3.27.100 \(\int (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx\) [2700]

Optimal. Leaf size=191 \[ -\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {5684677 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750}-\frac {84134 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875} \]

[Out]

2/45*(1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2)-5684677/10631250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155
^(1/2))*33^(1/2)-84134/5315625*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+62/1575*(2+3*x)^
(3/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)-347/39375*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-84134/354375*(1-2*x)^(1/
2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 191, normalized size of antiderivative = 1.00, 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 = {103, 159, 164, 114, 120} \begin {gather*} -\frac {84134 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875}-\frac {5684677 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750}+\frac {2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac {347 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{39375}-\frac {84134 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{354375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-84134*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/354375 - (347*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/
39375 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/1575 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)
^(3/2))/45 - (5684677*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3543750 - (84134*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1771875

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-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 159

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 164

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 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx &=\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {2}{45} \int \left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \, dx\\ &=\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {4 \int \frac {\left (-765-\frac {1041 x}{4}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{4725}\\ &=-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {4 \int \frac {\sqrt {3+5 x} \left (\frac {334107}{8}+\frac {126201 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{118125}\\ &=-\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {4 \int \frac {-\frac {10787703}{8}-\frac {17054031 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1063125}\\ &=-\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {462737 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1771875}+\frac {5684677 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3543750}\\ &=-\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {5684677 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750}-\frac {84134 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875}\\ \end {align*}

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Mathematica [A]
time = 5.82, size = 105, normalized size = 0.55 \begin {gather*} \frac {5684677 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (3 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-84697-359685 x+153000 x^2+472500 x^3\right )+581651 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{5315625 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5684677*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(
-84697 - 359685*x + 153000*x^2 + 472500*x^3) + 581651*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(53
15625*Sqrt[2])

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Maple [A]
time = 0.10, size = 153, normalized size = 0.80

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (425250000 x^{6}+2776422 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-5684677 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}-317371500 x^{4}-441589950 x^{3}-10447080 x^{2}[phone] x [phone]\right )}{10631250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(153\)
risch	\(\frac {\left (472500 x^{3}+153000 x^{2}-359685 x -84697\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{354375 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {3595901 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{38981250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5684677 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{38981250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(256\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3}-\frac {136 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{315}+\frac {7993 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7875}+\frac {84697 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{354375}+\frac {3595901 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{14883750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5684677 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14883750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(256\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/10631250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(425250000*x^6+2776422*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1
/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-5684677*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))[phone]*x^5-317371500*x^4-441589950*x^3-10447080*x^2+
82529670*x+15245460)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.21, size = 38, normalized size = 0.20 \begin {gather*} -\frac {1}{354375} \, {\left (472500 \, x^{3} + 153000 \, x^{2} - 359685 \, x - 84697\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/354375*(472500*x^3 + 153000*x^2 - 359685*x - 84697)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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