3.28.74 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx\) [2774]

Optimal. Leaf size=280 \[ -\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2257166048 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}} \]

[Out]

-2/39*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2)+230/1287*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2)-750410084
72/1752079329*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2257166048/1752079329*EllipticF(1
/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+1300/891*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-3347620
/1702701*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)[phone]/11918907*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)
[phone]/83432349*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/584026443*(1-2*x)^(1/2)*(3+5*x)^(1/
2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {2257166048 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {75041008472 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac {1300 (5 x+3)^{3/2} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}+\frac {75041008472 \sqrt {5 x+3} \sqrt {1-2 x}}{584026443 \sqrt {3 x+2}}+\frac {1079936248 \sqrt {5 x+3} \sqrt {1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac {23210828 \sqrt {5 x+3} \sqrt {1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac {3347620 \sqrt {5 x+3} \sqrt {1-2 x}}{1702701 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(15/2),x]

[Out]

(-3347620*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1702701*(2 + 3*x)^(7/2)) + (23210828*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(119
18907*(2 + 3*x)^(5/2)) + (1079936248*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(83432349*(2 + 3*x)^(3/2)) + (75041008472*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(584026443*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(39*(2 + 3*x)^(13/2
)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(1287*(2 + 3*x)^(11/2)) + (1300*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(891
*(2 + 3*x)^(9/2)) - (75041008472*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(53093313*Sqrt[33]) - (225
7166048*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(53093313*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)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {2}{39} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}-\frac {4 \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-\frac {2895}{2}+\frac {1995 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287}\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {8 \int \frac {\left (\frac {438345}{4}-149460 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{34749}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {16 \int \frac {\frac {15056835}{8}-\frac {10467525 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{5108103}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {1154474415}{8}-\frac {1305609075 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{178783605}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {64 \int \frac {\frac {100204281585}{16}-\frac {30373206975 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{3754455705}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {128 \int \frac {\frac {1336148092575}{16}+\frac {2110528363275 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26281189935}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {1128583024 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{53093313}+\frac {75041008472 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{584026443}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2257166048 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 9.03, size = 117, normalized size = 0.42 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (2532151719515+22577209892436 x+83893544414217 x^2+166295375376786 x^3+185457331738206 x^4+110328276131100 x^5+27352447588044 x^6\right )}{(2+3 x)^{13/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-604764298880 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{14016634632 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(15/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(2532151719515 + 22577209892436*x + 83893544414217*x^2 + 166295375376786*x^3
+ 185457331738206*x^4 + 110328276131100*x^5 + 27352447588044*x^6))/(2 + 3*x)^(13/2) + 1200656135552*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 604764298880*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1
4016634632*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs. \(2(208)=416\).
time = 0.10, size = 680, normalized size = 2.43

method	result	size
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {18134 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{5}}-\frac {7616 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2302911 \left (\frac {2}{3}+x \right )^{7}}+\frac {23210828 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321810489 \left (\frac {2}{3}+x \right )^{3}}+\frac {390100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137918781 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {750410084720}{584026443} x^{2}-\frac {75041008472}{584026443} x +\frac {75041008472}{194675481}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1079936248 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{750891141 \left (\frac {2}{3}+x \right )^{2}}+\frac {237537438680 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{12264555303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {375205042360 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{12264555303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(356\)
default	\(\frac {2 \left (27352447588044 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-13575160904184 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \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^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-54300643616736 \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}-90501072694560 \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}-80445397950720 \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] x^{8}[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}-40222698975360 \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}+3391905626697132 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}-10726053060096 \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}+5648532752247084 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 )-1191783673344 \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 )+4552278771338298 x^{5}+1346576472913014 x^{4}-567661448375343 x^{3}-611345718465195 x^{2}-195598433873379 x -22789365475635\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{1752079329 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\)	\(680\)

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)^(15/2),x,method=_RETURNVERBOSE)

[Out]

2/1752079329*(27352447588044*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^6*(2+3*x)^(1/2)*(-3-5*x)^(1
/2)*(1-2*x)^(1/2)-13575160904184*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^6*(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^5*(2+3*x)^(1/2)*(-
3-5*x)^(1/2)*(1-2*x)^(1/2)-54300643616736*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)-90501072694560*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)-80445397950720*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]*x^8+81044289149760*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)-40222698975360*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)+3391905626697132*x^7+216118104399
36*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)-1072605306
0096*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]
52247084*x^6+2401312271104*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))-1191783673344*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))+4552278771338298*x^5+1346576472913014*x^4-567661448375343*x^3-611345718465195*x^2-195598433873379*x
-22789365475635)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(13/2)

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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)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)

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Fricas [A]
time = 0.32, size = 90, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (27352447588044 \, x^{6} + 110328276131100 \, x^{5} + 185457331738206 \, x^{4} + 166295375376786 \, x^{3} + 83893544414217 \, x^{2} + 22577209892436 \, x + 2532151719515\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{584026443 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \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)^(15/2),x, algorithm="fricas")

[Out]

2/584026443*(27352447588044*x^6 + 110328276131100*x^5 + 185457331738206*x^4 + 166295375376786*x^3 + 8389354441
4217*x^2 + 22577209892436*x + 2532151719515)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2187*x^7 + 10206*x^6
+ 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**(15/2),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)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), 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 )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \end {gather*}

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)^(15/2),x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2)^(15/2), x)

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