3.2787 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac {509189 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8859375}+\frac {2}{45} (3 x+2)^{3/2} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {106 (3 x+2)^{3/2} \sqrt {5 x+3} (1-2 x)^{3/2}}{1575}+\frac {8878 (3 x+2)^{3/2} \sqrt {5 x+3} \sqrt {1-2 x}}{118125}+\frac {21547 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1771875}-\frac {8024546 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375} \]

[Out]

-8024546/26578125*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-509189/26578125*EllipticF(1/7
*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+106/1575*(1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)+2/45*(1-2
*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)+8878/118125*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+21547/1771875*(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 = 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 = {101, 154, 158, 113, 119} \[ \frac {2}{45} (3 x+2)^{3/2} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {106 (3 x+2)^{3/2} \sqrt {5 x+3} (1-2 x)^{3/2}}{1575}+\frac {8878 (3 x+2)^{3/2} \sqrt {5 x+3} \sqrt {1-2 x}}{118125}+\frac {21547 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1771875}-\frac {509189 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375}-\frac {8024546 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21547*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1771875 + (8878*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])
/118125 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/1575 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*Sqrt[3
+ 5*x])/45 - (8024546*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8859375 - (509189*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8859375

Rule 101

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 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 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]

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx &=\frac {2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {2}{45} \int \frac {\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} \sqrt {3+5 x}}{1575}+\frac {2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {4 \int \frac {\left (-\frac {5853}{2}-\frac {13317 x}{4}\right ) \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx}{4725}\\ &=\frac {8878 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{1575}+\frac {2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {8 \int \frac {\sqrt {2+3 x} \left (-\frac {545025}{8}+\frac {64641 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{354375}\\ &=\frac {21547 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1771875}+\frac {8878 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{1575}+\frac {2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {8 \int \frac {\frac {32249013}{16}+\frac {12036819 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5315625}\\ &=\frac {21547 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1771875}+\frac {8878 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{1575}+\frac {2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {5601079 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{17718750}+\frac {8024546 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{8859375}\\ &=\frac {21547 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1771875}+\frac {8878 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{1575}+\frac {2}{45} (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {8024546 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375}-\frac {509189 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 102, normalized size = 0.53 \[ \frac {754145 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (945000 x^3-1030500 x^2-113490 x+683887\right )[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{26578125 \sqrt {2}} \]

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]*(683887 - 113490*x - 1030500*x^2 + 945000*x^3) + 16049092*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 754145*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(265
78125*Sqrt[2])

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fricas [F]  time = 1.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}, x\right ) \]

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]

integral((12*x^3 - 4*x^2 - 5*x + 2)*sqrt(3*x + 2)*sqrt(-2*x + 1)/sqrt(5*x + 3), x)

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

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

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maple [C]  time = 0.01, size = 155, normalized size = 0.81 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-850500000 x^{6}[phone] x^{5}[phone] x^{4}-583495200 x^{3}-681204930 x^{2}[phone] x [phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+754145 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone]\right )}{53156250 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/53156250*(-2*x+1)^(1/2)*(3*x+2)^(1/2)*(5*x+3)^(1/2)*(-850500000*x^6+754145*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))[phone]*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))[phone]*x^5+1011636000*x^4-583495200*x^3-68
1204930*x^2+123188070*x+123099660)/(30*x^3+23*x^2-7*x-6)

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

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

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

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

[Out]

Timed out

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