3.28.55 \(\int (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx\) [2755]

Optimal. Leaf size=218 \[ -\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {326256461 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}-\frac {4738087 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}} \]

[Out]

106/2475*(1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2)+2/55*(1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2)-326256461/58
4718750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-4738087/292359375*EllipticF(1/7*21^(1/2
)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2866/86625*(2+3*x)^(3/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)+38729/2165625*(
3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-4738087/19490625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, 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 = {103, 159, 164, 114, 120} \begin {gather*} -\frac {4738087 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}}-\frac {326256461 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}+\frac {2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac {2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt {1-2 x}}{86625}+\frac {38729 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{2165625}-\frac {4738087 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{19490625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4738087*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (38729*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(
3/2))/2165625 + (2866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/86625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3
/2)*(3 + 5*x)^(3/2))/2475 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/55 - (326256461*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (4738087*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(8859375*Sqrt[33])

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)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx &=\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {2}{55} \int \left (-\frac {113}{2}-\frac {159 x}{2}\right ) (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x} \, dx\\ &=\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {4 \int \left (-2979-\frac {12897 x}{4}\right ) \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \, dx}{7425}\\ &=\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {8 \int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (-\frac {670815}{8}+\frac {348561 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{779625}\\ &=\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {57670353}{16}+\frac {42642783 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19490625}\\ &=-\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {8 \int \frac {-\frac {463899753}{4}-\frac {2936308149 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{175415625}\\ &=-\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {4738087 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{17718750}+\frac {326256461 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{194906250}\\ &=-\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {326256461 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}-\frac {4738087 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 6.56, size = 107, normalized size = 0.49 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (9437696+16294455 x-35750250 x^2-13702500 x^3+42525000 x^4\right )[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-169899590 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{292359375 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9437696 + 16294455*x - 35750250*x^2 - 13702500*x^3 + 42525000*x
^4) + 326256461*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 169899590*EllipticF[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2])/(292359375*Sqrt[2])

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Maple [A]
time = 0.10, size = 158, normalized size = 0.72

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-38272500000 x^{7}-17010000000 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-326256461 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}-29711102850 x^{3}-9525219690 x^{2}[phone] x [phone]\right )}{584718750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(158\)
risch	\(-\frac {\left (42525000 x^{4}-13702500 x^{3}-35750250 x^{2}[phone] x +9437696\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{19490625 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {103088834 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{1071984375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {326256461 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2143968750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(262\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {24 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {116 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{165}-\frac {31778 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{17325}+\frac {362099 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{433125}+\frac {9437696 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19490625}+\frac {103088834 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{409303125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {326256461 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{818606250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/584718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-38272500000*x^7-17010000000*x^6+156356871*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))-326256461*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+14779638000*x^4
-29711102850*x^3-9525219690*x^2+4914918060*x+1698785280)/(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)^(5/2)*(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)^(5/2), x)

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Fricas [A]
time = 0.21, size = 43, normalized size = 0.20 \begin {gather*} \frac {1}{19490625} \, {\left (42525000 \, x^{4} - 13702500 \, x^{3} - 35750250 \, x^{2} + 16294455 \, x + 9437696\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)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/19490625*(42525000*x^4 - 13702500*x^3 - 35750250*x^2 + 16294455*x + 9437696)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqr
t(-2*x + 1)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3060 deep

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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)^(5/2)*(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)^(5/2), x)

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

[Out]

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

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