3.28.17 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\) [2717]

Optimal. Leaf size=249 \[ -\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {1446357824 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}-\frac {43537016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}} \]

[Out]

-2/33*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2)-1446357824/224625555*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/3
3*1155^(1/2))*33^(1/2)-43537016/224625555*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+74/29
7*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-12872/43659*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+442076/15280
65*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/10696455*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+14463
57824/74875185*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {43537016 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}-\frac {1446357824 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 \sqrt {3 x+2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac {12872 \sqrt {1-2 x} \sqrt {5 x+3}}{43659 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(13/2),x]

[Out]

(-12872*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(7/2)) + (442076*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1528065*(
2 + 3*x)^(5/2)) + (20799916*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10696455*(2 + 3*x)^(3/2)) + (1446357824*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(74875185*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) + (74*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2))/(297*(2 + 3*x)^(9/2)) - (1446357824*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(6806835*Sqrt[33]) - (43537016*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6806835*Sqrt[33])

Rule 99

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 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 155

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 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {3+5 x} \left (-864+\frac {2235 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {67269}{4}+\frac {64875 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{130977}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {8968797}{8}+\frac {4973355 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{4584195}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {193192407}{4}+\frac {233999055 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{96268095}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {64 \int \frac {-\frac {10301685885}{16}-1016970345 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{673876665}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}+\frac {21768508 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6806835}+\frac {1446357824 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{74875185}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {1446357824 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}-\frac {43537016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 7.99, size = 112, normalized size = 0.45 \begin {gather*} \frac {\frac {24 \sqrt {2-4 x} \sqrt {3+5 x} \left (24398176891+180988667568 x+537061687749 x^2+797050394730 x^3+591671694906 x^4+175732475616 x^5\right )}{(2+3 x)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5823976480 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{898502220 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(13/2),x]

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(24398176891 + 180988667568*x + 537061687749*x^2 + 797050394730*x^3 + 5916716
94906*x^4 + 175732475616*x^5))/(2 + 3*x)^(11/2) + 11570862592*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2] - 5823976480*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(898502220*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(185)=370\).
time = 0.10, size = 587, normalized size = 2.36

method	result	size
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {296 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{5}}+\frac {8198 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3536379 \left (\frac {2}{3}+x \right )^{4}}+\frac {442076 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{41257755 \left (\frac {2}{3}+x \right )^{3}}+\frac {20799916 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96268095 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2892715648}{14975037} x^{2}-\frac {1446357824}{74875185} x +\frac {1446357824}{24958395}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {915705412 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{314475777 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1446357824 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{314475777 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(332\)
default	\(\frac {2 \left (175732475616 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-87280832826 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-290936109420 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-387914812560 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-258609875040 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{7}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-86203291680 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{6}[phone] \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 )-11493772224 \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 )[phone] x^{5}[phone] x^{4}-132608462283 x^{3}-3558643880307 x^{2}-1555703477439 x -219583592019\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{224625555 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\)	\(587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

2/224625555*(175732475616*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(-3-5*x)^(1/2)
*(1-2*x)^(1/2)-87280832826*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(-3-5*x)^(1/2
)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1
/2)*(1-2*x)^(1/2)-290936109420*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^
(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x
)^(1/2)*(1-2*x)^(1/2)-387914812560*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5
*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3
-5*x)^(1/2)*(1-2*x)^(1/2)-258609875040*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(
-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^7+173562938880*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))
*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-86203291680*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*
x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^6+23141725184*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))-11493772224*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))[phone]*x^5+13177956562506*x^4-132608462283*x
^3-3558643880307*x^2-1555703477439*x-219583592019)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(11/2)

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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)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(13/2), x)

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Fricas [A]
time = 0.22, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (175732475616 \, x^{5} + 591671694906 \, x^{4} + 797050394730 \, x^{3} + 537061687749 \, x^{2} + 180988667568 \, x + 24398176891\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{74875185 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

2/74875185*(175732475616*x^5 + 591671694906*x^4 + 797050394730*x^3 + 537061687749*x^2 + 180988667568*x + 24398
176891)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*
x + 64)

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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)*(3+5*x)**(3/2)/(2+3*x)**(13/2),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)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(13/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(13/2),x)

[Out]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(13/2), x)

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