + 1080992*EllipticE[ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]], -33/2] - 544355*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(131250*Sqrt[2])

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Maple [C]  time = 0.014, size = 150, normalized size = 1. \begin{align*}{\frac{1}{7875000\,{x}^{3}+6037500\,{x}^{2}-1837500\,x-1575000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 544355\,\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 ) -1080992\,\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}-21748500\,{x}^{4}-34377300\,{x}^{3}-12194070\,{x}^{2}+8712930\,x+4538340 \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)^(1/2),x)

[Out]

1/262500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(544355*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))-1080992*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-6075000*x^5-21748500*x^4-34377300*x^3-12194070*x^2+8712930*x+453834
0)/(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} \sqrt{-2 \, x + 1}}\,{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)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)/(sqrt(5*x + 3)*sqrt(-2*x + 1)), 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}}{10 \, x^{2} + 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)^(1/2)/(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)/(10*x^2 + 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)**(1/2)/(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} \sqrt{-2 \, x + 1}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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