3.28.66 \(\int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx\) [2766]

Optimal. Leaf size=218 \[ -\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}-\frac {12996374 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}} \]

[Out]

-829177897/1052493750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-12996374/526246875*Ellipt
icF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+326/7425*(1-2*x)^(3/2)*(3+5*x)^(5/2)*(2+3*x)^(1/2)+2/
55*(1-2*x)^(5/2)*(3+5*x)^(5/2)*(2+3*x)^(1/2)-78797/3898125*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+30362/779
625*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-12996374/35083125*(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 {12996374 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}}-\frac {829177897 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}}{7425}+\frac {30362 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{779625}-\frac {78797 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{3898125}-\frac {12996374 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{35083125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-12996374*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35083125 - (78797*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^
(3/2))/3898125 + (30362*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/779625 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*
x]*(3 + 5*x)^(5/2))/7425 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/55 - (829177897*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(31893750*Sqrt[33]) - (12996374*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/(15946875*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} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx &=\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{55} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-2809-\frac {15181 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{7425}\\ &=\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (-\frac {466273}{8}-\frac {236391 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{779625}\\ &=-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {48347127}{16}+\frac {19494561 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{11694375}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {1578296283}{16}-\frac {2487533691 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{105249375}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {6498187 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15946875}+\frac {829177897 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{350831250}\\ &=-\frac {12996374 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}-\frac {78797 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3898125}+\frac {30362 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{779625}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}}{7425}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {829177897 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{31893750 \sqrt {33}}-\frac {12996374 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15946875 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 6.49, size = 107, normalized size = 0.49 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (22517617+48272535 x-95024250 x^2-51502500 x^3+127575000 x^4\right )[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-400297555 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{526246875 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(22517617 + 48272535*x - 95024250*x^2 - 51502500*x^3 + 127575000
*x^4) + 829177897*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 400297555*EllipticF[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2])/(526246875*Sqrt[2])

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

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-114817500000 x^{7}-41674500000 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-829177897 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}-82799446950 x^{3}-22504288380 x^{2}[phone] x [phone]\right )}{1052493750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(158\)
risch	\(-\frac {\left (127575000 x^{4}-51502500 x^{3}-95024250 x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{35083125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {526098761 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3859143750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {829177897 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{3859143750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(262\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {40 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {436 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{297}-\frac {84466 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31185}+\frac {1072723 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{779625}+\frac {22517617 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35083125}+\frac {526098761 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1473491250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {829177897 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1473491250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1052493750*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-114817500000*x^7-41674500000*x^6+428880342*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))-829177897*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+34269426000*
x^4-82799446950*x^3-22504288380*x^2+13417755870*x+4053171060)/(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)*(3+5*x)^(3/2)*(2+3*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.28, size = 43, normalized size = 0.20 \begin {gather*} \frac {1}{35083125} \, {\left (127575000 \, x^{4} - 51502500 \, x^{3} - 95024250 \, x^{2} + 48272535 \, x + 22517617\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)*(3+5*x)^(3/2)*(2+3*x)^(1/2),x, algorithm="fricas")

[Out]

1/35083125*(127575000*x^4 - 51502500*x^3 - 95024250*x^2 + 48272535*x + 22517617)*sqrt(5*x + 3)*sqrt(3*x + 2)*s
qrt(-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)*(3+5*x)**(3/2)*(2+3*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)*(3+5*x)^(3/2)*(2+3*x)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(3*x + 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}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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