3.2651 \(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac {124 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2205}+\frac {4636 \sqrt {1-2 x} \sqrt {5 x+3}}{2205 \sqrt {3 x+2}}+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{315 (3 x+2)^{3/2}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}-\frac {4636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2205} \]

[Out]

-4636/6615*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-124/6615*EllipticF(1/7*21^(1/2)*(1-2
*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/15*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+74/315*(1-2*x)^(1/2)*(3+5*x
)^(1/2)/(2+3*x)^(3/2)+4636/2205*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, 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 = {97, 152, 158, 113, 119} \[ \frac {4636 \sqrt {1-2 x} \sqrt {5 x+3}}{2205 \sqrt {3 x+2}}+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{315 (3 x+2)^{3/2}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}-\frac {124 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2205}-\frac {4636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2205} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (74*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(315*(2 + 3*x)^(3/2))
+ (4636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2205*Sqrt[2 + 3*x]) - (4636*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/2205 - (124*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2205

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 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)]))

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{7/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {-\frac {1}{2}-10 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{3/2}}+\frac {4}{315} \int \frac {\frac {263}{2}-\frac {185 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{3/2}}+\frac {4636 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 \sqrt {2+3 x}}+\frac {8 \int \frac {\frac {7295}{4}+\frac {5795 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2205}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{3/2}}+\frac {4636 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 \sqrt {2+3 x}}+\frac {682 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2205}+\frac {4636 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2205}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{3/2}}+\frac {4636 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 \sqrt {2+3 x}}-\frac {4636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2205}-\frac {124 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2205}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 99, normalized size = 0.62 \[ \frac {2 \left (\sqrt {2} \left (2318 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-1295 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} \left (20862 x^2+28593 x+9643\right )}{(3 x+2)^{5/2}}\right )}{6615} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9643 + 28593*x + 20862*x^2))/(2 + 3*x)^(5/2) + Sqrt[2]*(2318*EllipticE[Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1295*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/6615

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fricas [F]  time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.04, size = 314, normalized size = 1.96 \[ \frac {2 \left (625860 x^{4}+920376 x^{3}-20862 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+11655 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+187311 x^{2}-27816 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+15540 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-228408 x -9272 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5180 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-86787\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{6615 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6615*(11655*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)
^(1/2)-20862*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^
(1/2)+15540*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/
2)-27816*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)+
5180*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-9272*2
^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))+625860*x^4+9
20376*x^3+187311*x^2-228408*x-86787)*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(10*x^2+x-3)/(3*x+2)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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