3.2769 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac {43214 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{6075}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 \sqrt {3 x+2}}+\frac {788}{135} \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}-\frac {43214 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1215}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075} \]

[Out]

-2/9*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(3/2)+116854/18225*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/
2))*33^(1/2)-43214/18225*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+230/27*(1-2*x)^(3/2)*(
3+5*x)^(3/2)/(2+3*x)^(1/2)+788/135*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-43214/1215*(1-2*x)^(1/2)*(2+3*x)^
(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 \sqrt {3 x+2}}+\frac {788}{135} \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}-\frac {43214 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1215}-\frac {43214 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-43214*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1215 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^(3/
2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(27*Sqrt[2 + 3*x]) + (788*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3
/2))/135 + (116854*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6075 - (43214*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6075

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 150

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {\left (-210-\frac {2955 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {8 \int \frac {\left (\frac {48285}{4}-\frac {324105 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2025}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {8 \int \frac {-\frac {338655}{8}-\frac {876405 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{18225}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {116854 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6075}+\frac {237677 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6075}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}-\frac {43214 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 107, normalized size = 0.56 \[ \frac {829885 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {5 x+3} \left (1620 x^3-3906 x^2-23538 x-13231\right )}{(3 x+2)^{3/2}}-116854 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{18225} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-13231 - 23538*x - 3906*x^2 + 1620*x^3))/(2 + 3*x)^(3/2) - 116854*Sqrt[2]*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 829885*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2])/18225

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fricas [F]  time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(27*x^3 + 54*x^2 + 36*x + 8), x
)

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

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)^(5/2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.02, size = 229, normalized size = 1.20 \[ -\frac {\left (-486000 x^{5}+1123200 x^{4}+7324380 x^{3}+4323900 x^{2}-350562 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2489655 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1721490 x -233708 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1659770 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1190790\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{18225 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18225*(2489655*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)-350562*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)+1659770*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))-233708*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))-
486000*x^5+1123200*x^4+7324380*x^3+4323900*x^2-1721490*x-1190790)*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(10*x^2+x-3)/(3
*x+2)^(3/2)

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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