3.27.71 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\) [2671]

Optimal. Leaf size=249 \[ -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}} \]

[Out]

-1305025844/1572378885*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-37904696/1572378885*Elli
pticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-118/2079*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-
2/33*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)-13022/305613*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+627806/
10696455*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/74875185*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)
[phone]/524126295*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.07, 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 {37904696 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {1305025844 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {5 x+3}}{524126295 \sqrt {3 x+2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac {13022 \sqrt {1-2 x} \sqrt {5 x+3}}{305613 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13022*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(7/2)) + (627806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10696455
*(2 + 3*x)^(5/2)) + (19417096*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(74875185*(2 + 3*x)^(3/2)) + (1305025844*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(524126295*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2079*(2 + 3*x)^(9/2)) - (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (1305025844*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(47647845*Sqrt[33]) - (37904696*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {4 \int \frac {\left (-\frac {189}{4}-\frac {5025 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{6237}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {8 \int \frac {-\frac {676497}{8}-185700 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{916839}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {16 \int \frac {\frac {1286433}{2}-\frac {14125635 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{32089365}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {32 \int \frac {\frac {687512943}{16}-\frac {109221165 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{673876665}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {64 \int \frac {\frac {2319498765}{4}+\frac {14681540745 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4717136655}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {18952348 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{47647845}+\frac {1305025844 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{524126295}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 5.23, size = 112, normalized size = 0.45 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (21813966691+162787885893 x+484598540169 x^2+719808574005 x^3+534040213536 x^4+158560640046 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 )-10873573760 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{12579031080 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(21813966691 + 162787885893*x + 484598540169*x^2 + 719808574005*x^3 + 5340402
13536*x^4 + 158560640046*x^5))/(2 + 3*x)^(11/2) + 20880413504*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2] - 10873573760*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(12579031080*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	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {538 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1515591 \left (\frac {2}{3}+x \right )^{5}}-\frac {93382 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{24754653 \left (\frac {2}{3}+x \right )^{4}}+\frac {627806 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{288804285 \left (\frac {2}{3}+x \right )^{3}}+\frac {19417096 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{673876665 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2610051688}{104825259} x^{2}-\frac {1305025844}{524126295} x +\frac {1305025844}{174708765}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {824710672 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2201330439 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1305025844 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2201330439 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(321\)
default	\(-\frac {2 \left (75989439306 \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}-158560640046 \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}[phone] \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}-528535466820 \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}[phone] \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}-704713955760 \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}[phone] \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}-469809303840 \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}-4756819201380 x^{7}[phone] \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}-156603101280 \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}-16496888326218 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 )-20880413504 \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 )-21769332100344 x^{5}-11891020005261 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1572378885 \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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/1572378885*(75989439306*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)-158560640046*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)[phone]*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)-528535466820*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)[phone]*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)-704713955760*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)[phone]*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)-469809303840*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)-4756819201380*x^7+75051298080*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)-156603101280*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)-16496888326218*x^6+10006839744*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))-20880413504*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))-21769332100344*x^5-11891020005261*x^4+140844968748
*x^3+3218604203112*x^2+1399649072964*x+196325700219)*(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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.16, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (158560640046 \, x^{5} + 534040213536 \, x^{4} + 719808574005 \, x^{3} + 484598540169 \, x^{2} + 162787885893 \, x + 21813966691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{524126295 \, {\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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

2/524126295*(158560640046*x^5 + 534040213536*x^4 + 719808574005*x^3 + 484598540169*x^2 + 162787885893*x + 2181
3966691)*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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(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 {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\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)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(13/2),x)

[Out]

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

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