3.28.63 \(\int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx\) [2763]

Optimal. Leaf size=249 \[ -\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}-\frac {247408648 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {7442032 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}} \]

[Out]

-247408648/2139291*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-7442032/2139291*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/33*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2)+10/99*(1-2*
x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+1900/2079*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+76492/14553*(1-2*x)^(
1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+3560432/101871*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/713097*(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 {7442032 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {247408648 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac {10 \sqrt {5 x+3} (1-2 x)^{3/2}}{99 (3 x+2)^{9/2}}+\frac {247408648 \sqrt {5 x+3} \sqrt {1-2 x}}{713097 \sqrt {3 x+2}}+\frac {3560432 \sqrt {5 x+3} \sqrt {1-2 x}}{101871 (3 x+2)^{3/2}}+\frac {76492 \sqrt {5 x+3} \sqrt {1-2 x}}{14553 (3 x+2)^{5/2}}+\frac {1900 \sqrt {5 x+3} \sqrt {1-2 x}}{2079 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (10*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(99*(2 + 3*x)^(9
/2)) + (1900*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2079*(2 + 3*x)^(7/2)) + (76492*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14553*
(2 + 3*x)^(5/2)) + (3560432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(101871*(2 + 3*x)^(3/2)) + (247408648*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(713097*Sqrt[2 + 3*x]) - (247408648*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sq
rt[33]) - (7442032*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*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} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} \left (-\frac {1035}{2}+\frac {585 x}{2}\right )}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {8 \int \frac {\frac {149805}{4}-51390 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{18711}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {16 \int \frac {2855520-\frac {12908025 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{654885}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {32 \int \frac {\frac {991125045}{8}-\frac {150205725 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{13752585}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}+\frac {64 \int \frac {1651972050+\frac {20875104675 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{96268095}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}+\frac {3721016 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{64827}+\frac {247408648 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{713097}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}-\frac {247408648 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {7442032 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.12, size = 115, normalized size = 0.46 \begin {gather*} \frac {\frac {24 \sqrt {1-2 x} \sqrt {3+5 x} \left (4174268813+30956769477 x+91862628912 x^2+136342955970 x^3+101209884912 x^4+30060150732 x^5\right )}{(2+3 x)^{11/2}}+32 \sqrt {2} \left (30926081 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-15576890 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{8557164} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4174268813 + 30956769477*x + 91862628912*x^2 + 136342955970*x^3 + 1012098849
12*x^4 + 30060150732*x^5))/(2 + 3*x)^(11/2) + 32*Sqrt[2]*(30926081*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2] - 15576890*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/8557164

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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	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8019 \left (\frac {2}{3}+x \right )^{5}}+\frac {568 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{56133 \left (\frac {2}{3}+x \right )^{4}}+\frac {76492 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{392931 \left (\frac {2}{3}+x \right )^{3}}+\frac {3560432 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{916839 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2474086480}{713097} x^{2}-\frac {247408648}{713097} x +\frac {247408648}{237699}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {783157120 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{14975037 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1237043240 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14975037 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(321\)
default	\(-\frac {2 \left (14919413652 \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}-30060150732 \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}-100200502440 \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}-133600669920 \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}-89067113280 \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}-901804521960 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}-29689037760 \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}-3126476999556 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 )-3958538368 \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 )-4123376977248 x^{5}-2254018771062 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2139291 \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)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/2139291*(14919413652*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)-30060150732*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)-100200502440*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)-133600669920*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)-89067113280*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)-901804521960*x^7+14735223360*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)-29689037760*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)-3126476999556*x^6+1964696448*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))-3958538368*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))-4123376977248*x^5-2254018771062*x^4+22795632684*x^3+60866528738
7*x^2+266088118854*x+37568419317)*(3+5*x)^(1/2)*(1-2*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)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.18, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (30060150732 \, x^{5} + 101209884912 \, x^{4} + 136342955970 \, x^{3} + 91862628912 \, x^{2} + 30956769477 \, x + 4174268813\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{713097 \, {\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)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

2/713097*(30060150732*x^5 + 101209884912*x^4 + 136342955970*x^3 + 91862628912*x^2 + 30956769477*x + 4174268813
)*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)**(5/2)*(3+5*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((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

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

[Out]

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

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